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|---|---|---|---|---|---|---|
V : Type u
instβΒ² : Category.{v, u} V
instβΒΉ : HasZeroMorphisms V
ΞΉ : Type u_1
c : ComplexShape ΞΉ
T : Type u_2
instβ : Category.{?u.78, u_2} T
C : HomologicalComplex (T β₯€ V) c
Xβ Yβ Zβ : T
hβ : Xβ βΆ Yβ
hβ : Yβ βΆ Zβ
β’ { obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.map
(hβ β« hβ) =
{ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.map
hβ β«
{ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.map
hβ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T β₯€ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ΞΉ : Type*} {c : ComplexShape ΞΉ}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
rw [(C.X i).map_id]
map_comp hβ hβ := by
| ext i | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
rw [(C.X i).map_id]
map_comp hβ hβ := by
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
instβΒ² : Category.{v, u} V
instβΒΉ : HasZeroMorphisms V
ΞΉ : Type u_1
c : ComplexShape ΞΉ
T : Type u_2
instβ : Category.{?u.78, u_2} T
C : HomologicalComplex (T β₯€ V) c
Xβ Yβ Zβ : T
hβ : Xβ βΆ Yβ
hβ : Yβ βΆ Zβ
i : ΞΉ
β’ Hom.f
({ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.map
(hβ β« hβ))
i =
Hom.f
({ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.map
hβ β«
{ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.map
hβ)
i | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T β₯€ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ΞΉ : Type*} {c : ComplexShape ΞΉ}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
rw [(C.X i).map_id]
map_comp hβ hβ := by
ext i
| dsimp | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
rw [(C.X i).map_id]
map_comp hβ hβ := by
ext i
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
instβΒ² : Category.{v, u} V
instβΒΉ : HasZeroMorphisms V
ΞΉ : Type u_1
c : ComplexShape ΞΉ
T : Type u_2
instβ : Category.{?u.78, u_2} T
C : HomologicalComplex (T β₯€ V) c
Xβ Yβ Zβ : T
hβ : Xβ βΆ Yβ
hβ : Yβ βΆ Zβ
i : ΞΉ
β’ (X C i).map (hβ β« hβ) = (X C i).map hβ β« (X C i).map hβ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T β₯€ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ΞΉ : Type*} {c : ComplexShape ΞΉ}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
rw [(C.X i).map_id]
map_comp hβ hβ := by
ext i
dsimp
| rw [Functor.map_comp] | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
rw [(C.X i).map_id]
map_comp hβ hβ := by
ext i
dsimp
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
instβΒ² : Category.{v, u} V
instβΒΉ : HasZeroMorphisms V
ΞΉ : Type u_1
c : ComplexShape ΞΉ
T : Type u_2
instβ : Category.{?u.4461, u_2} T
Xβ Yβ : HomologicalComplex (T β₯€ V) c
f : Xβ βΆ Yβ
t t' : T
g : t βΆ t'
β’ ((fun C => asFunctor C) Xβ).map g β« (fun t => Hom.mk fun i => (Hom.f f i).app t) t' =
(fun t => Hom.mk fun i => (Hom.f f i).app t) t β« ((fun C => asFunctor C) Yβ).map g | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T β₯€ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ΞΉ : Type*} {c : ComplexShape ΞΉ}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
rw [(C.X i).map_id]
map_comp hβ hβ := by
ext i
dsimp
rw [Functor.map_comp]
#align homological_complex.as_functor HomologicalComplex.asFunctor
-- TODO in fact, this is an equivalence of categories.
/-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T β₯€ V) c β₯€ T β₯€ HomologicalComplex V c where
obj C := C.asFunctor
map f :=
{ app := fun t =>
{ f := fun i => (f.f i).app t
comm' := fun i j _ => NatTrans.congr_app (f.comm i j) t }
naturality := fun t t' g => by
| ext i | /-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T β₯€ V) c β₯€ T β₯€ HomologicalComplex V c where
obj C := C.asFunctor
map f :=
{ app := fun t =>
{ f := fun i => (f.f i).app t
comm' := fun i j _ => NatTrans.congr_app (f.comm i j) t }
naturality := fun t t' g => by
| Mathlib.Algebra.Homology.Functor.63_0.DFL0Z3rvWsk3pdU | /-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T β₯€ V) c β₯€ T β₯€ HomologicalComplex V c where
obj C | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
instβΒ² : Category.{v, u} V
instβΒΉ : HasZeroMorphisms V
ΞΉ : Type u_1
c : ComplexShape ΞΉ
T : Type u_2
instβ : Category.{?u.4461, u_2} T
Xβ Yβ : HomologicalComplex (T β₯€ V) c
f : Xβ βΆ Yβ
t t' : T
g : t βΆ t'
i : ΞΉ
β’ Hom.f (((fun C => asFunctor C) Xβ).map g β« (fun t => Hom.mk fun i => (Hom.f f i).app t) t') i =
Hom.f ((fun t => Hom.mk fun i => (Hom.f f i).app t) t β« ((fun C => asFunctor C) Yβ).map g) i | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T β₯€ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ΞΉ : Type*} {c : ComplexShape ΞΉ}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T β₯€ V) c) :
T β₯€ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
rw [(C.X i).map_id]
map_comp hβ hβ := by
ext i
dsimp
rw [Functor.map_comp]
#align homological_complex.as_functor HomologicalComplex.asFunctor
-- TODO in fact, this is an equivalence of categories.
