from sympy.integrals.laplace import ( laplace_transform, inverse_laplace_transform, LaplaceTransform, InverseLaplaceTransform) 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.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.trigonometric import atan, cos, sin from sympy.functions.special.gamma_functions import lowergamma, gamma from sympy.functions.special.delta_functions import DiracDelta, Heaviside 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 a, b, c, = symbols('a, b, c', positive=True) t, w, x = symbols('t, w, x') f = Function('f') g = Function('g') # 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)*lowergamma(b + 1, a*s), 0, True)) assert LT(t**5/(t+a), t, s) == (120*a**5*lowergamma(-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(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, 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())