551 lines
28 KiB
Python
551 lines
28 KiB
Python
from sympy.integrals.laplace import (
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laplace_transform, inverse_laplace_transform,
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LaplaceTransform, InverseLaplaceTransform)
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from sympy.core.function import Function, expand_mul
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from sympy.core import EulerGamma, Subs, Derivative, diff
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from sympy.core.exprtools import factor_terms
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from sympy.core.numbers import I, oo, pi
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from sympy.core.singleton import S
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from sympy.core.symbol import Symbol, symbols
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from sympy.simplify.simplify import simplify
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from sympy.functions.elementary.complexes import Abs, re
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from sympy.functions.elementary.exponential import exp, log, exp_polar
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from sympy.functions.elementary.hyperbolic import cosh, sinh, coth, asinh
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import atan, cos, sin
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from sympy.functions.special.gamma_functions import lowergamma, gamma
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from sympy.functions.special.delta_functions import DiracDelta, Heaviside
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from sympy.functions.special.zeta_functions import lerchphi
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from sympy.functions.special.error_functions import (
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fresnelc, fresnels, erf, erfc, Ei, Ci, expint, E1)
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from sympy.functions.special.bessel import besseli, besselj, besselk, bessely
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from sympy.testing.pytest import slow, warns_deprecated_sympy
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from sympy.matrices import Matrix, eye
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from sympy.abc import s
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@slow
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def test_laplace_transform():
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LT = laplace_transform
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a, b, c, = symbols('a, b, c', positive=True)
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t, w, x = symbols('t, w, x')
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f = Function('f')
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g = Function('g')
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# Test whether `noconds=True` in `doit`:
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assert (2*LaplaceTransform(exp(t), t, s) - 1).doit() == -1 + 2/(s - 1)
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assert (LT(a*t+t**2+t**(S(5)/2), t, s) ==
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(a/s**2 + 2/s**3 + 15*sqrt(pi)/(8*s**(S(7)/2)), 0, True))
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assert LT(b/(t+a), t, s) == (-b*exp(-a*s)*Ei(-a*s), 0, True)
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assert (LT(1/sqrt(t+a), t, s) ==
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(sqrt(pi)*sqrt(1/s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True))
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assert (LT(sqrt(t)/(t+a), t, s) ==
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(-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
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0, True))
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assert (LT((t+a)**(-S(3)/2), t, s) ==
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(-2*sqrt(pi)*sqrt(s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + 2/sqrt(a),
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0, True))
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assert (LT(t**(S(1)/2)*(t+a)**(-1), t, s) ==
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(-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