/-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T β₯€ V) c β₯€ T β₯€ HomologicalComplex V c where
obj C := C.asFunctor
map f :=
{ app := fun t =>
{ f := fun i => (f.f i).app t
comm' := fun i j _ => NatTrans.congr_app (f.comm i j) t }
naturality := fun t t' g => by
ext i
| exact (f.f i).naturality g | /-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T β₯€ V) c β₯€ T β₯€ HomologicalComplex V c where
obj C := C.asFunctor
map f :=
{ app := fun t =>
{ f := fun i => (f.f i).app t
comm' := fun i j _ => NatTrans.congr_app (f.comm i j) t }
naturality := fun t t' g => by
ext i
| Mathlib.Algebra.Homology.Functor.63_0.DFL0Z3rvWsk3pdU | /-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T β₯€ V) c β₯€ T β₯€ HomologicalComplex V c where
obj C | Mathlib_Algebra_Homology_Functor |
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
| by_cases ha_zero : a = 0 | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : a = 0
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· | simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero] | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : a = 0
β’ limsup (fun x => β₯) f = β₯ | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
| exact limsup_const_bot | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : Β¬a = 0
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
| let g := fun x : ββ₯0β => a * x | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : Β¬a = 0
g : ββ₯0β β ββ₯0β := fun x => a * x
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
| have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β© | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : Β¬a = 0
g : ββ₯0β β ββ₯0β := fun x => a * x
x : ββ₯0β
β’ (fun x => aβ»ΒΉ * x) (g x) = x | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by | simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top] | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : Β¬a = 0
g : ββ₯0β β ββ₯0β := fun x => a * x
x : ββ₯0β
β’ g ((fun x => aβ»ΒΉ * x) x) = x | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
| simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top] | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : Β¬a = 0
g : ββ₯0β β ββ₯0β := fun x => a * x
hg_bij : Function.Bijective g
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
| have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1 | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : Β¬a = 0
g : ββ₯0β β ββ₯0β := fun x => a * x
hg_bij : Function.Bijective g
xβΒ² xβΒΉ : ββ₯0β
xβ : xβΒ² β€ xβΒΉ
β’ g xβΒ² β€ g xβΒΉ | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by | rwa [mul_le_mul_left ha_zero ha_top] | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : Β¬a = 0
g : ββ₯0β β ββ₯0β := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
| let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2 | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
ha_zero : Β¬a = 0
g : ββ₯0β β ββ₯0β := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
g_iso : ββ₯0β βo ββ₯0β := StrictMono.orderIsoOfSurjective g hg_mono (_ : Function.Surjective g)
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
| exact (OrderIso.limsup_apply g_iso).symm | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
| by_cases ha_top : a β β€ | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a β β€
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· | exact limsup_const_mul_of_ne_top ha_top | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : Β¬a β β€
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
| push_neg at ha_top | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
| by_cases hu : u =αΆ [f] 0 | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : u =αΆ [f] 0
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· | have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx] | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : u =αΆ [f] 0
x : Ξ±
hx : u x = OfNat.ofNat 0 x
β’ (fun x => a * u x) x = OfNat.ofNat 0 x | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by | simp [hx] | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : u =αΆ [f] 0
hau : (fun x => a * u x) =αΆ [f] 0
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
| simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot] | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : u =αΆ [f] 0
hau : (fun x => a * u x) =αΆ [f] 0
β’ β₯ = a * β₯ | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
| simp | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : Β¬u =αΆ [f] 0
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· | have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx] | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : Β¬u =αΆ [f] 0
β’ βαΆ (x : Ξ±) in f, β€ β€ if u x = 0 then 0 else β€ | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
| rw [EventuallyEq, not_eventually] at hu | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : βαΆ (x : Ξ±) in f, Β¬u x = OfNat.ofNat 0 x
β’ βαΆ (x : Ξ±) in f, β€ β€ if u x = 0 then 0 else β€ | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
| refine' hu.mono fun x hx => _ | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : βαΆ (x : Ξ±) in f, Β¬u x = OfNat.ofNat 0 x
x : Ξ±
hx : Β¬u x = OfNat.ofNat 0 x
β’ β€ β€ if u x = 0 then 0 else β€ | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
