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case e_a
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
n : β
s : Set β
xβ x : β
β’ (x - xβ) ^ (n + 1) β’ (β(n + 1)!)β»ΒΉ β’ iteratedDerivWithin (n + 1) f s xβ =
((βn !)β»ΒΉ * (βn + 1)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
| rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev] | @[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
| Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK | @[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
s : Set β
xβ x : β
β’ taylorWithinEval f 0 s xβ x = f xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
| dsimp only [taylorWithinEval] | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
| Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
s : Set β
xβ x : β
β’ (PolynomialModule.eval x) (taylorWithin f 0 s xβ) = f xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
| dsimp only [taylorWithin] | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
| Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
s : Set β
xβ x : β
β’ (PolynomialModule.eval x)
(β k in Finset.range (0 + 1),
(PolynomialModule.comp (Polynomial.X - Polynomial.C xβ))
((PolynomialModule.single β k) (taylorCoeffWithin f k s xβ))) =
f xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
| dsimp only [taylorCoeffWithin] | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
| Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
s : Set β
xβ x : β
β’ (PolynomialModule.eval x)
(β k in Finset.range (0 + 1),
(PolynomialModule.comp (Polynomial.X - Polynomial.C xβ))
((PolynomialModule.single β k) ((βk !)β»ΒΉ β’ iteratedDerivWithin k f s xβ))) =
f xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
| simp | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
| Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK | /-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
n : β
s : Set β
xβ : β
β’ taylorWithinEval f n s xβ xβ = f xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
| induction' n with k hk | /-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
| Mathlib.Analysis.Calculus.Taylor.106_0.INXnr4jrmq9RIjK | /-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ | Mathlib_Analysis_Calculus_Taylor |
case zero
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
s : Set β
xβ : β
β’ taylorWithinEval f Nat.zero s xβ xβ = f xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· | exact taylor_within_zero_eval _ _ _ _ | /-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· | Mathlib.Analysis.Calculus.Taylor.106_0.INXnr4jrmq9RIjK | /-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ | Mathlib_Analysis_Calculus_Taylor |
case succ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
s : Set β
xβ : β
k : β
hk : taylorWithinEval f k s xβ xβ = f xβ
β’ taylorWithinEval f (Nat.succ k) s xβ xβ = f xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
| simp [hk] | /-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
| Mathlib.Analysis.Calculus.Taylor.106_0.INXnr4jrmq9RIjK | /-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
n : β
s : Set β
xβ x : β
β’ taylorWithinEval f n s xβ x = β k in Finset.range (n + 1), ((βk !)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
| induction' n with k hk | theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
| Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK | theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ | Mathlib_Analysis_Calculus_Taylor |
case zero
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
s : Set β
xβ x : β
β’ taylorWithinEval f Nat.zero s xβ x =
β k in Finset.range (Nat.zero + 1), ((βk !)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· | simp | theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· | Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK | theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ | Mathlib_Analysis_Calculus_Taylor |
case succ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
s : Set β
xβ x : β
k : β
hk : taylorWithinEval f k s xβ x = β k in Finset.range (k + 1), ((βk !)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ
β’ taylorWithinEval f (Nat.succ k) s xβ x =
β k in Finset.range (Nat.succ k + 1), ((βk !)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
| rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] | theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
| Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK | theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ | Mathlib_Analysis_Calculus_Taylor |
case succ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
s : Set β
xβ x : β
k : β
hk : taylorWithinEval f k s xβ x = β k in Finset.range (k + 1), ((βk !)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ
β’ β k in Finset.range (k + 1), ((βk !)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ +
(((βk + 1) * βk !)β»ΒΉ * (x - xβ) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s xβ =
β x_1 in Finset.range (k + 1), ((βx_1 !)β»ΒΉ * (x - xβ) ^ x_1) β’ iteratedDerivWithin x_1 f s xβ +
((β(k + 1)!)β»ΒΉ * (x - xβ) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s xβ | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
| simp [Nat.factorial] | theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
| Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK | theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x : β
n : β
s : Set β
hs : UniqueDiffOn β s
hf : ContDiffOn β (βn) f s
β’ ContinuousOn (fun t => taylorWithinEval f n s t x) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
| simp_rw [taylor_within_apply] | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
| Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x : β
n : β
s : Set β
hs : UniqueDiffOn β s
hf : ContDiffOn β (βn) f s
β’ ContinuousOn (fun t => β k in Finset.range (n + 1), ((βk !)β»ΒΉ * (x - t) ^ k) β’ iteratedDerivWithin k f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
| refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
| Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x : β
n : β
s : Set β
hs : UniqueDiffOn β s
hf : ContDiffOn β (βn) f s
i : β
hi : i β Finset.range (n + 1)
β’ ContinuousOn (fun t => ((βi !)β»ΒΉ * (x - t) ^ i) β’ iteratedDerivWithin i f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
| refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
| Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x : β
n : β
s : Set β
hs : UniqueDiffOn β s
hf : ContDiffOn β (βn) f s
i : β
hi : i β Finset.range (n + 1)
β’ ContinuousOn (fun t => iteratedDerivWithin i f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
| rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
| Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x : β
n : β
s : Set β
hs : UniqueDiffOn β s
hf :
(β (m : β), βm β€ βn β ContinuousOn (iteratedDerivWithin m f s) s) β§
β (m : β), βm < βn β DifferentiableOn β (iteratedDerivWithin m f s) s
i : β
hi : i β Finset.range (n + 1)
β’ ContinuousOn (fun t => iteratedDerivWithin i f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
| cases' hf with hf_left | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
| Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s | Mathlib_Analysis_Calculus_Taylor |
case intro
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x : β
n : β
s : Set β
hs : UniqueDiffOn β s
i : β
hi : i β Finset.range (n + 1)
hf_left : β (m : β), βm β€ βn β ContinuousOn (iteratedDerivWithin m f s) s
rightβ : β (m : β), βm < βn β DifferentiableOn β (iteratedDerivWithin m f s) s
β’ ContinuousOn (fun t => iteratedDerivWithin i f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
| specialize hf_left i | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
| Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s | Mathlib_Analysis_Calculus_Taylor |
case intro
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x : β
n : β
s : Set β
hs : UniqueDiffOn β s
i : β
hi : i β Finset.range (n + 1)
rightβ : β (m : β), βm < βn β DifferentiableOn β (iteratedDerivWithin m f s) s
hf_left : βi β€ βn β ContinuousOn (iteratedDerivWithin i f s) s
β’ ContinuousOn (fun t => iteratedDerivWithin i f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
| simp only [Finset.mem_range] at hi | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
| Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s | Mathlib_Analysis_Calculus_Taylor |
case intro
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x : β
n : β
s : Set β
hs : UniqueDiffOn β s
i : β
rightβ : β (m : β), βm < βn β DifferentiableOn β (iteratedDerivWithin m f s) s
hf_left : βi β€ βn β ContinuousOn (iteratedDerivWithin i f s) s
hi : i < n + 1
β’ ContinuousOn (fun t => iteratedDerivWithin i f s t) s | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
| refine' hf_left _ | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
| Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s | Mathlib_Analysis_Calculus_Taylor |
case intro
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x : β
n : β
s : Set β
hs : UniqueDiffOn β s
i : β
rightβ : β (m : β), βm < βn β DifferentiableOn β (iteratedDerivWithin m f s) s
hf_left : βi β€ βn β ContinuousOn (iteratedDerivWithin i f s) s
hi : i < n + 1
β’ βi β€ βn | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
| simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
| Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK | /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
t x : β
n : β
β’ HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(βn + 1) * (x - t) ^ n) t | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
| simp_rw [sub_eq_neg_add] | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
| Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
t x : β
n : β
β’ HasDerivAt (fun y => (-y + x) ^ (n + 1)) (-(βn + 1) * (-t + x) ^ n) t | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
| rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc] | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
| Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
t x : β
n : β
β’ HasDerivAt (fun y => (-y + x) ^ (n + 1)) ((βn + 1) * (-t + x) ^ n * -1) t | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
| convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
| Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7.h.e'_5.h.e'_5
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
t x : β
n : β
β’ βn + 1 = β(n + 1) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
| simp only [Nat.cast_add, Nat.cast_one] | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
| Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK | /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y
β’ HasDerivWithinAt (fun z => (((βk + 1) * βk !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((βk + 1) * βk !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y)
t y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
| replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y
β’ HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
| convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y
β’ iteratedDerivWithin (k + 2) f s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
| rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y
β’ derivWithin (iteratedDerivWithin (k + 1) f s) s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
| exact (derivWithin_of_mem hs ht hf).symm | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
β’ HasDerivWithinAt (fun z => (((βk + 1) * βk !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((βk + 1) * βk !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y)
t y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
| have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
β’ HasDerivWithinAt (fun t => ((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) (-((βk !)β»ΒΉ * (x - y) ^ k)) t y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
| have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
β’ -((βk !)β»ΒΉ * (x - y) ^ k) = ((βk + 1) * βk !)β»ΒΉ * (-(βk + 1) * (x - y) ^ k) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
| field_simp | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
β’ -((x - y) ^ k * ((βk + 1) * βk !)) = (-1 + -βk) * (x - y) ^ k * βk ! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; | ring | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; | Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : -((βk !)β»ΒΉ * (x - y) ^ k) = ((βk + 1) * βk !)β»ΒΉ * (-(βk + 1) * (x - y) ^ k)
β’ HasDerivWithinAt (fun t => ((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) (-((βk !)β»ΒΉ * (x - y) ^ k)) t y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
| rw [this] | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : -((βk !)β»ΒΉ * (x - y) ^ k) = ((βk + 1) * βk !)β»ΒΉ * (-(βk + 1) * (x - y) ^ k)
β’ HasDerivWithinAt (fun t => ((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) (((βk + 1) * βk !)β»ΒΉ * (-(βk + 1) * (x - y) ^ k))
t y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
| exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) (-((βk !)β»ΒΉ * (x - y) ^ k)) t y