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0, True))
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assert (LT(1/(a*sqrt(t) + t**(3/2)), t, s) ==
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(pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True))
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assert (LT((t+a)**b, t, s) ==
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(s**(-b - 1)*exp(-a*s)*lowergamma(b + 1, a*s), 0, True))
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assert LT(t**5/(t+a), t, s) == (120*a**5*lowergamma(-5, a*s), 0, True)
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assert LT(exp(t), t, s) == (1/(s - 1), 1, True)
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assert LT(exp(2*t), t, s) == (1/(s - 2), 2, True)
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assert LT(exp(a*t), t, s) == (1/(s - a), a, True)
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assert LT(exp(a*(t-b)), t, s) == (exp(-a*b)/(-a + s), a, True)
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assert LT(t*exp(-a*(t)), t, s) == ((a + s)**(-2), -a, True)
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assert LT(t*exp(-a*(t-b)), t, s) == (exp(a*b)/(a + s)**2, -a, True)
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assert LT(b*t*exp(-a*t), t, s) == (b/(a + s)**2, -a, True)
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assert (LT(t**(S(7)/4)*exp(-8*t)/gamma(S(11)/4), t, s) ==
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((s + 8)**(-S(11)/4), -8, True))
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assert (LT(t**(S(3)/2)*exp(-8*t), t, s) ==
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(3*sqrt(pi)/(4*(s + 8)**(S(5)/2)), -8, True))
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assert LT(t**a*exp(-a*t), t, s) == ((a+s)**(-a-1)*gamma(a+1), -a, True)
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assert (LT(b*exp(-a*t**2), t, s) ==
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(sqrt(pi)*b*exp(s**2/(4*a))*erfc(s/(2*sqrt(a)))/(2*sqrt(a)),
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0, True))
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assert (LT(exp(-2*t**2), t, s) ==
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(sqrt(2)*sqrt(pi)*exp(s**2/8)*erfc(sqrt(2)*s/4)/4, 0, True))
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assert (LT(b*exp(2*t**2), t, s) ==
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(b*LaplaceTransform(exp(2*t**2), t, s), -oo, True))
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assert (LT(t*exp(-a*t**2), t, s) ==
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(1/(2*a) - s*erfc(s/(2*sqrt(a)))/(4*sqrt(pi)*a**(S(3)/2)),
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0, True))
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assert (LT(exp(-a/t), t, s) ==
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(2*sqrt(a)*sqrt(1/s)*besselk(1, 2*sqrt(a)*sqrt(s)), 0, True))
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assert LT(sqrt(t)*exp(-a/t), t, s, simplify=True) == (
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sqrt(pi)*(sqrt(a)*sqrt(s) + 1/S(2))*sqrt(s**(-3)) *
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exp(-2*sqrt(a)*sqrt(s)), 0, True)
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assert (LT(exp(-a/t)/sqrt(t), t, s) ==
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(sqrt(pi)*sqrt(1/s)*exp(-2*sqrt(a)*sqrt(s)), 0, True))
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assert (LT(exp(-a/t)/(t*sqrt(t)), t, s) ==
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(sqrt(pi)*sqrt(1/a)*exp(-2*sqrt(a)*sqrt(s)), 0, True))
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assert (
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LT(exp(-2*sqrt(a*t)), t, s) ==
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(1/s - sqrt(pi)*sqrt(a) * exp(a/s)*erfc(sqrt(a)*sqrt(1/s)) /
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s**(S(3)/2), 0, True))
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assert LT(exp(-2*sqrt(a*t))/sqrt(t), t, s) == (
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exp(a/s)*erfc(sqrt(a) * sqrt(1/s))*(sqrt(pi)*sqrt(1/s)), 0, True)
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assert (LT(t**4*exp(-2/t), t, s) ==
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(8*sqrt(2)*(1/s)**(S(5)/2)*besselk(5, 2*sqrt(2)*sqrt(s)),
