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1077 lines
49 KiB
1077 lines
49 KiB
from sympy.integrals.transforms import (mellin_transform,
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inverse_mellin_transform, laplace_transform,
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inverse_laplace_transform, fourier_transform, inverse_fourier_transform,
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sine_transform, inverse_sine_transform,
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cosine_transform, inverse_cosine_transform,
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hankel_transform, inverse_hankel_transform,
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LaplaceTransform, FourierTransform, SineTransform, CosineTransform,
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InverseLaplaceTransform, InverseFourierTransform,
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InverseSineTransform, InverseCosineTransform, IntegralTransformError)
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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.numbers import (I, Rational, oo, pi)
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from sympy.core.relational import Eq
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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.functions.combinatorial.factorials import factorial
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from sympy.functions.elementary.complexes import (Abs, re, unpolarify)
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from sympy.functions.elementary.exponential import (exp, exp_polar, log)
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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, atan2, cos, sin, tan)
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from sympy.functions.special.bessel import (besseli, besselj, besselk, bessely)
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from sympy.functions.special.delta_functions import Heaviside
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from sympy.functions.special.error_functions import (erf, erfc, expint, Ei)
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from sympy.functions.special.gamma_functions import gamma
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from sympy.functions.special.hyper import meijerg
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from sympy.simplify.gammasimp import gammasimp
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from sympy.simplify.hyperexpand import hyperexpand
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from sympy.simplify.trigsimp import trigsimp
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from sympy.testing.pytest import XFAIL, slow, skip, raises, warns_deprecated_sympy
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from sympy.matrices import Matrix, eye
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from sympy.abc import x, s, a, b, c, d
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nu, beta, rho = symbols('nu beta rho')
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def test_undefined_function():
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from sympy.integrals.transforms import MellinTransform
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f = Function('f')
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assert mellin_transform(f(x), x, s) == MellinTransform(f(x), x, s)
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assert mellin_transform(f(x) + exp(-x), x, s) == \
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(MellinTransform(f(x), x, s) + gamma(s + 1)/s, (0, oo), True)
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def test_free_symbols():
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f = Function('f')
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assert mellin_transform(f(x), x, s).free_symbols == {s}
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assert mellin_transform(f(x)*a, x, s).free_symbols == {s, a}
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def test_as_integral():
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from sympy.integrals.integrals import Integral
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f = Function('f')
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assert mellin_transform(f(x), x, s).rewrite('Integral') == \
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Integral(x**(s - 1)*f(x), (x, 0, oo))
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assert fourier_transform(f(x), x, s).rewrite('Integral') == \
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Integral(f(x)*exp(-2*I*pi*s*x), (x, -oo, oo))
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assert laplace_transform(f(x), x, s).rewrite('Integral') == \
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Integral(f(x)*exp(-s*x), (x, 0, oo))
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assert str(2*pi*I*inverse_mellin_transform(f(s), s, x, (a, b)).rewrite('Integral')) \
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== "Integral(f(s)/x**s, (s, _c - oo*I, _c + oo*I))"
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assert str(2*pi*I*inverse_laplace_transform(f(s), s, x).rewrite('Integral')) == \
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"Integral(f(s)*exp(s*x), (s, _c - oo*I, _c + oo*I))"
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assert inverse_fourier_transform(f(s), s, x).rewrite('Integral') == \
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Integral(f(s)*exp(2*I*pi*s*x), (s, -oo, oo))
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# NOTE this is stuck in risch because meijerint cannot handle it
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@slow
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@XFAIL
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def test_mellin_transform_fail():
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skip("Risch takes forever.")
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MT = mellin_transform
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bpos = symbols('b', positive=True)
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# bneg = symbols('b', negative=True)
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expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
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# TODO does not work with bneg, argument wrong. Needs changes to matching.