β’ HasDerivWithinAt (fun z => (((βk + 1) * βk !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((βk + 1) * βk !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y)
t y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
| convert this.smul hf using 1 | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) (-((βk !)β»ΒΉ * (x - y) ^ k)) t y
β’ (((βk + 1) * βk !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y =
(((βk + 1) * βk !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y +
-((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
| field_simp | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
k : β
s t : Set β
ht : UniqueDiffWithinAt β t y
hs : s β π[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) (-((βk !)β»ΒΉ * (x - y) ^ k)) t y
β’ ((x - y) ^ (k + 1) / ((βk + 1) * βk !)) β’ iteratedDerivWithin (k + 2) f s y -
((x - y) ^ k / βk !) β’ iteratedDerivWithin (k + 1) f s y =
((x - y) ^ (k + 1) / ((βk + 1) * βk !)) β’ iteratedDerivWithin (k + 2) f s y +
(-(x - y) ^ k / βk !) β’ iteratedDerivWithin (k + 1) f s y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
| rw [neg_div, neg_smul, sub_eq_add_neg] | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
| Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK | theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
n : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
hf : ContDiffOn β (βn) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y
β’ HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y)
s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
| induction' n with k hk | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case zero
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
hf : ContDiffOn β (βNat.zero) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin Nat.zero f s) s y
β’ HasDerivWithinAt (fun t => taylorWithinEval f Nat.zero s t x)
(((βNat.zero !)β»ΒΉ * (x - y) ^ Nat.zero) β’ iteratedDerivWithin (Nat.zero + 1) f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· | simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul] | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· | Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case zero
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
hf : ContDiffOn β (βNat.zero) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin Nat.zero f s) s y
β’ HasDerivWithinAt (fun t => f t) (iteratedDerivWithin (Nat.zero + 1) f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
| simp only [iteratedDerivWithin_zero] at hf' | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case zero
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
hf : ContDiffOn β (βNat.zero) f s
hf' : DifferentiableWithinAt β f s y
β’ HasDerivWithinAt (fun t => f t) (iteratedDerivWithin (Nat.zero + 1) f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
| rw [iteratedDerivWithin_one (hs_unique _ (h hy))] | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case zero
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
hf : ContDiffOn β (βNat.zero) f s
hf' : DifferentiableWithinAt β f s y
β’ HasDerivWithinAt (fun t => f t) (derivWithin f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
| norm_num | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case zero
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
hf : ContDiffOn β (βNat.zero) f s
hf' : DifferentiableWithinAt β f s y
β’ HasDerivWithinAt (fun t => f t) (derivWithin f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
| exact hf'.hasDerivWithinAt.mono h | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case succ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
k : β
hk :
ContDiffOn β (βk) f s β
DifferentiableWithinAt β (iteratedDerivWithin k f s) s y β
HasDerivWithinAt (fun t => taylorWithinEval f k s t x)
(((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) s' y
hf : ContDiffOn β (β(Nat.succ k)) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin (Nat.succ k) f s) s y
β’ HasDerivWithinAt (fun t => taylorWithinEval f (Nat.succ k) s t x)
(((β(Nat.succ k)!)β»ΒΉ * (x - y) ^ Nat.succ k) β’ iteratedDerivWithin (Nat.succ k + 1) f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
| simp_rw [Nat.add_succ, taylorWithinEval_succ] | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case succ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
k : β
hk :
ContDiffOn β (βk) f s β
DifferentiableWithinAt β (iteratedDerivWithin k f s) s y β
HasDerivWithinAt (fun t => taylorWithinEval f k s t x)
(((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) s' y
hf : ContDiffOn β (β(Nat.succ k)) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin (Nat.succ k) f s) s y
β’ HasDerivWithinAt
(fun t =>
taylorWithinEval f k s t x + (((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s t)
(((β(Nat.succ k)!)β»ΒΉ * (x - y) ^ Nat.succ k) β’ iteratedDerivWithin (Nat.succ (Nat.succ k + 0)) f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
| simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case succ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
k : β
hk :
ContDiffOn β (βk) f s β
DifferentiableWithinAt β (iteratedDerivWithin k f s) s y β
HasDerivWithinAt (fun t => taylorWithinEval f k s t x)
(((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) s' y
hf : ContDiffOn β (β(Nat.succ k)) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin (Nat.succ k) f s) s y
β’ HasDerivWithinAt
(fun t =>
taylorWithinEval f k s t x + (((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s t)
((((βk + 1) * βk !)β»ΒΉ * (x - y) ^ Nat.succ k) β’ iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
| have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case succ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
k : β
hk :
ContDiffOn β (βk) f s β
DifferentiableWithinAt β (iteratedDerivWithin k f s) s y β
HasDerivWithinAt (fun t => taylorWithinEval f k s t x)
(((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) s' y
hf : ContDiffOn β (β(Nat.succ k)) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin (Nat.succ k) f s) s y
coe_lt_succ : βk < β(Nat.succ k)
β’ HasDerivWithinAt
(fun t =>
taylorWithinEval f k s t x + (((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s t)
((((βk + 1) * βk !)β»ΒΉ * (x - y) ^ Nat.succ k) β’ iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
| have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case succ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
k : β
hk :
ContDiffOn β (βk) f s β
DifferentiableWithinAt β (iteratedDerivWithin k f s) s y β
HasDerivWithinAt (fun t => taylorWithinEval f k s t x)
(((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) s' y