| rw [Pi.zero_apply] at hx | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : βαΆ (x : Ξ±) in f, Β¬u x = OfNat.ofNat 0 x
x : Ξ±
hx : Β¬u x = 0
β’ β€ β€ if u x = 0 then 0 else β€ | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
| simp [hx] | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : Β¬u =αΆ [f] 0
hu_mul : βαΆ (x : Ξ±) in f, β€ β€ if u x = 0 then 0 else β€
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
| have h_top_le : (f.limsup fun x : Ξ± => ite (u x = 0) (0 : ββ₯0β) β€) = β€ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul) | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : Β¬u =αΆ [f] 0
hu_mul : βαΆ (x : Ξ±) in f, β€ β€ if u x = 0 then 0 else β€
h_top_le : limsup (fun x => if u x = 0 then 0 else β€) f = β€
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
have h_top_le : (f.limsup fun x : Ξ± => ite (u x = 0) (0 : ββ₯0β) β€) = β€ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul)
| have hfu : f.limsup u β 0 := mt limsup_eq_zero_iff.1 hu | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
have h_top_le : (f.limsup fun x : Ξ± => ite (u x = 0) (0 : ββ₯0β) β€) = β€ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul)
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u : Ξ± β ββ₯0β
a : ββ₯0β
ha_top : a = β€
hu : Β¬u =αΆ [f] 0
hu_mul : βαΆ (x : Ξ±) in f, β€ β€ if u x = 0 then 0 else β€
h_top_le : limsup (fun x => if u x = 0 then 0 else β€) f = β€
hfu : limsup u f β 0
β’ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
have h_top_le : (f.limsup fun x : Ξ± => ite (u x = 0) (0 : ββ₯0β) β€) = β€ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul)
have hfu : f.limsup u β 0 := mt limsup_eq_zero_iff.1 hu
| simp only [ha_top, top_mul', h_top_le, hfu, ite_false] | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
have h_top_le : (f.limsup fun x : Ξ± => ite (u x = 0) (0 : ββ₯0β) β€) = β€ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul)
have hfu : f.limsup u β 0 := mt limsup_eq_zero_iff.1 hu
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u v : Ξ± β ββ₯0β
β’ limsup (u * v) f β€ limsup (fun x => limsup u f * v x) f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
have h_top_le : (f.limsup fun x : Ξ± => ite (u x = 0) (0 : ββ₯0β) β€) = β€ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul)
have hfu : f.limsup u β 0 := mt limsup_eq_zero_iff.1 hu
simp only [ha_top, top_mul', h_top_le, hfu, ite_false]
#align ennreal.limsup_const_mul ENNReal.limsup_const_mul
theorem limsup_mul_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u * v) β€ f.limsup u * f.limsup v :=
calc
f.limsup (u * v) β€ f.limsup fun x => f.limsup u * v x := by
| refine limsup_le_limsup ?_ | theorem limsup_mul_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u * v) β€ f.limsup u * f.limsup v :=
calc
f.limsup (u * v) β€ f.limsup fun x => f.limsup u * v x := by
| Mathlib.Order.Filter.ENNReal.71_0.m2m9iHbAOFxzXph | theorem limsup_mul_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u * v) β€ f.limsup u * f.limsup v | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
f : Filter Ξ±
instβ : CountableInterFilter f
u v : Ξ± β ββ₯0β
β’ u * v β€αΆ [f] fun x => limsup u f * v x | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
have h_top_le : (f.limsup fun x : Ξ± => ite (u x = 0) (0 : ββ₯0β) β€) = β€ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul)
have hfu : f.limsup u β 0 := mt limsup_eq_zero_iff.1 hu
simp only [ha_top, top_mul', h_top_le, hfu, ite_false]
#align ennreal.limsup_const_mul ENNReal.limsup_const_mul
theorem limsup_mul_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u * v) β€ f.limsup u * f.limsup v :=
calc
f.limsup (u * v) β€ f.limsup fun x => f.limsup u * v x := by
refine limsup_le_limsup ?_
| filter_upwards [@eventually_le_limsup _ f _ u] with x hx using mul_le_mul' hx le_rfl | theorem limsup_mul_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u * v) β€ f.limsup u * f.limsup v :=
calc
f.limsup (u * v) β€ f.limsup fun x => f.limsup u * v x := by
refine limsup_le_limsup ?_
| Mathlib.Order.Filter.ENNReal.71_0.m2m9iHbAOFxzXph | theorem limsup_mul_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u * v) β€ f.limsup u * f.limsup v | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
fβ : Filter Ξ±
Ξ² : Type u_2
instβΒΉ : Countable Ξ²
f : Filter Ξ±
instβ : CountableInterFilter f
g : Filter Ξ²
u : Ξ± β Ξ² β ββ₯0β
β’ βαΆ (a : Ξ±) in f, β (b : Ξ²), u a b β€ limsup (fun a' => u a' b) f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
have h_top_le : (f.limsup fun x : Ξ± => ite (u x = 0) (0 : ββ₯0β) β€) = β€ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul)
have hfu : f.limsup u β 0 := mt limsup_eq_zero_iff.1 hu
simp only [ha_top, top_mul', h_top_le, hfu, ite_false]
#align ennreal.limsup_const_mul ENNReal.limsup_const_mul
theorem limsup_mul_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u * v) β€ f.limsup u * f.limsup v :=
calc
f.limsup (u * v) β€ f.limsup fun x => f.limsup u * v x := by
refine limsup_le_limsup ?_
filter_upwards [@eventually_le_limsup _ f _ u] with x hx using mul_le_mul' hx le_rfl
_ = f.limsup u * f.limsup v := limsup_const_mul
#align ennreal.limsup_mul_le ENNReal.limsup_mul_le
theorem limsup_add_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u + v) β€ f.limsup u + f.limsup v :=
sInf_le ((eventually_le_limsup u).mp
((eventually_le_limsup v).mono fun _ hxg hxf => add_le_add hxf hxg))