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assert MT(expr.subs(b, -bpos), x, s) == \
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((-1)**(a + 1)*2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(a + s)
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*gamma(1 - a - 2*s)/gamma(1 - s),
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(-re(a), -re(a)/2 + S.Half), True)
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expr = (sqrt(x + b**2) + b)**a
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assert MT(expr.subs(b, -bpos), x, s) == \
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(
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2**(a + 2*s)*a*bpos**(a + 2*s)*gamma(-a - 2*
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s)*gamma(a + s)/gamma(-s + 1),
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(-re(a), -re(a)/2), True)
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# Test exponent 1:
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assert MT(expr.subs({b: -bpos, a: 1}), x, s) == \
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(-bpos**(2*s + 1)*gamma(s)*gamma(-s - S.Half)/(2*sqrt(pi)),
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(-1, Rational(-1, 2)), True)
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def test_mellin_transform():
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from sympy.functions.elementary.miscellaneous import (Max, Min)
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MT = mellin_transform
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bpos = symbols('b', positive=True)
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# 8.4.2
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assert MT(x**nu*Heaviside(x - 1), x, s) == \
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(-1/(nu + s), (-oo, -re(nu)), True)
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assert MT(x**nu*Heaviside(1 - x), x, s) == \
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(1/(nu + s), (-re(nu), oo), True)
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assert MT((1 - x)**(beta - 1)*Heaviside(1 - x), x, s) == \
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(gamma(beta)*gamma(s)/gamma(beta + s), (0, oo), re(beta) > 0)
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assert MT((x - 1)**(beta - 1)*Heaviside(x - 1), x, s) == \
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(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
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(-oo, 1 - re(beta)), re(beta) > 0)
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assert MT((1 + x)**(-rho), x, s) == \
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(gamma(s)*gamma(rho - s)/gamma(rho), (0, re(rho)), True)
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assert MT(abs(1 - x)**(-rho), x, s) == (
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2*sin(pi*rho/2)*gamma(1 - rho)*
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cos(pi*(s - rho/2))*gamma(s)*gamma(rho-s)/pi,
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(0, re(rho)), re(rho) < 1)
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mt = MT((1 - x)**(beta - 1)*Heaviside(1 - x)
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+ a*(x - 1)**(beta - 1)*Heaviside(x - 1), x, s)
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assert mt[1], mt[2] == ((0, -re(beta) + 1), re(beta) > 0)
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assert MT((x**a - b**a)/(x - b), x, s)[0] == \
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pi*b**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s)))
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assert MT((x**a - bpos**a)/(x - bpos), x, s) == \
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(pi*bpos**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s))),
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(Max(0, -re(a)), Min(1, 1 - re(a))), True)
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expr = (sqrt(x + b**2) + b)**a
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assert MT(expr.subs(b, bpos), x, s) == \
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(-a*(2*bpos)**(a + 2*s)*gamma(s)*gamma(-a - 2*s)/gamma(-a - s + 1),
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(0, -re(a)/2), True)
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expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
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assert MT(expr.subs(b, bpos), x, s) == \
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(2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(s)
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*gamma(1 - a - 2*s)/gamma(1 - a - s),
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(0, -re(a)/2 + S.Half), True)
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# 8.4.2
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assert MT(exp(-x), x, s) == (gamma(s), (0, oo), True)
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assert MT(exp(-1/x), x, s) == (gamma(-s), (-oo, 0), True)
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# 8.4.5
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assert MT(log(x)**4*Heaviside(1 - x), x, s) == (24/s**5, (0, oo), True)
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assert MT(log(x)**3*Heaviside(x - 1), x, s) == (6/s**4, (-oo, 0), True)
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assert MT(log(x + 1), x, s) == (pi/(s*sin(pi*s)), (-1, 0), True)
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assert MT(log(1/x + 1), x, s) == (pi/(s*sin(pi*s)), (0, 1), True)
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assert MT(log(abs(1 - x)), x, s) == (pi/(s*tan(pi*s)), (-1, 0), True)
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assert MT(log(abs(1 - 1/x)), x, s) == (pi/(s*tan(pi*s)), (0, 1), True)
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# 8.4.14
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assert MT(erf(sqrt(x)), x, s) == \
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(-gamma(s + S.Half)/(sqrt(pi)*s), (Rational(-1, 2), 0), True)
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def test_mellin_transform2():
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MT = mellin_transform
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# TODO we cannot currently do these (needs summation of 3F2(-1))
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# this also implies that they cannot be written as a single g-function
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# (although this is possible)
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mt = MT(log(x)/(x + 1), x, s)
