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from sympy.integrals.laplace import (
laplace_transform, inverse_laplace_transform,
LaplaceTransform, InverseLaplaceTransform,
_laplace_deep_collect, laplace_correspondence,
laplace_initial_conds)
from sympy.core.function import Function, expand_mul
from sympy.core import EulerGamma, Subs, Derivative, diff
from sympy.core.exprtools import factor_terms
from sympy.core.numbers import I, oo, pi
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol, symbols
from sympy.simplify.simplify import simplify
from sympy.functions.elementary.complexes import Abs, re
from sympy.functions.elementary.exponential import exp, log, exp_polar
from sympy.functions.elementary.hyperbolic import cosh, sinh, coth, asinh
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import atan, cos, sin
from sympy.logic.boolalg import And
from sympy.functions.special.gamma_functions import (
lowergamma, gamma, uppergamma)
from sympy.functions.special.delta_functions import DiracDelta, Heaviside
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.zeta_functions import lerchphi
from sympy.functions.special.error_functions import (
fresnelc, fresnels, erf, erfc, Ei, Ci, expint, E1)
from sympy.functions.special.bessel import besseli, besselj, besselk, bessely
from sympy.testing.pytest import slow, warns_deprecated_sympy
from sympy.matrices import Matrix, eye
from sympy.abc import s
@slow
def test_laplace_transform():
LT = laplace_transform
ILT = inverse_laplace_transform
a, b, c = symbols('a, b, c', positive=True)
np = symbols('np', integer=True, positive=True)
t, w, x = symbols('t, w, x')
f = Function('f')
F = Function('F')
g = Function('g')
y = Function('y')
Y = Function('Y')
# Test helper functions
assert (
_laplace_deep_collect(exp((t+a)*(t+b)) +
besselj(2, exp((t+a)*(t+b)-t**2)), t) ==
exp(a*b + t**2 + t*(a + b)) + besselj(2, exp(a*b + t*(a + b))))
L = laplace_transform(diff(y(t), t, 3), t, s, noconds=True)
L = laplace_correspondence(L, {y: Y})
L = laplace_initial_conds(L, t, {y: [2, 4, 8, 16, 32]})
assert L == s**3*Y(s) - 2*s**2 - 4*s - 8
# Test whether `noconds=True` in `doit`:
assert (2*LaplaceTransform(exp(t), t, s) - 1).doit() == -1 + 2/(s - 1)
assert (LT(a*t+t**2+t**(S(5)/2), t, s) ==
(a/s**2 + 2/s**3 + 15*sqrt(pi)/(8*s**(S(7)/2)), 0, True))
assert LT(b/(t+a), t, s) == (-b*exp(-a*s)*Ei(-a*s), 0, True)
assert (LT(1/sqrt(t+a), t, s) ==
(sqrt(pi)*sqrt(1/s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True))
assert (LT(sqrt(t)/(t+a), t, s) ==
(-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
0, True))
assert (LT((t+a)**(-S(3)/2), t, s) ==
(-2*sqrt(pi)*sqrt(s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + 2/sqrt(a),
0, True))
assert (LT(t**(S(1)/2)*(t+a)**(-1), t, s) ==
(-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
0, True))
assert (LT(1/(a*sqrt(t) + t**(3/2)), t, s) ==
(pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True))
assert (LT((t+a)**b, t, s) ==
(s**(-b - 1)*exp(-a*s)*uppergamma(b + 1, a*s), 0, True))
assert LT(t**5/(t+a), t, s) == (120*a**5*uppergamma(-5, a*s), 0, True)
assert LT(exp(t), t, s) == (1/(s - 1), 1, True)
assert LT(exp(2*t), t, s) == (1/(s - 2), 2, True)
assert LT(exp(a*t), t, s) == (1/(s - a), a, True)
assert LT(exp(a*(t-b)), t, s) == (exp(-a*b)/(-a + s), a, True)
assert LT(t*exp(-a*(t)), t, s) == ((a + s)**(-2), -a, True)
assert LT(t*exp(-a*(t-b)), t, s) == (exp(a*b)/(a + s)**2, -a, True)
assert LT(b*t*exp(-a*t), t, s) == (b/(a + s)**2, -a, True)
assert LT(exp(-a*exp(-t)), t, s) == (lowergamma(s, a)/a**s, 0, True)
assert LT(exp(-a*exp(t)), t, s) == (a**s*uppergamma(-s, a), 0, True)
assert (LT(t**(S(7)/4)*exp(-8*t)/gamma(S(11)/4), t, s) ==
((s + 8)**(-S(11)/4), -8, True))
assert (LT(t**(S(3)/2)*exp(-8*t), t, s) ==
(3*sqrt(pi)/(4*(s + 8)**(S(5)/2)), -8, True))
assert LT(t**a*exp(-a*t), t, s) == ((a+s)**(-a-1)*gamma(a+1), -a, True)
assert (LT(b*exp(-a*t**2), t, s) ==
(sqrt(pi)*b*exp(s**2/(4*a))*erfc(s/(2*sqrt(a)))/(2*sqrt(a)),
0, True))