#align ennreal.limsup_add_le ENNReal.limsup_add_le
theorem limsup_liminf_le_liminf_limsup {Ξ²} [Countable Ξ²] {f : Filter Ξ±} [CountableInterFilter f]
{g : Filter Ξ²} (u : Ξ± β Ξ² β ββ₯0β) :
(f.limsup fun a : Ξ± => g.liminf fun b : Ξ² => u a b) β€
g.liminf fun b => f.limsup fun a => u a b :=
have h1 : βαΆ a in f, β b, u a b β€ f.limsup fun a' => u a' b := by
| rw [eventually_countable_forall] | theorem limsup_liminf_le_liminf_limsup {Ξ²} [Countable Ξ²] {f : Filter Ξ±} [CountableInterFilter f]
{g : Filter Ξ²} (u : Ξ± β Ξ² β ββ₯0β) :
(f.limsup fun a : Ξ± => g.liminf fun b : Ξ² => u a b) β€
g.liminf fun b => f.limsup fun a => u a b :=
have h1 : βαΆ a in f, β b, u a b β€ f.limsup fun a' => u a' b := by
| Mathlib.Order.Filter.ENNReal.86_0.m2m9iHbAOFxzXph | theorem limsup_liminf_le_liminf_limsup {Ξ²} [Countable Ξ²] {f : Filter Ξ±} [CountableInterFilter f]
{g : Filter Ξ²} (u : Ξ± β Ξ² β ββ₯0β) :
(f.limsup fun a : Ξ± => g.liminf fun b : Ξ² => u a b) β€
g.liminf fun b => f.limsup fun a => u a b | Mathlib_Order_Filter_ENNReal |
Ξ± : Type u_1
fβ : Filter Ξ±
Ξ² : Type u_2
instβΒΉ : Countable Ξ²
f : Filter Ξ±
instβ : CountableInterFilter f
g : Filter Ξ²
u : Ξ± β Ξ² β ββ₯0β
β’ β (i : Ξ²), βαΆ (x : Ξ±) in f, u x i β€ limsup (fun a' => u a' i) f | /-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order properties of extended non-negative reals
This file compiles filter-related results about `ββ₯0β` (see Data/Real/ENNReal.lean).
-/
open Filter ENNReal
namespace ENNReal
variable {Ξ± : Type*} {f : Filter Ξ±}
theorem eventually_le_limsup [CountableInterFilter f] (u : Ξ± β ββ₯0β) :
βαΆ y in f, u y β€ f.limsup u :=
_root_.eventually_le_limsup
#align ennreal.eventually_le_limsup ENNReal.eventually_le_limsup
theorem limsup_eq_zero_iff [CountableInterFilter f] {u : Ξ± β ββ₯0β} :
f.limsup u = 0 β u =αΆ [f] 0 :=
limsup_eq_bot
#align ennreal.limsup_eq_zero_iff ENNReal.limsup_eq_zero_iff
theorem limsup_const_mul_of_ne_top {u : Ξ± β ββ₯0β} {a : ββ₯0β} (ha_top : a β β€) :
(f.limsup fun x : Ξ± => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
Β· simp_rw [ha_zero, zero_mul, β ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ββ₯0β => a * x
have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
β¨fun x => aβ»ΒΉ * x,
β¨fun x => by simp [β mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [β mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]β©β©
have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
exact (OrderIso.limsup_apply g_iso).symm
#align ennreal.limsup_const_mul_of_ne_top ENNReal.limsup_const_mul_of_ne_top
theorem limsup_const_mul [CountableInterFilter f] {u : Ξ± β ββ₯0β} {a : ββ₯0β} :
f.limsup (a * u Β·) = a * f.limsup u := by
by_cases ha_top : a β β€
Β· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =αΆ [f] 0
Β· have hau : (a * u Β·) =αΆ [f] 0 := hu.mono fun x hx => by simp [hx]
simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, β ENNReal.bot_eq_zero,
limsup_const_bot]
simp
Β· have hu_mul : βαΆ x : Ξ± in f, β€ β€ ite (u x = 0) (0 : ββ₯0β) β€ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx]
have h_top_le : (f.limsup fun x : Ξ± => ite (u x = 0) (0 : ββ₯0β) β€) = β€ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul)
have hfu : f.limsup u β 0 := mt limsup_eq_zero_iff.1 hu
simp only [ha_top, top_mul', h_top_le, hfu, ite_false]
#align ennreal.limsup_const_mul ENNReal.limsup_const_mul
theorem limsup_mul_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u * v) β€ f.limsup u * f.limsup v :=
calc
f.limsup (u * v) β€ f.limsup fun x => f.limsup u * v x := by
refine limsup_le_limsup ?_
filter_upwards [@eventually_le_limsup _ f _ u] with x hx using mul_le_mul' hx le_rfl
_ = f.limsup u * f.limsup v := limsup_const_mul
#align ennreal.limsup_mul_le ENNReal.limsup_mul_le
theorem limsup_add_le [CountableInterFilter f] (u v : Ξ± β ββ₯0β) :
f.limsup (u + v) β€ f.limsup u + f.limsup v :=
sInf_le ((eventually_le_limsup u).mp
((eventually_le_limsup v).mono fun _ hxg hxf => add_le_add hxf hxg))
#align ennreal.limsup_add_le ENNReal.limsup_add_le
theorem limsup_liminf_le_liminf_limsup {Ξ²} [Countable Ξ²] {f : Filter Ξ±} [CountableInterFilter f]
{g : Filter Ξ²} (u : Ξ± β Ξ² β ββ₯0β) :
(f.limsup fun a : Ξ± => g.liminf fun b : Ξ² => u a b) β€
g.liminf fun b => f.limsup fun a => u a b :=
have h1 : βαΆ a in f, β b, u a b β€ f.limsup fun a' => u a' b := by
rw [eventually_countable_forall]
| exact fun b => ENNReal.eventually_le_limsup fun a => u a b | theorem limsup_liminf_le_liminf_limsup {Ξ²} [Countable Ξ²] {f : Filter Ξ±} [CountableInterFilter f]
{g : Filter Ξ²} (u : Ξ± β Ξ² β ββ₯0β) :
(f.limsup fun a : Ξ± => g.liminf fun b : Ξ² => u a b) β€
g.liminf fun b => f.limsup fun a => u a b :=
have h1 : βαΆ a in f, β b, u a b β€ f.limsup fun a' => u a' b := by
rw [eventually_countable_forall]
| Mathlib.Order.Filter.ENNReal.86_0.m2m9iHbAOFxzXph | theorem limsup_liminf_le_liminf_limsup {Ξ²} [Countable Ξ²] {f : Filter Ξ±} [CountableInterFilter f]
{g : Filter Ξ²} (u : Ξ± β Ξ² β ββ₯0β) :
(f.limsup fun a : Ξ± => g.liminf fun b : Ξ² => u a b) β€
g.liminf fun b => f.limsup fun a => u a b | Mathlib_Order_Filter_ENNReal |
u : β
hu : 0 < u.re
v : β
β’ IntervalIntegrable (fun x => βx ^ (u - 1) * (1 - βx) ^ (v - 1)) volume 0 (1 / 2) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
| apply IntervalIntegrable.mul_continuousOn | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hf