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assert mt[1:] == ((0, 1), True)
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assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
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mt = MT(log(x)**2/(x + 1), x, s)
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assert mt[1:] == ((0, 1), True)
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assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
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mt = MT(log(x)/(x + 1)**2, x, s)
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assert mt[1:] == ((0, 2), True)
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assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
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@slow
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def test_mellin_transform_bessel():
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from sympy.functions.elementary.miscellaneous import Max
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MT = mellin_transform
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# 8.4.19
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assert MT(besselj(a, 2*sqrt(x)), x, s) == \
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(gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, Rational(3, 4)), True)
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assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
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(2**a*gamma(-2*s + S.Half)*gamma(a/2 + s + S.Half)/(
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gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), (
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-re(a)/2 - S.Half, Rational(1, 4)), True)
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assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
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(2**a*gamma(a/2 + s)*gamma(-2*s + S.Half)/(
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gamma(-a/2 - s + S.Half)*gamma(a - 2*s + 1)), (
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-re(a)/2, Rational(1, 4)), True)
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assert MT(besselj(a, sqrt(x))**2, x, s) == \
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(gamma(a + s)*gamma(S.Half - s)
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/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
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(-re(a), S.Half), True)
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assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
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(gamma(s)*gamma(S.Half - s)
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/ (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
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(0, S.Half), True)
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# NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as
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# I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large)
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assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \
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(gamma(1 - s)*gamma(a + s - S.Half)
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/ (sqrt(pi)*gamma(Rational(3, 2) - s)*gamma(a - s + S.Half)),
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(S.Half - re(a), S.Half), True)
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assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \
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(4**s*gamma(1 - 2*s)*gamma((a + b)/2 + s)
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/ (gamma(1 - s + (b - a)/2)*gamma(1 - s + (a - b)/2)
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*gamma( 1 - s + (a + b)/2)),
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(-(re(a) + re(b))/2, S.Half), True)
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assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
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((Max(re(a), -re(a)), S.Half), True)
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# Section 8.4.20
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assert MT(bessely(a, 2*sqrt(x)), x, s) == \
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(-cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)/pi,
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(Max(-re(a)/2, re(a)/2), Rational(3, 4)), True)
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assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
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(-4**s*sin(pi*(a/2 - s))*gamma(S.Half - 2*s)
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* gamma((1 - a)/2 + s)*gamma((1 + a)/2 + s)
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/ (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)),
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(Max(-(re(a) + 1)/2, (re(a) - 1)/2), Rational(1, 4)), True)
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assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
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(-4**s*cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)*gamma(S.Half - 2*s)
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/ (sqrt(pi)*gamma(S.Half - s - a/2)*gamma(S.Half - s + a/2)),
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(Max(-re(a)/2, re(a)/2), Rational(1, 4)), True)
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assert MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s) == \
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(-cos(pi*s)*gamma(s)*gamma(a + s)*gamma(S.Half - s)
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/ (pi**S('3/2')*gamma(1 + a - s)),
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(Max(-re(a), 0), S.Half), True)
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assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \
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(-4**s*cos(pi*(a/2 - b/2 + s))*gamma(1 - 2*s)
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* gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s)
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/ (pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
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(Max((-re(a) + re(b))/2, (-re(a) - re(b))/2), S.Half), True)
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# NOTE bessely(a, sqrt(x))**2 and bessely(a, sqrt(x))*bessely(b, sqrt(x))
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# are a mess (no matter what way you look at it ...)