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0, True))
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assert LT(sinh(a*t), t, s) == (a/(-a**2 + s**2), a, True)
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assert (LT(b*sinh(a*t)**2, t, s) ==
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(2*a**2*b/(-4*a**2*s + s**3), 2*a, True))
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assert (LT(b*sinh(a*t)**2, t, s, simplify=True) ==
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(2*a**2*b/(s*(-4*a**2 + s**2)), 2*a, True))
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# The following line confirms that issue #21202 is solved
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assert LT(cosh(2*t), t, s) == (s/(-4 + s**2), 2, True)
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assert LT(cosh(a*t), t, s) == (s/(-a**2 + s**2), a, True)
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assert (LT(cosh(a*t)**2, t, s, simplify=True) ==
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((2*a**2 - s**2)/(s*(4*a**2 - s**2)), 2*a, True))
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assert (LT(sinh(x+3), x, s, simplify=True) ==
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((s*sinh(3) + cosh(3))/(s**2 - 1), 1, True))
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L, _, _ = LT(42*sin(w*t+x)**2, t, s)
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assert (
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L -
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21*(s**2 + s*(-s*cos(2*x) + 2*w*sin(2*x)) +
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4*w**2)/(s*(s**2 + 4*w**2))).simplify() == 0
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# The following line replaces the old test test_issue_7173()
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assert LT(sinh(a*t)*cosh(a*t), t, s, simplify=True) == (a/(-4*a**2 + s**2),
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2*a, True)
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assert LT(sinh(a*t)/t, t, s) == (log((a + s)/(-a + s))/2, a, True)
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assert (LT(t**(-S(3)/2)*sinh(a*t), t, s) ==
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(-sqrt(pi)*(sqrt(-a + s) - sqrt(a + s)), a, True))
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assert (LT(sinh(2*sqrt(a*t)), t, s) ==
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(sqrt(pi)*sqrt(a)*exp(a/s)/s**(S(3)/2), 0, True))
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assert (LT(sqrt(t)*sinh(2*sqrt(a*t)), t, s, simplify=True) ==
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((-sqrt(a)*s**(S(5)/2) + sqrt(pi)*s**2*(2*a + s)*exp(a/s) *
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erf(sqrt(a)*sqrt(1/s))/2)/s**(S(9)/2), 0, True))
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assert (LT(sinh(2*sqrt(a*t))/sqrt(t), t, s) ==
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(sqrt(pi)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/sqrt(s), 0, True))
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assert (LT(sinh(sqrt(a*t))**2/sqrt(t), t, s) ==
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(sqrt(pi)*(exp(a/s) - 1)/(2*sqrt(s)), 0, True))
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assert (LT(t**(S(3)/7)*cosh(a*t), t, s) ==
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(((a + s)**(-S(10)/7) + (-a+s)**(-S(10)/7))*gamma(S(10)/7)/2,
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a, True))
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assert (LT(cosh(2*sqrt(a*t)), t, s) ==
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(sqrt(pi)*sqrt(a)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/s**(S(3)/2) +
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1/s, 0, True))
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assert (LT(sqrt(t)*cosh(2*sqrt(a*t)), t, s) ==
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(sqrt(pi)*(a + s/2)*exp(a/s)/s**(S(5)/2), 0, True))
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assert (LT(cosh(2*sqrt(a*t))/sqrt(t), t, s) ==
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(sqrt(pi)*exp(a/s)/sqrt(s), 0, True))
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assert (LT(cosh(sqrt(a*t))**2/sqrt(t), t, s) ==