u : β
hu : 0 < u.re
v : β
β’ IntervalIntegrable (fun x => βx ^ (u - 1)) volume 0 (1 / 2) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· | refine' intervalIntegral.intervalIntegrable_cpow' _ | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hf
u : β
hu : 0 < u.re
v : β
β’ -1 < (u - 1).re | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
| rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right] | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg
u : β
hu : 0 < u.re
v : β
β’ ContinuousOn (fun x => (1 - βx) ^ (v - 1)) (uIcc 0 (1 / 2)) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· | apply ContinuousAt.continuousOn | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont
u : β
hu : 0 < u.re
v : β
β’ β x β uIcc 0 (1 / 2), ContinuousAt (fun x => (1 - βx) ^ (v - 1)) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
| intro x hx | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont
u : β
hu : 0 < u.re
v : β
x : β
hx : x β uIcc 0 (1 / 2)
β’ ContinuousAt (fun x => (1 - βx) ^ (v - 1)) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
| rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : β
hu : 0 < u.re
v : β
x : β
hx : x β uIcc 0 (1 / 2)
β’ 0 β€ 1 / 2 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by | positivity | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont
u : β
hu : 0 < u.re
v : β
x : β
hx : x β Icc 0 (1 / 2)
β’ ContinuousAt (fun x => (1 - βx) ^ (v - 1)) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
| apply ContinuousAt.cpow | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont.hf
u : β
hu : 0 < u.re
v : β
x : β
hx : x β Icc 0 (1 / 2)
β’ ContinuousAt (fun x => 1 - βx) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· | exact (continuous_const.sub continuous_ofReal).continuousAt | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont.hg
u : β
hu : 0 < u.re
v : β
x : β
hx : x β Icc 0 (1 / 2)
β’ ContinuousAt (fun x => v - 1) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· | exact continuousAt_const | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont.h0
u : β
hu : 0 < u.re
v : β
x : β
hx : x β Icc 0 (1 / 2)
β’ 0 < (1 - βx).re β¨ (1 - βx).im β 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· | rw [sub_re, one_re, ofReal_re, sub_pos] | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont.h0
u : β
hu : 0 < u.re
v : β
x : β
hx : x β Icc 0 (1 / 2)
β’ x < 1 β¨ (1 - βx).im β 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
| exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1)) | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : β
hu : 0 < u.re
v : β
x : β
hx : x β Icc 0 (1 / 2)
β’ 1 / 2 < 1 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by | norm_num | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
hu : 0 < u.re
hv : 0 < v.re
β’ IntervalIntegrable (fun x => βx ^ (u - 1) * (1 - βx) ^ (v - 1)) volume 0 1 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
| refine' (betaIntegral_convergent_left hu v).trans _ | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
hu : 0 < u.re
hv : 0 < v.re
β’ IntervalIntegrable (fun x => βx ^ (u - 1) * (1 - βx) ^ (v - 1)) volume (1 / 2) 1 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
| rw [IntervalIntegrable.iff_comp_neg] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
hu : 0 < u.re
hv : 0 < v.re
β’ IntervalIntegrable (fun x => β(-x) ^ (u - 1) * (1 - β(-x)) ^ (v - 1)) volume (-(1 / 2)) (-1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
| convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3
u v : β
hu : 0 < u.re
hv : 0 < v.re
β’ (fun x => β(-x) ^ (u - 1) * (1 - β(-x)) ^ (v - 1)) = fun x => β(x + 1) ^ (v - 1) * (1 - β(x + 1)) ^ (u - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· | ext1 x | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h
u v : β
hu : 0 < u.re
hv : 0 < v.re
x : β
β’ β(-x) ^ (u - 1) * (1 - β(-x)) ^ (v - 1) = β(x + 1) ^ (v - 1) * (1 - β(x + 1)) ^ (u - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
| conv_lhs => rw [mul_comm] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
hu : 0 < u.re
hv : 0 < v.re
x : β
| β(-x) ^ (u - 1) * (1 - β(-x)) ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => | rw [mul_comm] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
hu : 0 < u.re
hv : 0 < v.re
x : β
| β(-x) ^ (u - 1) * (1 - β(-x)) ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => | rw [mul_comm] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
hu : 0 < u.re
hv : 0 < v.re
x : β
| β(-x) ^ (u - 1) * (1 - β(-x)) ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => | rw [mul_comm] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h
u v : β
hu : 0 < u.re
hv : 0 < v.re
x : β
β’ (1 - β(-x)) ^ (v - 1) * β(-x) ^ (u - 1) = β(x + 1) ^ (v - 1) * (1 - β(x + 1)) ^ (u - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
| congr 2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h.e_a.e_a
u v : β
hu : 0 < u.re
hv : 0 < v.re
x : β
β’ 1 - β(-x) = β(x + 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· | push_cast | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h.e_a.e_a
u v : β
hu : 0 < u.re
hv : 0 < v.re
x : β
β’ 1 - -βx = βx + 1 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; | ring | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h.e_a.e_a
u v : β
hu : 0 < u.re
hv : 0 < v.re
x : β