assert (LT(exp(-2*t**2), t, s) ==
(sqrt(2)*sqrt(pi)*exp(s**2/8)*erfc(sqrt(2)*s/4)/4, 0, True))
assert (LT(b*exp(2*t**2), t, s) ==
(b*LaplaceTransform(exp(2*t**2), t, s), -oo, True))
assert (LT(t*exp(-a*t**2), t, s) ==
(1/(2*a) - s*erfc(s/(2*sqrt(a)))/(4*sqrt(pi)*a**(S(3)/2)),
0, True))
assert (LT(exp(-a/t), t, s) ==
(2*sqrt(a)*sqrt(1/s)*besselk(1, 2*sqrt(a)*sqrt(s)), 0, True))
assert LT(sqrt(t)*exp(-a/t), t, s, simplify=True) == (
sqrt(pi)*(sqrt(a)*sqrt(s) + 1/S(2))*sqrt(s**(-3)) *
exp(-2*sqrt(a)*sqrt(s)), 0, True)
assert (LT(exp(-a/t)/sqrt(t), t, s) ==
(sqrt(pi)*sqrt(1/s)*exp(-2*sqrt(a)*sqrt(s)), 0, True))
assert (LT(exp(-a/t)/(t*sqrt(t)), t, s) ==
(sqrt(pi)*sqrt(1/a)*exp(-2*sqrt(a)*sqrt(s)), 0, True))
assert (
LT(exp(-2*sqrt(a*t)), t, s) ==
(1/s - sqrt(pi)*sqrt(a) * exp(a/s)*erfc(sqrt(a)*sqrt(1/s)) /
s**(S(3)/2), 0, True))
assert LT(exp(-2*sqrt(a*t))/sqrt(t), t, s) == (
exp(a/s)*erfc(sqrt(a) * sqrt(1/s))*(sqrt(pi)*sqrt(1/s)), 0, True)
assert (LT(t**4*exp(-2/t), t, s) ==
(8*sqrt(2)*(1/s)**(S(5)/2)*besselk(5, 2*sqrt(2)*sqrt(s)),
0, True))
assert LT(sinh(a*t), t, s) == (a/(-a**2 + s**2), a, True)
assert (LT(b*sinh(a*t)**2, t, s) ==
(2*a**2*b/(-4*a**2*s + s**3), 2*a, True))
assert (LT(b*sinh(a*t)**2, t, s, simplify=True) ==
(2*a**2*b/(s*(-4*a**2 + s**2)), 2*a, True))
# The following line confirms that issue #21202 is solved
assert LT(cosh(2*t), t, s) == (s/(-4 + s**2), 2, True)
assert LT(cosh(a*t), t, s) == (s/(-a**2 + s**2), a, True)
assert (LT(cosh(a*t)**2, t, s, simplify=True) ==
((2*a**2 - s**2)/(s*(4*a**2 - s**2)), 2*a, True))
assert (LT(sinh(x+3), x, s, simplify=True) ==
((s*sinh(3) + cosh(3))/(s**2 - 1), 1, True))
L, _, _ = LT(42*sin(w*t+x)**2, t, s)
assert (
L -
21*(s**2 + s*(-s*cos(2*x) + 2*w*sin(2*x)) +
4*w**2)/(s*(s**2 + 4*w**2))).simplify() == 0
# The following line replaces the old test test_issue_7173()
assert LT(sinh(a*t)*cosh(a*t), t, s, simplify=True) == (a/(-4*a**2 + s**2),
2*a, True)
assert LT(sinh(a*t)/t, t, s) == (log((a + s)/(-a + s))/2, a, True)
assert (LT(t**(-S(3)/2)*sinh(a*t), t, s) ==
(-sqrt(pi)*(sqrt(-a + s) - sqrt(a + s)), a, True))
assert (LT(sinh(2*sqrt(a*t)), t, s) ==
(sqrt(pi)*sqrt(a)*exp(a/s)/s**(S(3)/2), 0, True))
assert (LT(sqrt(t)*sinh(2*sqrt(a*t)), t, s, simplify=True) ==
((-sqrt(a)*s**(S(5)/2) + sqrt(pi)*s**2*(2*a + s)*exp(a/s) *
erf(sqrt(a)*sqrt(1/s))/2)/s**(S(9)/2), 0, True))
assert (LT(sinh(2*sqrt(a*t))/sqrt(t), t, s) ==
(sqrt(pi)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/sqrt(s), 0, True))
assert (LT(sinh(sqrt(a*t))**2/sqrt(t), t, s) ==
(sqrt(pi)*(exp(a/s) - 1)/(2*sqrt(s)), 0, True))
assert (LT(t**(S(3)/7)*cosh(a*t), t, s) ==
(((a + s)**(-S(10)/7) + (-a+s)**(-S(10)/7))*gamma(S(10)/7)/2,
a, True))
assert (LT(cosh(2*sqrt(a*t)), t, s) ==
(sqrt(pi)*sqrt(a)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/s**(S(3)/2) +
1/s, 0, True))
assert (LT(sqrt(t)*cosh(2*sqrt(a*t)), t, s) ==
(sqrt(pi)*(a + s/2)*exp(a/s)/s**(S(5)/2), 0, True))
assert (LT(cosh(2*sqrt(a*t))/sqrt(t), t, s) ==
(sqrt(pi)*exp(a/s)/sqrt(s), 0, True))
assert (LT(cosh(sqrt(a*t))**2/sqrt(t), t, s) ==
(sqrt(pi)*(exp(a/s) + 1)/(2*sqrt(s)), 0, True))
assert LT(log(t), t, s, simplify=True) == (
(-log(s) - EulerGamma)/s, 0, True)
assert (LT(-log(t/a), t, s, simplify=True) ==
((log(a) + log(s) + EulerGamma)/s, 0, True))
assert LT(log(1+a*t), t, s) == (-exp(s/a)*Ei(-s/a)/s, 0, True)
assert (LT(log(t+a), t, s, simplify=True) ==
((s*log(a) - exp(s/a)*Ei(-s/a))/s**2, 0, True))
assert (LT(log(t)/sqrt(t), t, s, simplify=True) ==
(sqrt(pi)*(-log(s) - log(4) - EulerGamma)/sqrt(s), 0, True))
assert (LT(t**(S(5)/2)*log(t), t, s, simplify=True) ==
(sqrt(pi)*(-15*log(s) - log(1073741824) - 15*EulerGamma + 46) /
(8*s**(S(7)/2)), 0, True))
assert (LT(t**3*log(t), t, s, noconds=True, simplify=True) -
6*(-log(s) - S.EulerGamma + S(11)/6)/s**4).simplify() == S.Zero
assert (LT(log(t)**2, t, s, simplify=True) ==
(((log(s) + EulerGamma)**2 + pi**2/6)/s, 0, True))
assert (LT(exp(-a*t)*log(t), t, s, simplify=True) ==
((-log(a + s) - EulerGamma)/(a + s), -a, True))
assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True)
assert (LT(Abs(sin(a*t)), t, s) ==
(a*coth(pi*s/(2*a))/(a**2 + s**2), 0, True))
assert LT(sin(a*t)/t, t, s) == (atan(a/s), 0, True)
assert LT(sin(a*t)**2/t, t, s) == (log(4*a**2/s**2 + 1)/4, 0, True)
assert (LT(sin(a*t)**2/t**2, t, s) ==
(a*atan(2*a/s) - s*log(4*a**2/s**2 + 1)/4, 0, True))
assert (LT(sin(2*sqrt(a*t)), t, s) ==
(sqrt(pi)*sqrt(a)*exp(-a/s)/s**(S(3)/2), 0, True))
assert LT(sin(2*sqrt(a*t))/t, t, s) == (pi*erf(sqrt(a)*sqrt(1/s)), 0, True)
assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True)
assert (LT(cos(a*t)**2, t, s) ==
((2*a**2 + s**2)/(s*(4*a**2 + s**2)), 0, True))
assert (LT(sqrt(t)*cos(2*sqrt(a*t)), t, s, simplify=True) ==
(sqrt(pi)*(-a + s/2)*exp(-a/s)/s**(S(5)/2), 0, True))
assert (LT(cos(2*sqrt(a*t))/sqrt(t), t, s) ==
(sqrt(pi)*sqrt(1/s)*exp(-a/s), 0, True))
assert (LT(sin(a*t)*sin(b*t), t, s) ==
(2*a*b*s/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)), 0, True))
assert (LT(cos(a*t)*sin(b*t), t, s) ==
(b*(-a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
0, True))
assert (LT(cos(a*t)*cos(b*t), t, s) ==
(s*(a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
0, True))
assert (LT(-a*t*cos(a*t) + sin(a*t), t, s, simplify=True) ==
(2*a**3/(a**4 + 2*a**2*s**2 + s**4), 0, True))
assert LT(c*exp(-b*t)*sin(a*t), t, s) == (a *
c/(a**2 + (b + s)**2), -b, True)
assert LT(c*exp(-b*t)*cos(a*t), t, s) == (c*(b + s)/(a**2 + (b + s)**2),
-b, True)
L, plane, cond = LT(cos(x + 3), x, s, simplify=True)
assert plane == 0
assert L - (s*cos(3) - sin(3))/(s**2 + 1) == 0
# Error functions (laplace7.pdf)
assert LT(erf(a*t), t, s) == (exp(s**2/(4*a**2))*erfc(s/(2*a))/s, 0, True)
assert LT(erf(sqrt(a*t)), t, s) == (sqrt(a)/(s*sqrt(a + s)), 0, True)
assert (LT(exp(a*t)*erf(sqrt(a*t)), t, s, simplify=True) ==
(-sqrt(a)/(sqrt(s)*(a - s)), a, True))
assert (LT(erf(sqrt(a/t)/2), t, s, simplify=True) ==
(1/s - exp(-sqrt(a)*sqrt(s))/s, 0, True))
assert (LT(erfc(sqrt(a*t)), t, s, simplify=True) ==
(-sqrt(a)/(s*sqrt(a + s)) + 1/s, -a, True))
assert (LT(exp(a*t)*erfc(sqrt(a*t)), t, s) ==
(1/(sqrt(a)*sqrt(s) + s), 0, True))
assert LT(erfc(sqrt(a/t)/2), t, s) == (exp(-sqrt(a)*sqrt(s))/s, 0, True)
# Bessel functions (laplace8.pdf)
assert LT(besselj(0, a*t), t, s) == (1/sqrt(a**2 + s**2), 0, True)
assert (LT(besselj(1, a*t), t, s, simplify=True) ==
(a/(a**2 + s**2 + s*sqrt(a**2 + s**2)), 0, True))
assert (LT(besselj(2, a*t), t, s, simplify=True) ==
(a**2/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))**2), 0, True))
assert (LT(t*besselj(0, a*t), t, s) ==
(s/(a**2 + s**2)**(S(3)/2), 0, True))
assert (LT(t*besselj(1, a*t), t, s) ==
(a/(a**2 + s**2)**(S(3)/2), 0, True))
assert (LT(t**2*besselj(2, a*t), t, s) ==
(3*a**2/(a**2 + s**2)**(S(5)/2), 0, True))
assert LT(besselj(0, 2*sqrt(a*t)), t, s) == (exp(-a/s)/s, 0, True)
assert (LT(t**(S(3)/2)*besselj(3, 2*sqrt(a*t)), t, s) ==
(a**(S(3)/2)*exp(-a/s)/s**4, 0, True))
assert (LT(besselj(0, a*sqrt(t**2+b*t)), t, s, simplify=True) ==
(exp(b*(s - sqrt(a**2 + s**2)))/sqrt(a**2 + s**2), 0, True))
assert LT(besseli(0, a*t), t, s) == (1/sqrt(-a**2 + s**2), a, True)
assert (LT(besseli(1, a*t), t, s, simplify=True) ==
(a/(-a**2 + s**2 + s*sqrt(-a**2 + s**2)), a, True))
assert (LT(besseli(2, a*t), t, s, simplify=True) ==
(a**2/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))**2), a, True))
assert LT(t*besseli(0, a*t), t, s) == (s/(-a**2 + s**2)**(S(3)/2), a, True)
assert LT(t*besseli(1, a*t), t, s) == (a/(-a**2 + s**2)**(S(3)/2), a, True)
assert (LT(t**2*besseli(2, a*t), t, s) ==
(3*a**2/(-a**2 + s**2)**(S(5)/2), a, True))
assert (LT(t**(S(3)/2)*besseli(3, 2*sqrt(a*t)), t, s) ==
(a**(S(3)/2)*exp(a/s)/s**4, 0, True))
assert (LT(bessely(0, a*t), t, s) ==
(-2*asinh(s/a)/(pi*sqrt(a**2 + s**2)), 0, True))
assert (LT(besselk(0, a*t), t, s) ==
(log((s + sqrt(-a**2 + s**2))/a)/sqrt(-a**2 + s**2), -a, True))
assert (LT(sin(a*t)**4, t, s, simplify=True) ==
(24*a**4/(s*(64*a**4 + 20*a**2*s**2 + s**4)), 0, True))