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(sqrt(pi)*(exp(a/s) + 1)/(2*sqrt(s)), 0, True))
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assert LT(log(t), t, s, simplify=True) == (
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(-log(s) - EulerGamma)/s, 0, True)
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assert (LT(-log(t/a), t, s, simplify=True) ==
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((log(a) + log(s) + EulerGamma)/s, 0, True))
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assert LT(log(1+a*t), t, s) == (-exp(s/a)*Ei(-s/a)/s, 0, True)
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assert (LT(log(t+a), t, s, simplify=True) ==
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((s*log(a) - exp(s/a)*Ei(-s/a))/s**2, 0, True))
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assert (LT(log(t)/sqrt(t), t, s, simplify=True) ==
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(sqrt(pi)*(-log(s) - log(4) - EulerGamma)/sqrt(s), 0, True))
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assert (LT(t**(S(5)/2)*log(t), t, s, simplify=True) ==
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(sqrt(pi)*(-15*log(s) - log(1073741824) - 15*EulerGamma + 46) /
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(8*s**(S(7)/2)), 0, True))
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assert (LT(t**3*log(t), t, s, noconds=True, simplify=True) -
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6*(-log(s) - S.EulerGamma + S(11)/6)/s**4).simplify() == S.Zero
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assert (LT(log(t)**2, t, s, simplify=True) ==
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(((log(s) + EulerGamma)**2 + pi**2/6)/s, 0, True))
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assert (LT(exp(-a*t)*log(t), t, s, simplify=True) ==
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((-log(a + s) - EulerGamma)/(a + s), -a, True))
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assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True)
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assert (LT(Abs(sin(a*t)), t, s) ==
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(a*coth(pi*s/(2*a))/(a**2 + s**2), 0, True))
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assert LT(sin(a*t)/t, t, s) == (atan(a/s), 0, True)
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assert LT(sin(a*t)**2/t, t, s) == (log(4*a**2/s**2 + 1)/4, 0, True)
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assert (LT(sin(a*t)**2/t**2, t, s) ==
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(a*atan(2*a/s) - s*log(4*a**2/s**2 + 1)/4, 0, True))
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assert (LT(sin(2*sqrt(a*t)), t, s) ==
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(sqrt(pi)*sqrt(a)*exp(-a/s)/s**(S(3)/2), 0, True))
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assert LT(sin(2*sqrt(a*t))/t, t, s) == (pi*erf(sqrt(a)*sqrt(1/s)), 0, True)
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assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True)
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assert (LT(cos(a*t)**2, t, s) ==
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((2*a**2 + s**2)/(s*(4*a**2 + s**2)), 0, True))
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assert (LT(sqrt(t)*cos(2*sqrt(a*t)), t, s, simplify=True) ==
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(sqrt(pi)*(-a + s/2)*exp(-a/s)/s**(S(5)/2), 0, True))
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assert (LT(cos(2*sqrt(a*t))/sqrt(t), t, s) ==
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(sqrt(pi)*sqrt(1/s)*exp(-a/s), 0, True))
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assert (LT(sin(a*t)*sin(b*t), t, s) ==
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(2*a*b*s/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)), 0, True))
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assert (LT(cos(a*t)*sin(b*t), t, s) ==
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(b*(-a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
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0, True))
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assert (LT(cos(a*t)*cos(b*t), t, s) ==
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(s*(a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
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0, True))