β’ β(-x) = 1 - β(x + 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· | push_cast | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h.e_a.e_a
u v : β
hu : 0 < u.re
hv : 0 < v.re
x : β
β’ -βx = 1 - (βx + 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; | ring | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_5
u v : β
hu : 0 < u.re
hv : 0 < v.re
β’ -(1 / 2) = 1 / 2 - 1 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· | norm_num | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_6
u v : β
hu : 0 < u.re
hv : 0 < v.re
β’ -1 = 0 - 1 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· | norm_num | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
β’ betaIntegral v u = betaIntegral u v | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
| rw [betaIntegral, betaIntegral] | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
β’ β« (x : β) in 0 ..1, βx ^ (v - 1) * (1 - βx) ^ (u - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
| have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
this :
β« (x : β) in 0 ..1, β(-1 * x + 1) ^ (u - 1) * (1 - β(-1 * x + 1)) ^ (v - 1) =
(-1)β»ΒΉ β’ β« (x : β) in -1 * 0 + 1 ..-1 * 1 + 1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
β’ β« (x : β) in 0 ..1, βx ^ (v - 1) * (1 - βx) ^ (u - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
| rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
this :
β« (x : β) in 0 ..1, β(-1 * x + 1) ^ (u - 1) * (1 - β(-1 * x + 1)) ^ (v - 1) =
β« (x : β) in -1 * 1 + 1 ..-1 * 0 + 1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
β’ β« (x : β) in 0 ..1, βx ^ (v - 1) * (1 - βx) ^ (u - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
| simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
this :
β« (x : β) in 0 ..1, β(-1 * x + 1) ^ (u - 1) * (1 - β(-1 * x + 1)) ^ (v - 1) =
β« (x : β) in -1 * 1 + 1 ..-1 * 0 + 1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
β’ β« (x : β) in 0 ..1, βx ^ (v - 1) * (1 - βx) ^ (u - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
| simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
this : β« (x : β) in 0 ..1, (-βx + 1) ^ (u - 1) * βx ^ (v - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
β’ β« (x : β) in 0 ..1, βx ^ (v - 1) * (1 - βx) ^ (u - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
| conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm] | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
this : β« (x : β) in 0 ..1, (-βx + 1) ^ (u - 1) * βx ^ (v - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
| β« (x : β) in 0 ..1, (-βx + 1) ^ (u - 1) * βx ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => | arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm] | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
this : β« (x : β) in 0 ..1, (-βx + 1) ^ (u - 1) * βx ^ (v - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
| β« (x : β) in 0 ..1, (-βx + 1) ^ (u - 1) * βx ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => | arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm] | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
this : β« (x : β) in 0 ..1, (-βx + 1) ^ (u - 1) * βx ^ (v - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
| β« (x : β) in 0 ..1, (-βx + 1) ^ (u - 1) * βx ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => | arg 1 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
this : β« (x : β) in 0 ..1, (-βx + 1) ^ (u - 1) * βx ^ (v - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
| fun x => (-βx + 1) ^ (u - 1) * βx ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; | intro x | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h
u v : β
this : β« (x : β) in 0 ..1, (-βx + 1) ^ (u - 1) * βx ^ (v - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
x : β
| (-βx + 1) ^ (u - 1) * βx ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; | rw [add_comm, β sub_eq_add_neg, mul_comm] | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : β
this : β« (x : β) in 0 ..1, βx ^ (v - 1) * (1 - βx) ^ (u - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1)
β’ β« (x : β) in 0 ..1, βx ^ (v - 1) * (1 - βx) ^ (u - 1) = β« (x : β) in 0 ..1, βx ^ (u - 1) * (1 - βx) ^ (v - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
| exact this | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : β
hu : 0 < u.re
β’ betaIntegral u 1 = 1 / u | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
| simp_rw [betaIntegral, sub_self, cpow_zero, mul_one] | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : β
hu : 0 < u.re
β’ β« (x : β) in 0 ..1, βx ^ (u - 1) = 1 / u | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
| rw [integral_cpow (Or.inl _)] | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : β
hu : 0 < u.re
β’ (β1 ^ (u - 1 + 1) - β0 ^ (u - 1 + 1)) / (u - 1 + 1) = 1 / u | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· | rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel] | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : β
hu : 0 < u.re
β’ u - 1 + 1 β 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
| rw [sub_add_cancel] | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : β
hu : 0 < u.re
β’ u β 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
| contrapose! hu | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : β
hu : u = 0
β’ u.re β€ 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; | rw [hu, zero_re] | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : β