hf : ContDiffOn β (β(Nat.succ k)) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin (Nat.succ k) f s) s y
coe_lt_succ : βk < β(Nat.succ k)
hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s'
β’ HasDerivWithinAt
(fun t =>
taylorWithinEval f k s t x + (((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s t)
((((βk + 1) * βk !)β»ΒΉ * (x - y) ^ Nat.succ k) β’ iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
| specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case succ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
k : β
hf : ContDiffOn β (β(Nat.succ k)) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin (Nat.succ k) f s) s y
coe_lt_succ : βk < β(Nat.succ k)
hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s'
hk :
HasDerivWithinAt (fun t => taylorWithinEval f k s t x) (((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y)
s' y
β’ HasDerivWithinAt
(fun t =>
taylorWithinEval f k s t x + (((βk + 1) * βk !)β»ΒΉ * (x - t) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s t)
((((βk + 1) * βk !)β»ΒΉ * (x - y) ^ Nat.succ k) β’ iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y) s' y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
| convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
case h.e'_7
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
x y : β
s s' : Set β
hs'_unique : UniqueDiffWithinAt β s' y
hs_unique : UniqueDiffOn β s
hs' : s' β π[s] y
hy : y β s'
h : s' β s
k : β
hf : ContDiffOn β (β(Nat.succ k)) f s
hf' : DifferentiableWithinAt β (iteratedDerivWithin (Nat.succ k) f s) s y
coe_lt_succ : βk < β(Nat.succ k)
hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s'
hk :
HasDerivWithinAt (fun t => taylorWithinEval f k s t x) (((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y)
s' y
β’ (((βk + 1) * βk !)β»ΒΉ * (x - y) ^ Nat.succ k) β’ iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y =
((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y +
((((βk + 1) * βk !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((βk !)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
| exact (add_sub_cancel'_right _ _).symm | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
| Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK | /-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f g g' : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
| rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ© | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
| Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | Mathlib_Analysis_Calculus_Taylor |
case intro.intro
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f g g' : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y =
(taylorWithinEval f n (Icc xβ x) x x - taylorWithinEval f n (Icc xβ x) xβ x) * g' y
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
| use y, hy | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
| Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f g g' : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y =
(taylorWithinEval f n (Icc xβ x) x x - taylorWithinEval f n (Icc xβ x) xβ x) * g' y
β’ f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
| simp only [taylorWithinEval_self] at h | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
| Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f g g' : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y =
(f x - taylorWithinEval f n (Icc xβ x) xβ x) * g' y
β’ f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
| rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
| Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f g g' : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
y : β
hy : y β Ioo xβ x
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y * (g x - g xβ) / g' y =
f x - taylorWithinEval f n (Icc xβ x) xβ x
β’ f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
| rw [β h] | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
| Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f g g' : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
y : β
hy : y β Ioo xβ x
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y * (g x - g xβ) / g' y =
f x - taylorWithinEval f n (Icc xβ x) xβ x
β’ ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y * (g x - g xβ) / g' y =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
| field_simp [g'_ne y hy] | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
| Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f g g' : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
y : β
hy : y β Ioo xβ x
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y * (g x - g xβ) / g' y =
f x - taylorWithinEval f n (Icc xβ x) xβ x
β’ (x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc xβ x) y * (g x - g xβ) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc xβ x) y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
| ring | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
| Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
| have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
β’ ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
| refine' Continuous.continuousOn _ | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
β’ Continuous fun t => (x - t) ^ (n + 1) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
| exact (continuous_const.sub continuous_id').pow _ | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
| have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne' | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
β’ β y β Ioo xβ x, (x - y) ^ n β 0 | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
| intro y hy | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
y : β
hy : y β Ioo xβ x
β’ (x - y) ^ n β 0 | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
| refine' pow_ne_zero _ _ | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
y : β
hy : y β Ioo xβ x
β’ x - y β 0 | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
| rw [mem_Ioo] at hy | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
y : β
hy : xβ < y β§ y < x
β’ x - y β 0 | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
| rw [sub_ne_zero] | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
y : β
hy : xβ < y β§ y < x
β’ x β y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
| exact hy.2.ne' | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
xy_ne : β y β Ioo xβ x, (x - y) ^ n β 0
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
| have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
xy_ne : β y β Ioo xβ x, (x - y) ^ n β 0