# Test general rules and unevaluated forms
# These all also test whether issue #7219 is solved.
assert LT(Heaviside(t-1)*cos(t-1), t, s) == (s*exp(-s)/(s**2 + 1), 0, True)
assert LT(a*f(t), t, w) == (a*LaplaceTransform(f(t), t, w), -oo, True)
assert (LT(a*Heaviside(t+1)*f(t+1), t, s) ==
(a*LaplaceTransform(f(t + 1), t, s), -oo, True))
assert (LT(a*Heaviside(t-1)*f(t-1), t, s) ==
(a*LaplaceTransform(f(t), t, s)*exp(-s), -oo, True))
assert (LT(b*f(t/a), t, s) ==
(a*b*LaplaceTransform(f(t), t, a*s), -oo, True))
assert LT(exp(-f(x)*t), t, s) == (1/(s + f(x)), -re(f(x)), True)
assert (LT(exp(-a*t)*f(t), t, s) ==
(LaplaceTransform(f(t), t, a + s), -oo, True))
assert (LT(exp(-a*t)*erfc(sqrt(b/t)/2), t, s) ==
(exp(-sqrt(b)*sqrt(a + s))/(a + s), -a, True))
assert (LT(sinh(a*t)*f(t), t, s) ==
(LaplaceTransform(f(t), t, -a + s)/2 -
LaplaceTransform(f(t), t, a + s)/2, -oo, True))
assert (LT(sinh(a*t)*t, t, s, simplify=True) ==
(2*a*s/(a**4 - 2*a**2*s**2 + s**4), a, True))
assert (LT(cosh(a*t)*f(t), t, s) ==
(LaplaceTransform(f(t), t, -a + s)/2 +
LaplaceTransform(f(t), t, a + s)/2, -oo, True))
assert (LT(cosh(a*t)*t, t, s, simplify=True) ==
(1/(2*(a + s)**2) + 1/(2*(a - s)**2), a, True))
assert (LT(sin(a*t)*f(t), t, s, simplify=True) ==
(I*(-LaplaceTransform(f(t), t, -I*a + s) +
LaplaceTransform(f(t), t, I*a + s))/2, -oo, True))
assert (LT(sin(f(t)), t, s) ==
(LaplaceTransform(sin(f(t)), t, s), -oo, True))
assert (LT(sin(a*t)*t, t, s, simplify=True) ==
(2*a*s/(a**4 + 2*a**2*s**2 + s**4), 0, True))
assert (LT(cos(a*t)*f(t), t, s) ==
(LaplaceTransform(f(t), t, -I*a + s)/2 +
LaplaceTransform(f(t), t, I*a + s)/2, -oo, True))
assert (LT(cos(a*t)*t, t, s, simplify=True) ==
((-a**2 + s**2)/(a**4 + 2*a**2*s**2 + s**4), 0, True))
L, plane, _ = LT(sin(a*t+b)**2*f(t), t, s)
assert plane == -oo
assert (
-L + (
LaplaceTransform(f(t), t, s)/2 -
LaplaceTransform(f(t), t, -2*I*a + s)*exp(2*I*b)/4 -
LaplaceTransform(f(t), t, 2*I*a + s)*exp(-2*I*b)/4)) == 0
L = LT(sin(a*t+b)**2*f(t), t, s, noconds=True)
assert (
laplace_correspondence(L, {f: F}) ==
F(s)/2 - F(-2*I*a + s)*exp(2*I*b)/4 -
F(2*I*a + s)*exp(-2*I*b)/4)
L, plane, _ = LT(sin(a*t)**3*cosh(b*t), t, s)
assert plane == b
assert (
-L - 3*a/(8*(9*a**2 + b**2 + 2*b*s + s**2)) -
3*a/(8*(9*a**2 + b**2 - 2*b*s + s**2)) +
3*a/(8*(a**2 + b**2 + 2*b*s + s**2)) +
3*a/(8*(a**2 + b**2 - 2*b*s + s**2))).simplify() == 0
assert (LT(t**2*exp(-t**2), t, s) ==
(sqrt(pi)*s**2*exp(s**2/4)*erfc(s/2)/8 - s/4 +
sqrt(pi)*exp(s**2/4)*erfc(s/2)/4, 0, True))
assert (LT((a*t**2 + b*t + c)*f(t), t, s) ==
(a*Derivative(LaplaceTransform(f(t), t, s), (s, 2)) -
b*Derivative(LaplaceTransform(f(t), t, s), s) +
c*LaplaceTransform(f(t), t, s), -oo, True))
assert (LT(t**np*g(t), t, s) ==
((-1)**np*Derivative(LaplaceTransform(g(t), t, s), (s, np)),
-oo, True))
# The following tests check whether _piecewise_to_heaviside works:
x1 = Piecewise((0, t <= 0), (1, t <= 1), (0, True))
X1 = LT(x1, t, s)[0]
assert X1 == 1/s - exp(-s)/s
y1 = ILT(X1, s, t)
assert y1 == Heaviside(t) - Heaviside(t - 1)
x1 = Piecewise((0, t <= 0), (t, t <= 1), (2-t, t <= 2), (0, True))
X1 = LT(x1, t, s)[0].simplify()
assert X1 == (exp(2*s) - 2*exp(s) + 1)*exp(-2*s)/s**2
y1 = ILT(X1, s, t)
assert (
-y1 + t*Heaviside(t) + (t - 2)*Heaviside(t - 2) -
2*(t - 1)*Heaviside(t - 1)).simplify() == 0
x1 = Piecewise((exp(t), t <= 0), (1, t <= 1), (exp(-(t)), True))
X1 = LT(x1, t, s)[0]
assert X1 == exp(-1)*exp(-s)/(s + 1) + 1/s - exp(-s)/s
y1 = ILT(X1, s, t)
assert y1 == (
exp(-1)*exp(1 - t)*Heaviside(t - 1) + Heaviside(t) - Heaviside(t - 1))
x1 = Piecewise((0, x <= 0), (1, x <= 1), (0, True))
X1 = LT(x1, t, s)[0]
assert X1 == Piecewise((0, x <= 0), (1, x <= 1), (0, True))/s
x1 = [
a*Piecewise((1, And(t > 1, t <= 3)), (2, True)),
a*Piecewise((1, And(t >= 1, t <= 3)), (2, True)),
a*Piecewise((1, And(t >= 1, t < 3)), (2, True)),
a*Piecewise((1, And(t > 1, t < 3)), (2, True))]
for x2 in x1:
assert LT(x2, t, s)[0].expand() == 2*a/s - a*exp(-s)/s + a*exp(-3*s)/s
assert (
LT(Piecewise((1, Eq(t, 1)), (2, True)), t, s)[0] ==
LaplaceTransform(Piecewise((1, Eq(t, 1)), (2, True)), t, s))