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assert MT(bessely(a, sqrt(x))**2, x, s)[1:] == \
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((Max(-re(a), 0, re(a)), S.Half), True)
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# Section 8.4.22
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# TODO we can't do any of these (delicate cancellation)
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# Section 8.4.23
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assert MT(besselk(a, 2*sqrt(x)), x, s) == \
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(gamma(
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s - a/2)*gamma(s + a/2)/2, (Max(-re(a)/2, re(a)/2), oo), True)
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assert MT(besselj(a, 2*sqrt(2*sqrt(x)))*besselk(
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a, 2*sqrt(2*sqrt(x))), x, s) == (4**(-s)*gamma(2*s)*
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gamma(a/2 + s)/(2*gamma(a/2 - s + 1)), (Max(0, -re(a)/2), oo), True)
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# TODO bessely(a, x)*besselk(a, x) is a mess
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assert MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
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(gamma(s)*gamma(
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a + s)*gamma(-s + S.Half)/(2*sqrt(pi)*gamma(a - s + 1)),
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(Max(-re(a), 0), S.Half), True)
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assert MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
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(2**(2*s - 1)*gamma(-2*s + 1)*gamma(-a/2 + b/2 + s)* \
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gamma(a/2 + b/2 + s)/(gamma(-a/2 + b/2 - s + 1)* \
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gamma(a/2 + b/2 - s + 1)), (Max(-re(a)/2 - re(b)/2, \
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re(a)/2 - re(b)/2), S.Half), True)
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# TODO products of besselk are a mess
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mt = MT(exp(-x/2)*besselk(a, x/2), x, s)
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mt0 = gammasimp(trigsimp(gammasimp(mt[0].expand(func=True))))
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assert mt0 == 2*pi**Rational(3, 2)*cos(pi*s)*gamma(S.Half - s)/(
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(cos(2*pi*a) - cos(2*pi*s))*gamma(-a - s + 1)*gamma(a - s + 1))
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assert mt[1:] == ((Max(-re(a), re(a)), oo), True)
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# TODO exp(x/2)*besselk(a, x/2) [etc] cannot currently be done
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# TODO various strange products of special orders
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@slow
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def test_expint():
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from sympy.functions.elementary.miscellaneous import Max
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from sympy.functions.special.error_functions import (Ci, E1, Ei, Si)
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from sympy.functions.special.zeta_functions import lerchphi
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from sympy.simplify.simplify import simplify
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aneg = Symbol('a', negative=True)
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u = Symbol('u', polar=True)
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assert mellin_transform(E1(x), x, s) == (gamma(s)/s, (0, oo), True)
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assert inverse_mellin_transform(gamma(s)/s, s, x,
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(0, oo)).rewrite(expint).expand() == E1(x)
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assert mellin_transform(expint(a, x), x, s) == \
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(gamma(s)/(a + s - 1), (Max(1 - re(a), 0), oo), True)
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# XXX IMT has hickups with complicated strips ...
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assert simplify(unpolarify(
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inverse_mellin_transform(gamma(s)/(aneg + s - 1), s, x,
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(1 - aneg, oo)).rewrite(expint).expand(func=True))) == \
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expint(aneg, x)
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assert mellin_transform(Si(x), x, s) == \
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(-2**s*sqrt(pi)*gamma(s/2 + S.Half)/(
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2*s*gamma(-s/2 + 1)), (-1, 0), True)
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assert inverse_mellin_transform(-2**s*sqrt(pi)*gamma((s + 1)/2)
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/(2*s*gamma(-s/2 + 1)), s, x, (-1, 0)) \
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== Si(x)
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assert mellin_transform(Ci(sqrt(x)), x, s) == \
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(-2**(2*s - 1)*sqrt(pi)*gamma(s)/(s*gamma(-s + S.Half)), (0, 1), True)
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assert inverse_mellin_transform(
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-4**s*sqrt(pi)*gamma(s)/(2*s*gamma(-s + S.Half)),
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s, u, (0, 1)).expand() == Ci(sqrt(u))
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# TODO LT of Si, Shi, Chi is a mess ...