hg' : β y β Ioo xβ x, -(βn + 1) * (x - y) ^ n β 0
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
| rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ© | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
case intro.intro
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
xy_ne : β y β Ioo xβ x, (x - y) ^ n β 0
hg' : β y β Ioo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β Ioo xβ x
h :
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (Icc xβ x) y
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
| use y, hy | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
xy_ne : β y β Ioo xβ x, (x - y) ^ n β 0
hg' : β y β Ioo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β Ioo xβ x
h :
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (Icc xβ x) y
β’ f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
| simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
xy_ne : β y β Ioo xβ x, (x - y) ^ n β 0
hg' : β y β Ioo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β Ioo xβ x
h :
f x - taylorWithinEval f n (Icc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (Icc xβ x) y
β’ f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
| rw [h, neg_div, β div_neg, neg_mul, neg_neg] | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
xy_ne : β y β Ioo xβ x, (x - y) ^ n β 0
hg' : β y β Ioo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β Ioo xβ x
h :
f x - taylorWithinEval f n (Icc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (Icc xβ x) y
β’ ((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1) / ((βn + 1) * (x - y) ^ n)) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y =
iteratedDerivWithin (n + 1) f (Icc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
| field_simp [xy_ne y hy, Nat.factorial] | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
| Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc xβ x)
xy_ne : β y β Ioo xβ x, (x - y) ^ n β 0
hg' : β y β Ioo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β Ioo xβ x
h :
f x - taylorWithinEval f n (Icc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (Icc xβ x) y
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc xβ x) y * ((βn + 1) * βn !) =
iteratedDerivWithin (n + 1) f (Icc xβ x) y * (x - xβ) ^ (n + 1) * (βn ! * ((βn + 1) * (x - y) ^ n)) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; | ring | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; | Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / βn ! * (x - xβ) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
| have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity) | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
| Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
β’ Continuous id | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by | continuity | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn id (Icc xβ x)
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / βn ! * (x - xβ) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
| have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _ | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
| Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn id (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt id ((fun x => 1) x_1) x_1
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / βn ! * (x - xβ) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
| rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ© | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
| Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn id (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt id ((fun x => 1) x_1) x_1
xβΒΉ : β
xβ : xβΒΉ β Ioo xβ x
β’ (fun x => 1) xβΒΉ β 0 | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by | simp | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by | Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) | Mathlib_Analysis_Calculus_Taylor |
case intro.intro
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn id (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt id ((fun x => 1) x_1) x_1
y : β
hy : y β Ioo xβ x
h :
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / (fun x => 1) y) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y
β’ β x' β Ioo xβ x,
f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / βn ! * (x - xβ) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
| use y, hy | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
| Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn id (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt id ((fun x => 1) x_1) x_1
y : β
hy : y β Ioo xβ x
h :
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / (fun x => 1) y) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y
β’ f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) y * (x - y) ^ n / βn ! * (x - xβ) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
| rw [h] | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
| Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn id (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt id ((fun x => 1) x_1) x_1
y : β
hy : y β Ioo xβ x
h :
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / (fun x => 1) y) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y
β’ ((x - y) ^ n / βn ! * (id x - id xβ) / (fun x => 1) y) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y =
iteratedDerivWithin (n + 1) f (Icc xβ x) y * (x - y) ^ n / βn ! * (x - xβ) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
| field_simp [n.factorial_ne_zero] | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
| Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) | Mathlib_Analysis_Calculus_Taylor |
case right
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β β
x xβ : β
n : β
hx : xβ < x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn id (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt id ((fun x => 1) x_1) x_1
y : β
hy : y β Ioo xβ x
h :
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / (fun x => 1) y) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) y
β’ (x - y) ^ n * (x - xβ) * iteratedDerivWithin (n + 1) f (Icc xβ x) y =
iteratedDerivWithin (n + 1) f (Icc xβ x) y * (x - y) ^ n * (x - xβ) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
| ring | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
| Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / βn ! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
| rcases eq_or_lt_of_le hab with (rfl | h) | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case inl
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a C x : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hx : x β Icc a a