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assert (LT(-a*t*cos(a*t) + sin(a*t), t, s, simplify=True) ==
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(2*a**3/(a**4 + 2*a**2*s**2 + s**4), 0, True))
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assert LT(c*exp(-b*t)*sin(a*t), t, s) == (a *
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c/(a**2 + (b + s)**2), -b, True)
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assert LT(c*exp(-b*t)*cos(a*t), t, s) == (c*(b + s)/(a**2 + (b + s)**2),
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-b, True)
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L, plane, cond = LT(cos(x + 3), x, s, simplify=True)
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assert plane == 0
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assert L - (s*cos(3) - sin(3))/(s**2 + 1) == 0
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# Error functions (laplace7.pdf)
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assert LT(erf(a*t), t, s) == (exp(s**2/(4*a**2))*erfc(s/(2*a))/s, 0, True)
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assert LT(erf(sqrt(a*t)), t, s) == (sqrt(a)/(s*sqrt(a + s)), 0, True)
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assert (LT(exp(a*t)*erf(sqrt(a*t)), t, s, simplify=True) ==
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(-sqrt(a)/(sqrt(s)*(a - s)), a, True))
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assert (LT(erf(sqrt(a/t)/2), t, s, simplify=True) ==
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(1/s - exp(-sqrt(a)*sqrt(s))/s, 0, True))
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assert (LT(erfc(sqrt(a*t)), t, s, simplify=True) ==
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(-sqrt(a)/(s*sqrt(a + s)) + 1/s, -a, True))
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assert (LT(exp(a*t)*erfc(sqrt(a*t)), t, s) ==
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(1/(sqrt(a)*sqrt(s) + s), 0, True))
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assert LT(erfc(sqrt(a/t)/2), t, s) == (exp(-sqrt(a)*sqrt(s))/s, 0, True)
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# Bessel functions (laplace8.pdf)
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assert LT(besselj(0, a*t), t, s) == (1/sqrt(a**2 + s**2), 0, True)
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assert (LT(besselj(1, a*t), t, s, simplify=True) ==
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(a/(a**2 + s**2 + s*sqrt(a**2 + s**2)), 0, True))
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assert (LT(besselj(2, a*t), t, s, simplify=True) ==
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(a**2/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))**2), 0, True))
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assert (LT(t*besselj(0, a*t), t, s) ==
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(s/(a**2 + s**2)**(S(3)/2), 0, True))
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assert (LT(t*besselj(1, a*t), t, s) ==
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(a/(a**2 + s**2)**(S(3)/2), 0, True))
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assert (LT(t**2*besselj(2, a*t), t, s) ==
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(3*a**2/(a**2 + s**2)**(S(5)/2), 0, True))
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assert LT(besselj(0, 2*sqrt(a*t)), t, s) == (exp(-a/s)/s, 0, True)
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assert (LT(t**(S(3)/2)*besselj(3, 2*sqrt(a*t)), t, s) ==
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(a**(S(3)/2)*exp(-a/s)/s**4, 0, True))
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assert (LT(besselj(0, a*sqrt(t**2+b*t)), t, s, simplify=True) ==
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(exp(b*(s - sqrt(a**2 + s**2)))/sqrt(a**2 + s**2), 0, True))
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assert LT(besseli(0, a*t), t, s) == (1/sqrt(-a**2 + s**2), a, True)
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assert (LT(besseli(1, a*t), t, s, simplify=True) ==
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(a/(-a**2 + s**2 + s*sqrt(-a**2 + s**2)), a, True))
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assert (LT(besseli(2, a*t), t, s, simplify=True) ==
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(a**2/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))**2), a, True))