# The following lines test whether _laplace_transform successfully
# removes Heaviside(1) before processing espressions. It fails if
# Heaviside(t) remains because then meijerg functions will appear.
X1 = 1/sqrt(a*s**2-b)
x1 = ILT(X1, s, t)
Y1 = LT(x1, t, s)[0]
Z1 = (Y1**2/X1**2).simplify()
assert Z1 == 1
# The following two lines test whether issues #5813 and #7176 are solved.
assert (LT(diff(f(t), (t, 1)), t, s, noconds=True) ==
s*LaplaceTransform(f(t), t, s) - f(0))
assert (LT(diff(f(t), (t, 3)), t, s, noconds=True) ==
s**3*LaplaceTransform(f(t), t, s) - s**2*f(0) -
s*Subs(Derivative(f(t), t), t, 0) -
Subs(Derivative(f(t), (t, 2)), t, 0))
# Issue #7219
assert (LT(diff(f(x, t, w), t, 2), t, s) ==
(s**2*LaplaceTransform(f(x, t, w), t, s) - s*f(x, 0, w) -
Subs(Derivative(f(x, t, w), t), t, 0), -oo, True))
# Issue #23307
assert (LT(10*diff(f(t), (t, 1)), t, s, noconds=True) ==
10*s*LaplaceTransform(f(t), t, s) - 10*f(0))
assert (LT(a*f(b*t)+g(c*t), t, s, noconds=True) ==
a*LaplaceTransform(f(t), t, s/b)/b +
LaplaceTransform(g(t), t, s/c)/c)
assert inverse_laplace_transform(
f(w), w, t, plane=0) == InverseLaplaceTransform(f(w), w, t, 0)
assert (LT(f(t)*g(t), t, s, noconds=True) ==
LaplaceTransform(f(t)*g(t), t, s))
# Issue #24294
assert (LT(b*f(a*t), t, s, noconds=True) ==
b*LaplaceTransform(f(t), t, s/a)/a)
assert LT(3*exp(t)*Heaviside(t), t, s) == (3/(s - 1), 1, True)
assert (LT(2*sin(t)*Heaviside(t), t, s, simplify=True) ==
(2/(s**2 + 1), 0, True))
# Issue #25293
assert (
LT((1/(t-1))*sin(4*pi*(t-1))*DiracDelta(t-1) *
(Heaviside(t-1/4) - Heaviside(t-2)), t, s)[0] == 4*pi*exp(-s))
# additional basic tests from wikipedia
assert (LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) ==
((c + s)**(-b - 1)*exp(-a*s)*gamma(b + 1), -c, True))
assert (
LT((exp(2*t)-1)*exp(-b-t)*Heaviside(t)/2, t, s, noconds=True,
simplify=True) ==
exp(-b)/(s**2 - 1))
# DiracDelta function: standard cases
assert LT(DiracDelta(t), t, s) == (1, -oo, True)
assert LT(DiracDelta(a*t), t, s) == (1/a, -oo, True)
assert LT(DiracDelta(t/42), t, s) == (42, -oo, True)
assert LT(DiracDelta(t+42), t, s) == (0, -oo, True)
assert (LT(DiracDelta(t)+DiracDelta(t-42), t, s) ==
(1 + exp(-42*s), -oo, True))
assert (LT(DiracDelta(t)-a*exp(-a*t), t, s, simplify=True) ==
(s/(a + s), -a, True))
assert (
LT(exp(-t)*(DiracDelta(t)+DiracDelta(t-42)), t, s, simplify=True) ==
(exp(-42*s - 42) + 1, -oo, True))
assert LT(f(t)*DiracDelta(t-42), t, s) == (f(42)*exp(-42*s), -oo, True)
assert LT(f(t)*DiracDelta(b*t-a), t, s) == (f(a/b)*exp(-a*s/b)/b,
-oo, True)
assert LT(f(t)*DiracDelta(b*t+a), t, s) == (0, -oo, True)
# SingularityFunction
assert LT(SingularityFunction(t, a, -1), t, s)[0] == exp(-a*s)
assert LT(SingularityFunction(t, a, 1), t, s)[0] == exp(-a*s)/s**2
assert LT(SingularityFunction(t, a, x), t, s)[0] == (
LaplaceTransform(SingularityFunction(t, a, x), t, s))
# Collection of cases that cannot be fully evaluated and/or would catch
# some common implementation errors
assert (LT(DiracDelta(t**2), t, s, noconds=True) ==
LaplaceTransform(DiracDelta(t**2), t, s))
assert LT(DiracDelta(t**2 - 1), t, s) == (exp(-s)/2, -oo, True)
assert LT(DiracDelta(t*(1 - t)), t, s) == (1 - exp(-s), -oo, True)
assert (LT((DiracDelta(t) + 1)*(DiracDelta(t - 1) + 1), t, s) ==
(LaplaceTransform(DiracDelta(t)*DiracDelta(t - 1), t, s) +
1 + exp(-s) + 1/s, 0, True))
assert LT(DiracDelta(2*t-2*exp(a)), t, s) == (exp(-s*exp(a))/2, -oo, True)
assert LT(DiracDelta(-2*t+2*exp(a)), t, s) == (exp(-s*exp(a))/2, -oo, True)
# Heaviside tests
assert LT(Heaviside(t), t, s) == (1/s, 0, True)
assert LT(Heaviside(t - a), t, s) == (exp(-a*s)/s, 0, True)
assert LT(Heaviside(t-1), t, s) == (exp(-s)/s, 0, True)
assert LT(Heaviside(2*t-4), t, s) == (exp(-2*s)/s, 0, True)
assert LT(Heaviside(2*t+4), t, s) == (1/s, 0, True)
assert (LT(Heaviside(-2*t+4), t, s, simplify=True) ==
(1/s - exp(-2*s)/s, 0, True))
assert (LT(g(t)*Heaviside(t - w), t, s) ==
(LaplaceTransform(g(t)*Heaviside(t - w), t, s), -oo, True))
assert (
LT(Heaviside(t-a)*g(t), t, s) ==
(LaplaceTransform(g(a + t), t, s)*exp(-a*s), -oo, True))
assert (
LT(Heaviside(t+a)*g(t), t, s) ==
(LaplaceTransform(g(t), t, s), -oo, True))
assert (
LT(Heaviside(-t+a)*g(t), t, s) ==
(LaplaceTransform(g(t), t, s) -
LaplaceTransform(g(a + t), t, s)*exp(-a*s), -oo, True))
assert (
LT(Heaviside(-t-a)*g(t), t, s) == (0, 0, True))
# Fresnel functions
assert (laplace_transform(fresnels(t), t, s, simplify=True) ==
((-sin(s**2/(2*pi))*fresnels(s/pi) +
sqrt(2)*sin(s**2/(2*pi) + pi/4)/2 -
cos(s**2/(2*pi))*fresnelc(s/pi))/s, 0, True))
assert (laplace_transform(fresnelc(t), t, s, simplify=True) ==
((sin(s**2/(2*pi))*fresnelc(s/pi) -
cos(s**2/(2*pi))*fresnels(s/pi) +
sqrt(2)*cos(s**2/(2*pi) + pi/4)/2)/s, 0, True))
# Matrix tests
Mt = Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]])
Ms = Matrix([[1/(s - 1), (s + 1)**(-2)],
[(s + 1)**(-2), 1/(s - 1)]])
# The default behaviour for Laplace transform of a Matrix returns a Matrix
# of Tuples and is deprecated:
with warns_deprecated_sympy():
Ms_conds = Matrix(
[[(1/(s - 1), 1, True), ((s + 1)**(-2), -1, True)],
[((s + 1)**(-2), -1, True), (1/(s - 1), 1, True)]])
with warns_deprecated_sympy():
assert LT(Mt, t, s) == Ms_conds
# The new behavior is to return a tuple of a Matrix and the convergence
# conditions for the matrix as a whole:
assert LT(Mt, t, s, legacy_matrix=False) == (Ms, 1, True)
# With noconds=True the transformed matrix is returned without conditions
# either way:
assert LT(Mt, t, s, noconds=True) == Ms
assert LT(Mt, t, s, legacy_matrix=False, noconds=True) == Ms
@slow
def test_inverse_laplace_transform():
s = symbols('s')
k, n, t = symbols('k, n, t', real=True)
a, b, c, d = symbols('a, b, c, d', positive=True)
f = Function('f')
F = Function('F')
def ILT(g):
return inverse_laplace_transform(g, s, t)
def ILTS(g):
return inverse_laplace_transform(g, s, t, simplify=True)
def ILTF(g):
return laplace_correspondence(
inverse_laplace_transform(g, s, t), {f: F})