hC : β y β Icc a a, βiteratedDerivWithin (n + 1) f (Icc a a) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc a a) a xβ β€ C * (x - a) ^ (n + 1) / βn ! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· | rw [Icc_self, mem_singleton_iff] at hx | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case inl
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a C x : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hx : x = a
hC : β y β Icc a a, βiteratedDerivWithin (n + 1) f (Icc a a) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc a a) a xβ β€ C * (x - a) ^ (n + 1) / βn ! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
| simp [hx] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case inr
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
β’ βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / βn ! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
| have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h) | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case inr
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
β’ βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / βn ! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
| have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> linarith
-- Estimate the iterated derivative by `C`
Β· exact hC y β¨hay, hyx.le.trans hx.2β© | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
β’ β y β Ico a x, β((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€ (βn !)β»ΒΉ * |x - a| ^ n * C | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
| rintro y β¨hay, hyxβ© | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case intro
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : β
hay : a β€ y
hyx : y < x
β’ β((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€ (βn !)β»ΒΉ * |x - a| ^ n * C | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
| rw [norm_smul, Real.norm_eq_abs] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case intro
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : β
hay : a β€ y
hyx : y < x
β’ |(βn !)β»ΒΉ * (x - y) ^ n| * βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ (βn !)β»ΒΉ * |x - a| ^ n * C | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
| gcongr | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case intro.hβ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : β
hay : a β€ y
hyx : y < x
β’ |(βn !)β»ΒΉ * (x - y) ^ n| β€ (βn !)β»ΒΉ * |x - a| ^ n | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· | rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case intro.hβ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : β
hay : a β€ y
hyx : y < x
β’ (βn !)β»ΒΉ * |x - y| ^ n β€ (βn !)β»ΒΉ * |x - a| ^ n | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
| gcongr | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case intro.hβ.h.hab
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : β
hay : a β€ y
hyx : y < x
β’ |x - y| β€ |x - a| | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
| rw [abs_of_nonneg, abs_of_nonneg] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case intro.hβ.h.hab
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : β
hay : a β€ y
hyx : y < x
β’ x - y β€ x - a | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> | linarith | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case intro.hβ.h.hab
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : β
hay : a β€ y
hyx : y < x
β’ 0 β€ x - a | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> | linarith | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case intro.hβ.h.hab
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : β
hay : a β€ y
hyx : y < x
β’ 0 β€ x - y | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> | linarith | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case intro.hβ
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
y : β
hay : a β€ y
hyx : y < x
β’ βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> linarith
-- Estimate the iterated derivative by `C`
Β· | exact hC y β¨hay, hyx.le.trans hx.2β© | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> linarith
-- Estimate the iterated derivative by `C`
Β· | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
case inr
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
h' : β y β Ico a x, β((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€ (βn !)β»ΒΉ * |x - a| ^ n * C
β’ βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / βn ! | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> linarith
-- Estimate the iterated derivative by `C`
Β· exact hC y β¨hay, hyx.le.trans hx.2β©
-- Apply the mean value theorem for vector valued functions:
| have A : β t β Icc a x, HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((βn !)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t := by
intro t ht
have I : Icc a x β Icc a b := Icc_subset_Icc_right hx.2
exact (has_deriv_within_taylorWithinEval_at_Icc x h (I ht) hf.of_succ hf').mono I | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> linarith
-- Estimate the iterated derivative by `C`
Β· exact hC y β¨hay, hyx.le.trans hx.2β©
-- Apply the mean value theorem for vector valued functions:
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
π : Type u_1
E : Type u_2
F : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace β E
f : β β E
a b C x : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
h : a < b
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)
h' : β y β Ico a x, β((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€ (βn !)β»ΒΉ * |x - a| ^ n * C
β’ β t β Icc a x,
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((βn !)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
/-!
# Taylor's theorem
This file defines the Taylor polynomial of a real function `f : β β E`,
where `E` is a normed vector space over `β` and proves Taylor's theorem,
which states that if `f` is sufficiently smooth, then
`f` can be approximated by the Taylor polynomial up to an explicit error term.
## Main definitions
* `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin`
* `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin`
## Main statements
* `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term
* `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder
* `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder
* `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a
polynomial bound on the remainder
## TODO
* the Peano form of the remainder
* the integral form of the remainder