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assert LT(t*besseli(0, a*t), t, s) == (s/(-a**2 + s**2)**(S(3)/2), a, True)
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assert LT(t*besseli(1, a*t), t, s) == (a/(-a**2 + s**2)**(S(3)/2), a, True)
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assert (LT(t**2*besseli(2, a*t), t, s) ==
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(3*a**2/(-a**2 + s**2)**(S(5)/2), a, True))
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assert (LT(t**(S(3)/2)*besseli(3, 2*sqrt(a*t)), t, s) ==
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(a**(S(3)/2)*exp(a/s)/s**4, 0, True))
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assert (LT(bessely(0, a*t), t, s) ==
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(-2*asinh(s/a)/(pi*sqrt(a**2 + s**2)), 0, True))
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assert (LT(besselk(0, a*t), t, s) ==
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(log((s + sqrt(-a**2 + s**2))/a)/sqrt(-a**2 + s**2), -a, True))
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assert (LT(sin(a*t)**4, t, s, simplify=True) ==
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(24*a**4/(s*(64*a**4 + 20*a**2*s**2 + s**4)), 0, True))
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# Test general rules and unevaluated forms
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# These all also test whether issue #7219 is solved.
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assert LT(Heaviside(t-1)*cos(t-1), t, s) == (s*exp(-s)/(s**2 + 1), 0, True)
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assert LT(a*f(t), t, w) == (a*LaplaceTransform(f(t), t, w), -oo, True)
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assert (LT(a*Heaviside(t+1)*f(t+1), t, s) ==
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(a*LaplaceTransform(f(t + 1), t, s), -oo, True))
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assert (LT(a*Heaviside(t-1)*f(t-1), t, s) ==
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(a*LaplaceTransform(f(t), t, s)*exp(-s), -oo, True))
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assert (LT(b*f(t/a), t, s) ==
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(a*b*LaplaceTransform(f(t), t, a*s), -oo, True))
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assert LT(exp(-f(x)*t), t, s) == (1/(s + f(x)), -re(f(x)), True)
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assert (LT(exp(-a*t)*f(t), t, s) ==
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(LaplaceTransform(f(t), t, a + s), -oo, True))
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assert (LT(exp(-a*t)*erfc(sqrt(b/t)/2), t, s) ==
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(exp(-sqrt(b)*sqrt(a + s))/(a + s), -a, True))
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assert (LT(sinh(a*t)*f(t), t, s) ==
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(LaplaceTransform(f(t), t, -a + s)/2 -
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LaplaceTransform(f(t), t, a + s)/2, -oo, True))
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assert (LT(sinh(a*t)*t, t, s, simplify=True) ==
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(2*a*s/(a**4 - 2*a**2*s**2 + s**4), a, True))
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assert (LT(cosh(a*t)*f(t), t, s) ==
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(LaplaceTransform(f(t), t, -a + s)/2 +
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LaplaceTransform(f(t), t, a + s)/2, -oo, True))
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assert (LT(cosh(a*t)*t, t, s, simplify=True) ==
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(1/(2*(a + s)**2) + 1/(2*(a - s)**2), a, True))
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assert (LT(sin(a*t)*f(t), t, s, simplify=True) ==
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(I*(-LaplaceTransform(f(t), t, -I*a + s) +
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LaplaceTransform(f(t), t, I*a + s))/2, -oo, True))
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assert (LT(sin(f(t)), t, s) ==
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(LaplaceTransform(sin(f(t)), t, s), -oo, True))
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assert (LT(sin(a*t)*t, t, s, simplify=True) ==