# Tests for the rules in Bateman54.
# Section 4.1: Some of the Laplace transform rules can also be used well
# in the inverse transform.
assert ILTF(exp(-a*s)*F(s)) == f(-a + t)
assert ILTF(k*F(s-a)) == k*f(t)*exp(-a*t)
assert ILTF(diff(F(s), s, 3)) == -t**3*f(t)
assert ILTF(diff(F(s), s, 4)) == t**4*f(t)
# Section 5.1: Most rules are impractical for a computer algebra system.
# Section 5.2: Rational functions
assert ILT(2) == 2*DiracDelta(t)
assert ILT(1/s) == Heaviside(t)
assert ILT(1/s**2) == t*Heaviside(t)
assert ILT(1/s**5) == t**4*Heaviside(t)/24
assert ILT(1/s**n) == t**(n - 1)*Heaviside(t)/gamma(n)
assert ILT(a/(a + s)) == a*exp(-a*t)*Heaviside(t)
assert ILT(s/(a + s)) == -a*exp(-a*t)*Heaviside(t) + DiracDelta(t)
assert (ILT(b*s/(s+a)**2) ==
b*(-a*t*exp(-a*t)*Heaviside(t) + exp(-a*t)*Heaviside(t)))
assert (ILTS(c/((s+a)*(s+b))) ==
c*(exp(a*t) - exp(b*t))*exp(-t*(a + b))*Heaviside(t)/(a - b))
assert (ILTS(c*s/((s+a)*(s+b))) ==
c*(a*exp(b*t) - b*exp(a*t))*exp(-t*(a + b))*Heaviside(t)/(a - b))
assert ILTS(s/(a + s)**3) == t*(-a*t + 2)*exp(-a*t)*Heaviside(t)/2
assert ILTS(1/(s*(a + s)**3)) == (
-a**2*t**2 - 2*a*t + 2*exp(a*t) - 2)*exp(-a*t)*Heaviside(t)/(2*a**3)
assert ILT(1/(s*(a + s)**n)) == (
Heaviside(t)*lowergamma(n, a*t)/(a**n*gamma(n)))
assert ILT((s-a)**(-b)) == t**(b - 1)*exp(a*t)*Heaviside(t)/gamma(b)
assert ILT((a + s)**(-2)) == t*exp(-a*t)*Heaviside(t)
assert ILT((a + s)**(-5)) == t**4*exp(-a*t)*Heaviside(t)/24
assert ILT(s**2/(s**2 + 1)) == -sin(t)*Heaviside(t) + DiracDelta(t)
assert ILT(1 - 1/(s**2 + 1)) == -sin(t)*Heaviside(t) + DiracDelta(t)
assert ILT(a/(a**2 + s**2)) == sin(a*t)*Heaviside(t)
assert ILT(s/(s**2 + a**2)) == cos(a*t)*Heaviside(t)
assert ILT(b/(b**2 + (a + s)**2)) == exp(-a*t)*sin(b*t)*Heaviside(t)
assert (ILT(b*s/(b**2 + (a + s)**2)) ==
b*(-a*exp(-a*t)*sin(b*t)/b + exp(-a*t)*cos(b*t))*Heaviside(t))
assert ILT(1/(s**2*(s**2 + 1))) == t*Heaviside(t) - sin(t)*Heaviside(t)
assert (ILTS(c*s/(d**2*(s+a)**2+b**2)) ==
c*(-a*d*sin(b*t/d) + b*cos(b*t/d))*exp(-a*t)*Heaviside(t)/(b*d**2))
assert ILTS((b*s**2 + d)/(a**2 + s**2)**2) == (
2*a**2*b*sin(a*t) + (a**2*b - d)*(a*t*cos(a*t) -
sin(a*t)))*Heaviside(t)/(2*a**3)
assert ILTS(b/(s**2-a**2)) == b*sinh(a*t)*Heaviside(t)/a
assert (ILT(b/(s**2-a**2)) ==
b*(exp(a*t)*Heaviside(t)/(2*a) - exp(-a*t)*Heaviside(t)/(2*a)))
assert ILTS(b*s/(s**2-a**2)) == b*cosh(a*t)*Heaviside(t)
assert (ILT(b/(s*(s+a))) ==
b*(Heaviside(t)/a - exp(-a*t)*Heaviside(t)/a))
# Issue #24424
assert (ILTS((s + 8)/((s + 2)*(s**2 + 2*s + 10))) ==
((8*sin(3*t) - 9*cos(3*t))*exp(t) + 9)*exp(-2*t)*Heaviside(t)/15)
# Issue #8514; this is not important anymore, since this function
# is not solved by integration anymore
assert (ILT(1/(a*s**2+b*s+c)) ==
2*exp(-b*t/(2*a))*sin(t*sqrt(4*a*c - b**2)/(2*a)) *
Heaviside(t)/sqrt(4*a*c - b**2))
# Section 5.3: Irrational algebraic functions
assert ( # (1)
ILT(1/sqrt(s)/(b*s-a)) ==
exp(a*t/b)*Heaviside(t)*erf(sqrt(a)*sqrt(t)/sqrt(b))/(sqrt(a)*sqrt(b)))