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assert laplace_transform(Ci(x), x, s) == (-log(1 + s**2)/2/s, 0, True)
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assert laplace_transform(expint(a, x), x, s) == \
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(lerchphi(s*exp_polar(I*pi), 1, a), 0, re(a) > S.Zero)
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assert laplace_transform(expint(1, x), x, s) == (log(s + 1)/s, 0, True)
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assert laplace_transform(expint(2, x), x, s) == \
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((s - log(s + 1))/s**2, 0, True)
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assert inverse_laplace_transform(-log(1 + s**2)/2/s, s, u).expand() == \
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Heaviside(u)*Ci(u)
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assert inverse_laplace_transform(log(s + 1)/s, s, x).rewrite(expint) == \
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Heaviside(x)*E1(x)
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assert inverse_laplace_transform((s - log(s + 1))/s**2, s,
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x).rewrite(expint).expand() == \
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(expint(2, x)*Heaviside(x)).rewrite(Ei).rewrite(expint).expand()
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@slow
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def test_inverse_mellin_transform():
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from sympy.core.function import expand
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from sympy.functions.elementary.miscellaneous import (Max, Min)
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from sympy.functions.elementary.trigonometric import cot
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from sympy.simplify.powsimp import powsimp
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from sympy.simplify.simplify import simplify
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IMT = inverse_mellin_transform
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assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
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|
assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1/x)
|
|
assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \
|
|
(x**2 + 1)*Heaviside(1 - x)/(4*x)
|
|
|
|
# test passing "None"
|
|
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
|
|
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
|
|
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
|
|
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
|
|
|
|
# test expansion of sums
|
|
assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1)*exp(-x)/x
|
|
|
|
# test factorisation of polys
|
|
r = symbols('r', real=True)
|
|
assert IMT(1/(s**2 + 1), s, exp(-x), (None, oo)
|
|
).subs(x, r).rewrite(sin).simplify() \
|
|
== sin(r)*Heaviside(1 - exp(-r))
|
|
|
|
# test multiplicative substitution
|
|
_a, _b = symbols('a b', positive=True)
|
|
assert IMT(_b**(-s/_a)*factorial(s/_a)/s, s, x, (0, oo)) == exp(-_b*x**_a)
|
|
assert IMT(factorial(_a/_b + s/_b)/(_a + s), s, x, (-_a, oo)) == x**_a*exp(-x**_b)
|
|
|
|
def simp_pows(expr):
|
|
return simplify(powsimp(expand_mul(expr, deep=False), force=True)).replace(exp_polar, exp)
|
|
|
|
# Now test the inverses of all direct transforms tested above
|
|
|
|
# Section 8.4.2
|
|
nu = symbols('nu', real=True)
|
|
assert IMT(-1/(nu + s), s, x, (-oo, None)) == x**nu*Heaviside(x - 1)
|
|
assert IMT(1/(nu + s), s, x, (None, oo)) == x**nu*Heaviside(1 - x)
|
|
assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \
|
|
== (1 - x)**(beta - 1)*Heaviside(1 - x)
|
|
assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
|
|
s, x, (-oo, None))) \
|
|
== (x - 1)**(beta - 1)*Heaviside(x - 1)
|
|
assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \
|
|
== (1/(x + 1))**rho
|
|
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
|
|
*gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
|
|
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
|
|
== (x**c - d**c)/(x - d)
|
|
|
|
assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s)
|
|
*gamma(-c/2 - s)/gamma(1 - c - s),
|
|
s, x, (0, -re(c)/2))) == \
|
|
(1 + sqrt(x + 1))**c
|
|
assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s)
|
|
/gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \
|
|
b**(a - 1)*(sqrt(1 + x/b**2) + 1)**(a - 1)*(b**2*sqrt(1 + x/b**2) +
|
|
b**2 + x)/(b**2 + x)
|
|
assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s)
|
|
/ gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \
|
|
b**c*(sqrt(1 + x/b**2) + 1)**c
|
|
|
|
# Section 8.4.5
|
|
assert IMT(24/s**5, s, x, (0, oo)) == log(x)**4*Heaviside(1 - x)
|
|
assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \
|
|
log(x)**3*Heaviside(x - 1)
|
|
assert IMT(pi/(s*sin(pi*s)), s, x, (-1, 0)) == log(x + 1)
|
|
assert IMT(pi/(s*sin(pi*s/2)), s, x, (-2, 0)) == log(x**2 + 1)
|
|
assert IMT(pi/(s*sin(2*pi*s)), s, x, (Rational(-1, 2), 0)) == log(sqrt(x) + 1)
|
|
assert IMT(pi/(s*sin(pi*s)), s, x, (0, 1)) == log(1 + 1/x)
|
|
|
|
# TODO
|
|
def mysimp(expr):
|
|
from sympy.core.function import expand
|
|
from sympy.simplify.powsimp import powsimp
|
|
from sympy.simplify.simplify import logcombine
|
|
return expand(
|
|
powsimp(logcombine(expr, force=True), force=True, deep=True),
|
|
force=True).replace(exp_polar, exp)
|
|
|
|
assert mysimp(mysimp(IMT(pi/(s*tan(pi*s)), s, x, (-1, 0)))) in [
|
|
log(1 - x)*Heaviside(1 - x) + log(x - 1)*Heaviside(x - 1),
|
|
log(x)*Heaviside(x - 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
|
|
1)*Heaviside(-x + 1)]
|
|
# test passing cot
|
|
assert mysimp(IMT(pi*cot(pi*s)/s, s, x, (0, 1))) in [
|
|
log(1/x - 1)*Heaviside(1 - x) + log(1 - 1/x)*Heaviside(x - 1),
|
|
-log(x)*Heaviside(-x + 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
|
|
1)*Heaviside(-x + 1), ]
|
|
|
|
# 8.4.14
|
|
assert IMT(-gamma(s + S.Half)/(sqrt(pi)*s), s, x, (Rational(-1, 2), 0)) == \
|
|
erf(sqrt(x))
|
|
|
|
# 8.4.19
|
|
assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, Rational(3, 4)))) \
|
|
== besselj(a, 2*sqrt(x))
|
|
assert simplify(IMT(2**a*gamma(S.Half - 2*s)*gamma(s + (a + 1)/2)
|
|
/ (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
|
|
s, x, (-(re(a) + 1)/2, Rational(1, 4)))) == \
|
|
sin(sqrt(x))*besselj(a, sqrt(x))
|
|
assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(S.Half - 2*s)
|
|
/ (gamma(S.Half - s - a/2)*gamma(1 - 2*s + a)),
|
|
s, x, (-re(a)/2, Rational(1, 4)))) == \
|
|
cos(sqrt(x))*besselj(a, sqrt(x))
|
|
# TODO this comes out as an amazing mess, but simplifies nicely
|
|
assert simplify(IMT(gamma(a + s)*gamma(S.Half - s)
|
|
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
|
|
s, x, (-re(a), S.Half))) == \
|
|
besselj(a, sqrt(x))**2
|
|
assert simplify(IMT(gamma(s)*gamma(S.Half - s)
|
|
/ (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
|
|
s, x, (0, S.Half))) == \
|
|
besselj(-a, sqrt(x))*besselj(a, sqrt(x))
|
|
assert simplify(IMT(4**s*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
|
|
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
|
|
*gamma(a/2 + b/2 - s + 1)),
|
|
s, x, (-(re(a) + re(b))/2, S.Half))) == \
|
|
besselj(a, sqrt(x))*besselj(b, sqrt(x))