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(2*a*s/(a**4 + 2*a**2*s**2 + s**4), 0, True))
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|
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, 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))
|
|
# 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))
|
|
# 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)
|
|
# 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))
|
|
# 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():
|
|
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')
|
|
|
|
def simp_hyp(expr):
|
|
return factor_terms(expand_mul(expr)).rewrite(sin)
|
|
|
|
assert ILT(1, s, t) == DiracDelta(t)
|
|
assert ILT(1/s, s, t) == Heaviside(t)
|
|
assert ILT(a/(a + s), s, t) == a*exp(-a*t)*Heaviside(t)
|
|
assert ILT(s/(a + s), s, t) == -a*exp(-a*t)*Heaviside(t) + DiracDelta(t)
|
|
assert (ILT(s/(a + s)**3, s, t, simplify=True) ==
|
|
t*(-a*t + 4)*exp(-a*t)*Heaviside(t)/2)
|
|
assert (ILT(1/(s*(a + s)**3), s, t, simplify=True) ==
|
|
(-a**2*t**2 - 4*a*t + 4*exp(a*t) - 4) *
|
|
exp(-a*t)*Heaviside(t)/(2*a**3))
|
|
assert ILT(1/(s*(a + s)**n), s, t) == (
|
|
Heaviside(t)*lowergamma(n, a*t)/(a**n*gamma(n)))
|
|
assert ILT((s-a)**(-b), s, t) == t**(b - 1)*exp(a*t)*Heaviside(t)/gamma(b)
|
|
assert ILT((a + s)**(-2), s, t) == t*exp(-a*t)*Heaviside(t)
|
|
assert ILT((a + s)**(-5), s, t) == t**4*exp(-a*t)*Heaviside(t)/24
|
|
assert ILT(a/(a**2 + s**2), s, t) == sin(a*t)*Heaviside(t)
|
|
assert ILT(s/(s**2 + a**2), s, t) == cos(a*t)*Heaviside(t)
|
|
assert ILT(b/(b**2 + (a + s)**2), s, t) == exp(-a*t)*sin(b*t)*Heaviside(t)
|
|
assert (ILT(b*s/(b**2 + (a + s)**2), s, t) ==
|
|
b*(-a*exp(-a*t)*sin(b*t)/b + exp(-a*t)*cos(b*t))*Heaviside(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))
|
|
assert (ILT(1/(s**2*(s**2 + 1)), s, t) ==
|
|
t*Heaviside(t) - sin(t)*Heaviside(t))
|
|
assert ILT(s**2/(s**2 + 1), s, t) == -sin(t)*Heaviside(t) + DiracDelta(t)
|
|
assert ILT(1 - 1/(s**2 + 1), s, t) == -sin(t)*Heaviside(t) + DiracDelta(t)
|
|
assert ILT(1/s**2, s, t) == t*Heaviside(t)
|
|
assert ILT(1/s**5, s, t) == t**4*Heaviside(t)/24
|
|
assert ILT(1/s**n, s, t) == t**(n - 1)*Heaviside(t)/gamma(n)
|
|
# Issue #24424
|
|
assert (ILT((s + 8)/((s + 2)*(s**2 + 2*s + 10)), s, t, simplify=True) ==
|
|
((8*sin(3*t) - 9*cos(3*t))*exp(t) + 9)*exp(-2*t)*Heaviside(t)/15)
|
|
assert simp_hyp(ILT(a/(s**2 - a**2), s, t)) == sinh(a*t)*Heaviside(t)
|
|
assert simp_hyp(ILT(s/(s**2 - a**2), s, t)) == cosh(a*t)*Heaviside(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(1/(s**2*(s**2 + 1)), s, t, simplify=True) ==
|
|
(t - sin(t))*Heaviside(t))
|
|
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)]]))
|
|
# New tests for rules
|
|
assert (ILT(b/(s**2-a**2), s, t, simplify=True) ==
|
|
b*sinh(a*t)*Heaviside(t)/a)
|
|
assert (ILT(b/(s**2-a**2), s, t) ==
|
|
b*(exp(a*t)*Heaviside(t)/(2*a) - exp(-a*t)*Heaviside(t)/(2*a)))
|
|
assert (ILT(b*s/(s**2-a**2), s, t, simplify=True) ==
|
|
b*cosh(a*t)*Heaviside(t))
|
|
assert (ILT(b/(s*(s+a)), s, t) ==
|
|
b*(Heaviside(t)/a - exp(-a*t)*Heaviside(t)/a))
|
|
assert (ILT(b*s/(s+a)**2, s, t) ==
|
|
b*(-a*t*exp(-a*t)*Heaviside(t) + exp(-a*t)*Heaviside(t)))
|
|
assert (ILT(c/((s+a)*(s+b)), s, t, simplify=True) ==
|
|
c*(exp(a*t) - exp(b*t))*exp(-t*(a + b))*Heaviside(t)/(a - b))
|
|
assert (ILT(c*s/((s+a)*(s+b)), s, t, simplify=True) ==
|
|
c*(a*exp(b*t) - b*exp(a*t))*exp(-t*(a + b))*Heaviside(t)/(a - b))
|
|
assert (ILT(c*s/(d**2*(s+a)**2+b**2), s, t, simplify=True) ==
|
|
c*(-a*d*sin(b*t/d) + b*cos(b*t/d))*exp(-a*t)*Heaviside(t)/(b*d**2))
|
|
# Test time_diff rule
|
|
assert (ILT(s**42*f(s), s, t) ==
|
|
Derivative(InverseLaplaceTransform(f(s), s, t, None), (t, 42)))
|
|
assert (ILT((b*s**2 + d)/(a**2 + s**2)**2, s, t, simplify=True) ==
|
|
(a**3*b*t*cos(a*t) + 5*a**2*b*sin(a*t) - a*d*t*cos(a*t) +
|
|
d*sin(a*t))*Heaviside(t)/(2*a**3))
|
|
assert ILT(cos(s), s, t) == InverseLaplaceTransform(cos(s), s, t, None)
|
|
# Rules for testing different DiracDelta cases
|
|
assert ILT(2, s, t) == 2*DiracDelta(t)
|
|
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
|
|
# old test for Issue 8514, is not important anymore since this function
|
|
# is not solved by integration anymore
|
|
assert (ILT(1/(a*s**2+b*s+c), s, t) ==
|
|
2*exp(-b*t/(2*a))*sin(t*sqrt(4*a*c - b**2)/(2*a)) *
|
|
Heaviside(t)/sqrt(4*a*c - b**2))
|
|
|
|
|
|
@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())
|