assert ( # (2)
ILT(1/sqrt(k*s)/(c*s-a)/s) ==
(-2*c*sqrt(t)/(sqrt(pi)*a) +
c**(S(3)/2)*exp(a*t/c)*erf(sqrt(a)*sqrt(t)/sqrt(c))/a**(S(3)/2)) *
Heaviside(t)/(c*sqrt(k)))
assert ( # (4)
ILT(1/(sqrt(c*s)+a)) == (-a*exp(a**2*t/c)*erfc(a*sqrt(t)/sqrt(c))/c +
1/(sqrt(pi)*sqrt(c)*sqrt(t)))*Heaviside(t))
assert ( # (5)
ILT(a/s/(b*sqrt(s)+a)) ==
(-exp(a**2*t/b**2)*erfc(a*sqrt(t)/b) + 1)*Heaviside(t))
assert ( # (6)
ILT((a-b)*sqrt(s)/(sqrt(s)+sqrt(a))/(s-b)) ==
(sqrt(a)*sqrt(b)*exp(b*t)*erfc(sqrt(b)*sqrt(t)) +
a*exp(a*t)*erfc(sqrt(a)*sqrt(t)) - b*exp(b*t))*Heaviside(t))
assert ( # (7)
ILT(1/sqrt(s)/(sqrt(b*s)+a)) ==
exp(a**2*t/b)*Heaviside(t)*erfc(a*sqrt(t)/sqrt(b))/sqrt(b))
assert ( # (8)
ILT(a**2/(sqrt(s)+a)/s**(S(3)/2)) ==
(2*a*sqrt(t)/sqrt(pi) + exp(a**2*t)*erfc(a*sqrt(t)) - 1) *
Heaviside(t))
assert ( # (9)
ILT((a-b)*sqrt(b)/(s-b)/sqrt(s)/(sqrt(s)+sqrt(a))) ==
(sqrt(a)*exp(b*t)*erf(sqrt(b)*sqrt(t)) +
sqrt(b)*exp(a*t)*erfc(sqrt(a)*sqrt(t)) -
sqrt(b)*exp(b*t))*Heaviside(t))
assert ( # (10)
ILT(1/(sqrt(s)+sqrt(a))**2) ==
(-2*sqrt(a)*sqrt(t)/sqrt(pi) +
(-2*a*t + 1)*(erf(sqrt(a)*sqrt(t)) -
1)*exp(a*t) + 1)*Heaviside(t))
assert ( # (11)
ILT(1/(sqrt(s)+sqrt(a))**2/s) ==
((2*t - 1/a)*exp(a*t)*erfc(sqrt(a)*sqrt(t)) + 1/a -
2*sqrt(t)/(sqrt(pi)*sqrt(a)))*Heaviside(t))
assert ( # (12)
ILT(1/(sqrt(s)+a)**2/sqrt(s)) ==
(-2*a*t*exp(a**2*t)*erfc(a*sqrt(t)) +
2*sqrt(t)/sqrt(pi))*Heaviside(t))
assert ( # (13)
ILT(1/(sqrt(s)+a)**3) ==
(-a*t*(2*a**2*t + 3)*exp(a**2*t)*erfc(a*sqrt(t)) +
2*sqrt(t)*(a**2*t + 1)/sqrt(pi))*Heaviside(t))
x = (
- ILT(sqrt(s)/(sqrt(s)+a)**3) +
2*(sqrt(pi)*a**2*t*(-2*sqrt(pi)*erfc(a*sqrt(t)) +
2*exp(-a**2*t)/(a*sqrt(t))) *
(-a**4*t**2 - 5*a**2*t/2 - S.Half) * exp(a**2*t)/2 +
sqrt(pi)*a*sqrt(t)*(a**2*t + 1)/2) *
Heaviside(t)/(pi*a**2*t)).simplify()
assert ( # (14)
x == 0)
x = (
- ILT(1/sqrt(s)/(sqrt(s)+a)**3) +
Heaviside(t)*(sqrt(t)*((2*a**2*t + 1) *
(sqrt(pi)*a*sqrt(t)*exp(a**2*t) *
erfc(a*sqrt(t)) - 1) + 1) /
(sqrt(pi)*a))).simplify()
assert ( # (15)
x == 0)
assert ( # (16)
factor_terms(ILT(3/(sqrt(s)+a)**4)) ==
3*(-2*a**3*t**(S(5)/2)*(2*a**2*t + 5)/(3*sqrt(pi)) +
t*(4*a**4*t**2 + 12*a**2*t + 3)*exp(a**2*t) *
erfc(a*sqrt(t))/3)*Heaviside(t))
assert ( # (17)
ILT((sqrt(s)-a)/(s*(sqrt(s)+a))) ==
(2*exp(a**2*t)*erfc(a*sqrt(t))-1)*Heaviside(t))
assert ( # (18)
ILT((sqrt(s)-a)**2/(s*(sqrt(s)+a)**2)) == (
1 + 8*a**2*t*exp(a**2*t)*erfc(a*sqrt(t)) -
8/sqrt(pi)*a*sqrt(t))*Heaviside(t))
assert ( # (19)
ILT((sqrt(s)-a)**3/(s*(sqrt(s)+a)**3)) == Heaviside(t)*(
2*(8*a**4*t**2+8*a**2*t+1)*exp(a**2*t) *
erfc(a*sqrt(t))-8/sqrt(pi)*a*sqrt(t)*(2*a**2*t+1)-1))
assert ( # (22)
ILT(sqrt(s+a)/(s+b)) == Heaviside(t)*(
exp(-a*t)/sqrt(t)/sqrt(pi) +
sqrt(a-b)*exp(-b*t)*erf(sqrt(a-b)*sqrt(t))))
assert ( # (23)
ILT(1/sqrt(s+b)/(s+a)) == Heaviside(t)*(
1/sqrt(b-a)*exp(-a*t)*erf(sqrt(b-a)*sqrt(t))))
assert ( # (35)
ILT(1/sqrt(s**2+a**2)) == Heaviside(t)*(
besselj(0, a*t)))
assert ( # (44)
ILT(1/sqrt(s**2-a**2)) == Heaviside(t)*(
besseli(0, a*t)))