|
|
|
|
# Section 8.4.20
|
|
# TODO this can be further simplified!
|
|
assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) *
|
|
gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) /
|
|
(pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
|
|
s, x,
|
|
(Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), S.Half))) == \
|
|
besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) -
|
|
besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b)
|
|
# TODO more
|
|
|
|
# for coverage
|
|
|
|
assert IMT(pi/cos(pi*s), s, x, (0, S.Half)) == sqrt(x)/(x + 1)
|
|
|
|
|
|
@slow
|
|
def test_laplace_transform():
|
|
from sympy import lowergamma
|
|
from sympy.functions.special.delta_functions import DiracDelta
|
|
from sympy.functions.special.error_functions import (fresnelc, fresnels)
|
|
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 rule-base evaluation according to
|
|
# http://eqworld.ipmnet.ru/en/auxiliary/inttrans/
|
|
# Power-law functions (laplace2.pdf)
|
|
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)
|
|
# Exponential functions (laplace3.pdf)
|
|
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)
|
|
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) ==\
|
|
(sqrt(pi)*(2*sqrt(a)*sqrt(s) + 1)*sqrt(s**(-3))*exp(-2*sqrt(a)*\
|
|
sqrt(s))/2, 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)
|
|
# Hyperbolic functions (laplace4.pdf)
|
|
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**2 + s**3),
|
|
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) == ((-2*a**2 + s**2)/(-4*a**2*s**2 + s**3),
|
|
2*a, True)
|
|
assert LT(sinh(x + 3), x, s) == (
|
|
(-s + (s + 1)*exp(6) + 1)*exp(-3)/(s - 1)/(s + 1)/2, 0, Abs(s) > 1)
|
|
# The following line replaces the old test test_issue_7173()
|
|
assert LT(sinh(a*t)*cosh(a*t), t, s) == (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) ==\
|
|
(-sqrt(a)/s**2 + sqrt(pi)*(a + s/2)*exp(a/s)*erf(sqrt(a)*\
|
|
sqrt(1/s))/s**(S(5)/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)
|
|
# logarithmic functions (laplace5.pdf)
|
|
assert LT(log(t), t, s) == (-log(s+S.EulerGamma)/s, 0, True)
|
|
assert LT(log(t/a), t, s) == (-log(a*s + 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) == ((log(a) - exp(s/a)*Ei(-s/a)/s)/s, 0, True)
|
|
assert LT(log(t)/sqrt(t), t, s) ==\
|
|
(sqrt(pi)*(-log(s) - 2*log(2) - S.EulerGamma)/sqrt(s), 0, True)
|
|
assert LT(t**(S(5)/2)*log(t), t, s) ==\
|
|
(15*sqrt(pi)*(-log(s)-2*log(2)-S.EulerGamma+S(46)/15)/(8*s**(S(7)/2)),
|
|
0, True)
|
|
assert (LT(t**3*log(t), t, s, noconds=True)-6*(-log(s) - S.EulerGamma\
|
|
+ S(11)/6)/s**4).simplify() == S.Zero
|
|
assert LT(log(t)**2, t, s) ==\
|
|
(((log(s) + EulerGamma)**2 + pi**2/6)/s, 0, True)
|
|
assert LT(exp(-a*t)*log(t), t, s) ==\
|
|
((-log(a + s) - S.EulerGamma)/(a + s), -a, True)
|
|
# Trigonometric functions (laplace6.pdf)
|
|
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) ==\
|
|
(sqrt(pi)*(-2*a + s)*exp(-a/s)/(2*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(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) == ((b + s)*c/(a**2 + (b + s)**2),
|
|
-b, True)
|
|
assert LT(cos(x + 3), x, s) == ((s*cos(3) - sin(3))/(s**2 + 1), 0, True)
|
|
# 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) ==\
|
|
(sqrt(a)/(sqrt(s)*(-a + s)), a, True)
|
|
assert LT(erf(sqrt(a/t)/2), t, s) == ((1-exp(-sqrt(a)*sqrt(s)))/s, 0, True)
|
|
assert LT(erfc(sqrt(a*t)), t, s) ==\
|
|
((-sqrt(a) + sqrt(a + s))/(s*sqrt(a + s)), 0, 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) ==\
|
|
(a/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))), 0, True)
|
|
assert LT(besselj(2, a*t), t, s) ==\
|
|