hu : 0 < u.re
β’ -1 < (u - 1).re | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· | rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel] | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
β’ β« (x : β) in 0 ..a, βx ^ (s - 1) * (βa - βx) ^ (t - 1) = βa ^ (s + t - 1) * betaIntegral s t | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
| have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne' | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
β’ β« (x : β) in 0 ..a, βx ^ (s - 1) * (βa - βx) ^ (t - 1) = βa ^ (s + t - 1) * betaIntegral s t | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
| rw [betaIntegral] | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
β’ β« (x : β) in 0 ..a, βx ^ (s - 1) * (βa - βx) ^ (t - 1) =
βa ^ (s + t - 1) * β« (x : β) in 0 ..1, βx ^ (s - 1) * (1 - βx) ^ (t - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
| have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc] | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
β’ βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1)) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
| rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc] | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
β’ s + t - 1 = 1 + (s - 1) + (t - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by | abel | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
β’ s + t - 1 = 1 + (s - 1) + (t - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by | abel | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
β’ β« (x : β) in 0 ..a, βx ^ (s - 1) * (βa - βx) ^ (t - 1) =
βa ^ (s + t - 1) * β« (x : β) in 0 ..1, βx ^ (s - 1) * (1 - βx) ^ (t - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
| rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div] | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
β’ β« (x : β) in 0 ..a, βx ^ (s - 1) * (βa - βx) ^ (t - 1) =
β« (x : β) in 0 ..a, βa ^ (s - 1) * βa ^ (t - 1) * (β(x / a) ^ (s - 1) * (1 - β(x / a)) ^ (t - 1)) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
| simp_rw [intervalIntegral.integral_of_le ha.le] | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
β’ β« (x : β) in Ioc 0 a, βx ^ (s - 1) * (βa - βx) ^ (t - 1) βvolume =
β« (x : β) in Ioc 0 a, βa ^ (s - 1) * βa ^ (t - 1) * (β(x / a) ^ (s - 1) * (1 - β(x / a)) ^ (t - 1)) βvolume | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
| refine' set_integral_congr measurableSet_Ioc fun x hx => _ | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
x : β
hx : x β Ioc 0 a
β’ βx ^ (s - 1) * (βa - βx) ^ (t - 1) = βa ^ (s - 1) * βa ^ (t - 1) * (β(x / a) ^ (s - 1) * (1 - β(x / a)) ^ (t - 1)) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
| rw [mul_mul_mul_comm] | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
x : β
hx : x β Ioc 0 a
β’ βx ^ (s - 1) * (βa - βx) ^ (t - 1) = βa ^ (s - 1) * β(x / a) ^ (s - 1) * (βa ^ (t - 1) * (1 - β(x / a)) ^ (t - 1)) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
| congr 1 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
x : β
hx : x β Ioc 0 a
β’ βx ^ (s - 1) = βa ^ (s - 1) * β(x / a) ^ (s - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· | rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha'] | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
x : β
hx : x β Ioc 0 a
β’ (βa - βx) ^ (t - 1) = βa ^ (t - 1) * (1 - β(x / a)) ^ (t - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha']
Β· | rw [(by norm_cast : (1 : β) - β(x / a) = β(1 - x / a)), β
mul_cpow_ofReal_nonneg ha.le (sub_nonneg.mpr <| (div_le_one ha).mpr hx.2)] | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha']
Β· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
x : β
hx : x β Ioc 0 a
β’ 1 - β(x / a) = β(1 - x / a) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha']
Β· rw [(by | norm_cast | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha']
Β· rw [(by | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
x : β
hx : x β Ioc 0 a
β’ (βa - βx) ^ (t - 1) = (βa * β(1 - x / a)) ^ (t - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha']
Β· rw [(by norm_cast : (1 : β) - β(x / a) = β(1 - x / a)), β
mul_cpow_ofReal_nonneg ha.le (sub_nonneg.mpr <| (div_le_one ha).mpr hx.2)]
| push_cast | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha']
Β· rw [(by norm_cast : (1 : β) - β(x / a) = β(1 - x / a)), β
mul_cpow_ofReal_nonneg ha.le (sub_nonneg.mpr <| (div_le_one ha).mpr hx.2)]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a
s t : β
a : β
ha : 0 < a
ha' : βa β 0
A : βa ^ (s + t - 1) = βa * (βa ^ (s - 1) * βa ^ (t - 1))
x : β
hx : x β Ioc 0 a
β’ (βa - βx) ^ (t - 1) = (βa * (1 - βx / βa)) ^ (t - 1) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha']
Β· rw [(by norm_cast : (1 : β) - β(x / a) = β(1 - x / a)), β
mul_cpow_ofReal_nonneg ha.le (sub_nonneg.mpr <| (div_le_one ha).mpr hx.2)]
push_cast
| rw [mul_sub, mul_one, mul_div_cancel' _ ha'] | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha']
Β· rw [(by norm_cast : (1 : β) - β(x / a) = β(1 - x / a)), β
mul_cpow_ofReal_nonneg ha.le (sub_nonneg.mpr <| (div_le_one ha).mpr hx.2)]
push_cast
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : β
hs : 0 < s.re
ht : 0 < t.re
β’ Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
#align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
/-!