* Generalization to higher dimensions
## Tags
Taylor polynomial, Taylor's theorem
-/
open scoped BigOperators Interval Topology Nat
open Set
variable {π E F : Type*}
variable [NormedAddCommGroup E] [NormedSpace β E]
/-- The `k`th coefficient of the Taylor polynomial. -/
noncomputable def taylorCoeffWithin (f : β β E) (k : β) (s : Set β) (xβ : β) : E :=
(k ! : β)β»ΒΉ β’ iteratedDerivWithin k f s xβ
#align taylor_coeff_within taylorCoeffWithin
/-- The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$β_{k=0}^n \frac{(x - xβ)^k}{k!} f^{(k)}(xβ),$$
where $f^{(k)}(xβ)$ denotes the iterated derivative in the set `s`. -/
noncomputable def taylorWithin (f : β β E) (n : β) (s : Set β) (xβ : β) : PolynomialModule β E :=
(Finset.range (n + 1)).sum fun k =>
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β k (taylorCoeffWithin f k s xβ))
#align taylor_within taylorWithin
/-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `β β E`-/
noncomputable def taylorWithinEval (f : β β E) (n : β) (s : Set β) (xβ x : β) : E :=
PolynomialModule.eval x (taylorWithin f n s xβ)
#align taylor_within_eval taylorWithinEval
theorem taylorWithin_succ (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithin f (n + 1) s xβ = taylorWithin f n s xβ +
PolynomialModule.comp (Polynomial.X - Polynomial.C xβ)
(PolynomialModule.single β (n + 1) (taylorCoeffWithin f (n + 1) s xβ)) := by
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
#align taylor_within_succ taylorWithin_succ
@[simp]
theorem taylorWithinEval_succ (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f (n + 1) s xβ x = taylorWithinEval f n s xβ x +
(((n + 1 : β) * n !)β»ΒΉ * (x - xβ) ^ (n + 1)) β’ iteratedDerivWithin (n + 1) f s xβ := by
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [β mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one,
mul_inv_rev]
#align taylor_within_eval_succ taylorWithinEval_succ
/-- The Taylor polynomial of order zero evaluates to `f x`. -/
@[simp]
theorem taylor_within_zero_eval (f : β β E) (s : Set β) (xβ x : β) :
taylorWithinEval f 0 s xβ x = f xβ := by
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
#align taylor_within_zero_eval taylor_within_zero_eval
/-- Evaluating the Taylor polynomial at `x = xβ` yields `f x`. -/
@[simp]
theorem taylorWithinEval_self (f : β β E) (n : β) (s : Set β) (xβ : β) :
taylorWithinEval f n s xβ xβ = f xβ := by
induction' n with k hk
Β· exact taylor_within_zero_eval _ _ _ _
simp [hk]
#align taylor_within_eval_self taylorWithinEval_self
theorem taylor_within_apply (f : β β E) (n : β) (s : Set β) (xβ x : β) :
taylorWithinEval f n s xβ x =
β k in Finset.range (n + 1), ((k ! : β)β»ΒΉ * (x - xβ) ^ k) β’ iteratedDerivWithin k f s xβ := by
induction' n with k hk
Β· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
#align taylor_within_apply taylor_within_apply
/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylorWithinEval f n s xβ x` is continuous in `xβ`. -/
theorem continuousOn_taylorWithinEval {f : β β E} {x : β} {n : β} {s : Set β}
(hs : UniqueDiffOn β s) (hf : ContDiffOn β n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by
simp_rw [taylor_within_apply]
refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _
refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp only [Finset.mem_range] at hi
refine' hf_left _
simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]
#align continuous_on_taylor_within_eval continuousOn_taylorWithinEval
/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/
theorem monomial_has_deriv_aux (t x : β) (n : β) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add]
rw [β neg_one_mul, mul_comm (-1 : β), mul_assoc, mul_comm (-1 : β), β mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
#align monomial_has_deriv_aux monomial_has_deriv_aux
theorem hasDerivWithinAt_taylor_coeff_within {f : β β E} {x y : β} {k : β} {s t : Set β}
(ht : UniqueDiffWithinAt β t y) (hs : s β π[t] y)
(hf : DifferentiableWithinAt β (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : β) * k !)β»ΒΉ * (x - z) ^ (k + 1)) β’ iteratedDerivWithin (k + 1) f s z)
((((k + 1 : β) * k !)β»ΒΉ * (x - y) ^ (k + 1)) β’ iteratedDerivWithin (k + 2) f s y -
((k ! : β)β»ΒΉ * (x - y) ^ k) β’ iteratedDerivWithin (k + 1) f s y) t y := by
replace hf :
HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
exact (derivWithin_of_mem hs ht hf).symm
have : HasDerivWithinAt (fun t => ((k + 1 : β) * k !)β»ΒΉ * (x - t) ^ (k + 1))
(-((k ! : β)β»ΒΉ * (x - y) ^ k)) t y := by
-- Commuting the factors:
have : -((k ! : β)β»ΒΉ * (x - y) ^ k) = ((k + 1 : β) * k !)β»ΒΉ * (-(k + 1) * (x - y) ^ k) := by
field_simp; ring
rw [this]
exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
convert this.smul hf using 1
field_simp
rw [neg_div, neg_smul, sub_eq_add_neg]
#align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for arbitrary sets -/
theorem hasDerivWithinAt_taylorWithinEval {f : β β E} {x y : β} {n : β} {s s' : Set β}
(hs'_unique : UniqueDiffWithinAt β s' y) (hs_unique : UniqueDiffOn β s) (hs' : s' β π[s] y)
(hy : y β s') (h : s' β s) (hf : ContDiffOn β n f s)
(hf' : DifferentiableWithinAt β (iteratedDerivWithin n f s) s y) :
HasDerivWithinAt (fun t => taylorWithinEval f n s t x)
(((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f s y) s' y := by
induction' n with k hk
Β· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
norm_num
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, taylorWithinEval_succ]
simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]
have coe_lt_succ : (k : WithTop β) < k.succ := Nat.cast_lt.2 k.lt_succ_self
have hdiff : DifferentiableOn β (iteratedDerivWithin k f s) s' :=
(hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h
specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')
convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique
(nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1
exact (add_sub_cancel'_right _ _).symm
#align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for open intervals -/
theorem taylorWithinEval_hasDerivAt_Ioo {f : β β E} {a b t : β} (x : β) {n : β} (hx : a < b)
(ht : t β Ioo a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) :
HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) t :=
have h_nhds : Ioo a b β π t := isOpen_Ioo.mem_nhds ht
have h_nhds' : Ioo a b β π[Icc a b] t := nhdsWithin_le_nhds h_nhds
(hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht
Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds
#align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo
/-- Calculate the derivative of the Taylor polynomial with respect to `xβ`.