# Miscellaneous tests
# Can _inverse_laplace_time_shift deal with positive exponents?
assert (
- ILT((s**2*exp(2*s) + 4*exp(s) - 4)*exp(-2*s)/(s*(s**2 + 1))) +
cos(t)*Heaviside(t) + 4*cos(t - 2)*Heaviside(t - 2) -
4*cos(t - 1)*Heaviside(t - 1) - 4*Heaviside(t - 2) +
4*Heaviside(t - 1)).simplify() == 0
@slow
def test_inverse_laplace_transform_old():
from sympy.functions.special.delta_functions import DiracDelta
ILT = inverse_laplace_transform
a, b, c, d = symbols('a b c d', positive=True)
n, r = symbols('n, r', real=True)
t, z = symbols('t z')
f = Function('f')
F = Function('F')
def simp_hyp(expr):
return factor_terms(expand_mul(expr)).rewrite(sin)
L = ILT(F(s), s, t)
assert laplace_correspondence(L, {f: F}) == f(t)
assert ILT(exp(-a*s)/s, s, t) == Heaviside(-a + t)
assert ILT(exp(-a*s)/(b + s), s, t) == exp(-b*(-a + t))*Heaviside(-a + t)
assert (ILT((b + s)/(a**2 + (b + s)**2), s, t) ==
exp(-b*t)*cos(a*t)*Heaviside(t))
assert (ILT(exp(-a*s)/s**b, s, t) ==
(-a + t)**(b - 1)*Heaviside(-a + t)/gamma(b))
assert (ILT(exp(-a*s)/sqrt(s**2 + 1), s, t) ==
Heaviside(-a + t)*besselj(0, a - t))
assert ILT(1/(s*sqrt(s + 1)), s, t) == Heaviside(t)*erf(sqrt(t))
# TODO sinh/cosh shifted come out a mess. also delayed trig is a mess
# TODO should this simplify further?
assert (ILT(exp(-a*s)/s**b, s, t) ==
(t - a)**(b - 1)*Heaviside(t - a)/gamma(b))
assert (ILT(exp(-a*s)/sqrt(1 + s**2), s, t) ==
Heaviside(t - a)*besselj(0, a - t)) # note: besselj(0, x) is even
# XXX ILT turns these branch factor into trig functions ...
assert (
simplify(ILT(a**b*(s + sqrt(s**2 - a**2))**(-b)/sqrt(s**2 - a**2),
s, t).rewrite(exp)) ==
Heaviside(t)*besseli(b, a*t))
assert (
ILT(a**b*(s + sqrt(s**2 + a**2))**(-b)/sqrt(s**2 + a**2),
s, t, simplify=True).rewrite(exp) ==
Heaviside(t)*besselj(b, a*t))
assert ILT(1/(s*sqrt(s + 1)), s, t) == Heaviside(t)*erf(sqrt(t))
# TODO can we make erf(t) work?
assert (ILT((s * eye(2) - Matrix([[1, 0], [0, 2]])).inv(), s, t) ==
Matrix([[exp(t)*Heaviside(t), 0], [0, exp(2*t)*Heaviside(t)]]))
# Test time_diff rule
assert (ILT(s**42*f(s), s, t) ==
Derivative(InverseLaplaceTransform(f(s), s, t, None), (t, 42)))
assert ILT(cos(s), s, t) == InverseLaplaceTransform(cos(s), s, t, None)
# Rules for testing different DiracDelta cases
assert (ILT(2*exp(3*s) - 5*exp(-7*s), s, t) ==
2*InverseLaplaceTransform(exp(3*s), s, t, None) -
5*DiracDelta(t - 7))
a = cos(sin(7)/2)
assert ILT(a*exp(-3*s), s, t) == a*DiracDelta(t - 3)
assert ILT(exp(2*s), s, t) == InverseLaplaceTransform(exp(2*s), s, t, None)
r = Symbol('r', real=True)
assert ILT(exp(r*s), s, t) == InverseLaplaceTransform(exp(r*s), s, t, None)
# Rules for testing whether Heaviside(t) is treated properly in diff rule
assert ILT(s**2/(a**2 + s**2), s, t) == (
-a*sin(a*t)*Heaviside(t) + DiracDelta(t))
assert ILT(s**2*(f(s) + 1/(a**2 + s**2)), s, t) == (
-a*sin(a*t)*Heaviside(t) + DiracDelta(t) +
Derivative(InverseLaplaceTransform(f(s), s, t, None), (t, 2)))
# Rules from the previous test_inverse_laplace_transform_delta_cond():
assert (ILT(exp(r*s), s, t, noconds=False) ==
(InverseLaplaceTransform(exp(r*s), s, t, None), True))
# inversion does not exist: verify it doesn't evaluate to DiracDelta
for z in (Symbol('z', extended_real=False),
Symbol('z', imaginary=True, zero=False)):
f = ILT(exp(z*s), s, t, noconds=False)
f = f[0] if isinstance(f, tuple) else f
assert f.func != DiracDelta
@slow
def test_expint():
x = Symbol('x')
a = Symbol('a')
u = Symbol('u', polar=True)
# TODO LT of Si, Shi, Chi is a mess ...
assert laplace_transform(Ci(x), x, s) == (-log(1 + s**2)/2/s, 0, True)
assert (laplace_transform(expint(a, x), x, s, simplify=True) ==
(lerchphi(s*exp_polar(I*pi), 1, a), 0, re(a) > S.Zero))
assert (laplace_transform(expint(1, x), x, s, simplify=True) ==
(log(s + 1)/s, 0, True))
assert (laplace_transform(expint(2, x), x, s, simplify=True) ==
((s - log(s + 1))/s**2, 0, True))
assert (inverse_laplace_transform(-log(1 + s**2)/2/s, s, u).expand() ==
Heaviside(u)*Ci(u))
assert (
inverse_laplace_transform(log(s + 1)/s, s, x,
simplify=True).rewrite(expint) ==
Heaviside(x)*E1(x))
assert (
inverse_laplace_transform(
(s - log(s + 1))/s**2, s, x,
simplify=True).rewrite(expint).expand() ==
(expint(2, x)*Heaviside(x)).rewrite(Ei).rewrite(expint).expand())
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