# The Beta function, and further properties of the Gamma function
In this file we define the Beta integral, relate Beta and Gamma functions, and prove some
refined properties of the Gamma function using these relations.
## Results on the Beta function
* `Complex.betaIntegral`: the Beta function `Ξ(u, v)`, where `u`, `v` are complex with positive
real part.
* `Complex.Gamma_mul_Gamma_eq_betaIntegral`: the formula
`Gamma u * Gamma v = Gamma (u + v) * betaIntegral u v`.
## Results on the Gamma function
* `Complex.Gamma_ne_zero`: for all `s : β` with `s β {-n : n β β}` we have `Ξ s β 0`.
* `Complex.GammaSeq_tendsto_Gamma`: for all `s`, the limit as `n β β` of the sequence
`n β¦ n ^ s * n! / (s * (s + 1) * ... * (s + n))` is `Ξ(s)`.
* `Complex.Gamma_mul_Gamma_one_sub`: Euler's reflection formula
`Gamma s * Gamma (1 - s) = Ο / sin Ο s`.
* `Complex.differentiable_one_div_Gamma`: the function `1 / Ξ(s)` is differentiable everywhere.
* `Complex.Gamma_mul_Gamma_add_half`: Legendre's duplication formula
`Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt Ο`.
* `Real.Gamma_ne_zero`, `Real.GammaSeq_tendsto_Gamma`,
`Real.Gamma_mul_Gamma_one_sub`, `Real.Gamma_mul_Gamma_add_half`: real versions of the above.
-/
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set Real MeasureTheory
open scoped Nat Topology BigOperators Real
section BetaIntegral
/-! ## The Beta function -/
namespace Complex
/-- The Beta function `Ξ (u, v)`, defined as `β« x:β in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. -/
noncomputable def betaIntegral (u v : β) : β :=
β« x : β in (0)..1, (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)
#align complex.beta_integral Complex.betaIntegral
/-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continuousOn
Β· refine' intervalIntegral.intervalIntegrable_cpow' _
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuous_const.sub continuous_ofReal).continuousAt
Β· exact continuousAt_const
Β· rw [sub_re, one_re, ofReal_re, sub_pos]
exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : β) < 1))
#align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left
/-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
#align complex.beta_integral_convergent Complex.betaIntegral_convergent
theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, β sub_eq_add_neg, mul_comm]
exact this
#align complex.beta_integral_symm Complex.betaIntegral_symm
theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
#align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right
theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.integral_const_mul, β real_smul, β zero_div a, β
div_self ha.ne', β intervalIntegral.integral_comp_div _ ha.ne', zero_div]
simp_rw [intervalIntegral.integral_of_le ha.le]
refine' set_integral_congr measurableSet_Ioc fun x hx => _
rw [mul_mul_mul_comm]
congr 1
Β· rw [β mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha']
Β· rw [(by norm_cast : (1 : β) - β(x / a) = β(1 - x / a)), β
mul_cpow_ofReal_nonneg ha.le (sub_nonneg.mpr <| (div_le_one ha).mpr hx.2)]
push_cast
rw [mul_sub, mul_one, mul_div_cancel' _ ha']
#align complex.beta_integral_scaled Complex.betaIntegral_scaled
/-- Relation between Beta integral and Gamma function. -/
theorem Gamma_mul_Gamma_eq_betaIntegral {s t : β} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t := by
-- Note that we haven't proved (yet) that the Gamma function has no zeroes, so we can't formulate
-- this as a formula for the Beta function.
| have conv_int := integral_posConvolution
(GammaIntegral_convergent hs) (GammaIntegral_convergent ht) (ContinuousLinearMap.mul β β) | /-- Relation between Beta integral and Gamma function. -/
theorem Gamma_mul_Gamma_eq_betaIntegral {s t : β} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t := by
-- Note that we haven't proved (yet) that the Gamma function has no zeroes, so we can't formulate
-- this as a formula for the Beta function.
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.134_0.in2QiCFW52coQT2 | /-- Relation between Beta integral and Gamma function. -/
theorem Gamma_mul_Gamma_eq_betaIntegral {s t : β} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
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