Version for closed intervals -/
theorem has_deriv_within_taylorWithinEval_at_Icc {f : β β E} {a b t : β} (x : β) {n : β}
(hx : a < b) (ht : t β Icc a b) (hf : ContDiffOn β n f (Icc a b))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b)) :
HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t :=
hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx)
self_mem_nhdsWithin ht rfl.subset hf (hf' t ht)
#align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc
/-! ### Taylor's theorem with mean value type remainder estimate -/
/-- **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`, and `g` is a differentiable function on
`Ioo xβ x` and continuous on `Icc xβ x`. Then there exists an `x' β Ioo xβ x` such that
$$f(x) - (P_n f)(xβ, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(xβ)}{g' x'},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$. -/
theorem taylor_mean_remainder {f : β β β} {g g' : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x))
(gcont : ContinuousOn g (Icc xβ x))
(gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt g (g' x_1) x_1)
(g'_ne : β x_1 : β, x_1 β Ioo xβ x β g' x_1 β 0) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / n ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' := by
-- We apply the mean value theorem
rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc xβ x) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc xβ x) t) hx
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with β¨y, hy, hβ©
use y, hy
-- The rest is simplifications and trivial calculations
simp only [taylorWithinEval_self] at h
rw [mul_comm, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h
rw [β h]
field_simp [g'_ne y hy]
ring
#align taylor_mean_remainder taylor_mean_remainder
/-- **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - xβ)^{n+1}}{(n+1)!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_lagrange {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - xβ) ^ (n + 1) / (n + 1)! := by
have gcont : ContinuousOn (fun t : β => (x - t) ^ (n + 1)) (Icc xβ x) := by
refine' Continuous.continuousOn _
exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`
have xy_ne : β y : β, y β Ioo xβ x β (x - y) ^ n β 0 := by
intro y hy
refine' pow_ne_zero _ _
rw [mem_Ioo] at hy
rw [sub_ne_zero]
exact hy.2.ne'
have hg' : β y : β, y β Ioo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)
-- We apply the general theorem with g(t) = (x - t)^(n+1)
rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with
β¨y, hy, hβ©
use y, hy
simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h
rw [h, neg_div, β div_neg, neg_mul, neg_neg]
field_simp [xy_ne y hy, Nat.factorial]; ring
#align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange
/-- **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc xβ x` and
`n+1`-times differentiable on the open set `Ioo xβ x`. Then there exists an `x' β Ioo xβ x` such
that $$f(x) - (P_n f)(xβ, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-xβ)}{n!},$$
where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated
derivative. -/
theorem taylor_mean_remainder_cauchy {f : β β β} {x xβ : β} {n : β} (hx : xβ < x)
(hf : ContDiffOn β n f (Icc xβ x))
(hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)) :
β x' β Ioo xβ x, f x - taylorWithinEval f n (Icc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (Icc xβ x) x' * (x - x') ^ n / n ! * (x - xβ) := by
have gcont : ContinuousOn id (Icc xβ x) := Continuous.continuousOn (by continuity)
have gdiff : β x_1 : β, x_1 β Ioo xβ x β HasDerivAt id ((fun _ : β => (1 : β)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with β¨y, hy, hβ©
use y, hy
rw [h]
field_simp [n.factorial_ne_zero]
ring
#align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy
/-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> linarith
-- Estimate the iterated derivative by `C`
Β· exact hC y β¨hay, hyx.le.trans hx.2β©
-- Apply the mean value theorem for vector valued functions:
have A : β t β Icc a x, HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((βn !)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t := by
| intro t ht | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! := by
rcases eq_or_lt_of_le hab with (rfl | h)
Β· rw [Icc_self, mem_singleton_iff] at hx
simp [hx]
-- The nth iterated derivative is differentiable
have hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Icc a b) :=
hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self)
(uniqueDiffOn_Icc h)
-- We can uniformly bound the derivative of the Taylor polynomial
have h' : β y β Ico a x,
β((n ! : β)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) yβ β€
(n ! : β)β»ΒΉ * |x - a| ^ n * C := by
rintro y β¨hay, hyxβ©
rw [norm_smul, Real.norm_eq_abs]
gcongr
Β· rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]
gcongr
rw [abs_of_nonneg, abs_of_nonneg] <;> linarith
-- Estimate the iterated derivative by `C`
Β· exact hC y β¨hay, hyx.le.trans hx.2β©
-- Apply the mean value theorem for vector valued functions:
have A : β t β Icc a x, HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x)
(((βn !)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t := by
| Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/
theorem taylor_mean_remainder_bound {f : β β E} {a b C x : β} {n : β} (hab : a β€ b)
(hf : ContDiffOn β (n + 1) f (Icc a b)) (hx : x β Icc a b)
(hC : β y β Icc a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C) :
βf x - taylorWithinEval f n (Icc a b) a xβ β€ C * (x - a) ^ (n + 1) / n ! | Mathlib_Analysis_Calculus_Taylor |
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