(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) ==\
|
|
(exp(b*s - b*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) ==\
|
|
(a/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))), a, True)
|
|
assert LT(besseli(2, a*t), t, s) ==\
|
|
(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))/sqrt(-a**2 + s**2), a, True)
|
|
assert LT(sin(a*t)**8, t, s) ==\
|
|
(40320*a**8/(s*(147456*a**8 + 52480*a**6*s**2 + 4368*a**4*s**4 +\
|
|
120*a**2*s**6 + s**8)), 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)
|
|
assert LT(a*Heaviside(t+1)*f(t+1), t, s) ==\
|
|
a*LaplaceTransform(f(t + 1)*Heaviside(t + 1), t, s)
|
|
assert LT(a*Heaviside(t-1)*f(t-1), t, s) ==\
|
|
a*LaplaceTransform(f(t), t, s)*exp(-s)
|
|
assert LT(b*f(t/a), t, s) == a*b*LaplaceTransform(f(t), t, a*s)
|
|
assert LT(exp(-f(x)*t), t, s) == (1/(s + f(x)), -f(x), True)
|
|
assert LT(exp(-a*t)*f(t), t, s) == LaplaceTransform(f(t), t, a + s)
|
|
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
|
|
assert LT(sinh(a*t)*t, t, s) ==\
|
|
(-1/(2*(a + s)**2) + 1/(2*(-a + s)**2), 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
|
|
assert LT(cosh(a*t)*t, t, s) ==\
|
|
(1/(2*(a + s)**2) + 1/(2*(-a + s)**2), a, True)
|
|
assert LT(sin(a*t)*f(t), t, s) ==\
|
|
I*(-LaplaceTransform(f(t), t, -I*a + s) +\
|
|
LaplaceTransform(f(t), t, I*a + s))/2
|
|
assert LT(sin(a*t)*t, t, s) ==\
|
|
(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
|
|
assert LT(cos(a*t)*t, t, s) ==\
|
|
((-a**2 + s**2)/(a**4 + 2*a**2*s**2 + s**4), 0, True)
|
|
# The following two lines test whether issues #5813 and #7176 are solved.
|
|
assert LT(diff(f(t), (t, 1)), t, s) == s*LaplaceTransform(f(t), t, s)\
|
|
- f(0)
|
|
assert LT(diff(f(t), (t, 3)), t, s) == 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)
|
|
assert LT(a*f(b*t)+g(c*t), t, s) == 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) == LaplaceTransform(f(t)*g(t), t, s)
|
|
|
|
# additional basic tests from wikipedia
|
|
assert LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) == \
|
|
((s + c)**(-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) \
|
|
== exp(-b)/(s**2 - 1)
|
|
|
|
# DiracDelta function: standard cases
|
|
assert LT(DiracDelta(t), t, s) == (1, 0, True)
|
|
assert LT(DiracDelta(a*t), t, s) == (1/a, 0, True)
|
|
assert LT(DiracDelta(t/42), t, s) == (42, 0, True)
|
|
assert LT(DiracDelta(t+42), t, s) == (0, 0, True)
|
|
assert LT(DiracDelta(t)+DiracDelta(t-42), t, s) == \
|
|
(1 + exp(-42*s), 0, True)
|
|
assert LT(DiracDelta(t)-a*exp(-a*t), t, s) == (s/(a + s), 0, True)
|
|
assert LT(exp(-t)*(DiracDelta(t)+DiracDelta(t-42)), t, s) == \
|
|
(exp(-42*s - 42) + 1, -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) == 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) == \
|
|
LaplaceTransform(DiracDelta(-t**2 + t), t, s)
|
|
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, 0, True)
|
|
assert LT(DiracDelta(-2*t+2*exp(a)), t, s) == (exp(-s*exp(a))/2, 0, 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 - 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) == ((1 - exp(-2*s))/s, 0, True)
|
|
|
|
# Fresnel functions
|
|
assert laplace_transform(fresnels(t), t, s) == \
|
|
((-sin(s**2/(2*pi))*fresnels(s/pi) + sin(s**2/(2*pi))/2 -
|
|
cos(s**2/(2*pi))*fresnelc(s/pi) + cos(s**2/(2*pi))/2)/s, 0, True)
|
|
assert laplace_transform(fresnelc(t), t, s) == (
|
|
((2*sin(s**2/(2*pi))*fresnelc(s/pi) - 2*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 tranform 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.core.exprtools import factor_terms
|
|
from sympy.functions.special.delta_functions import DiracDelta
|
|
from sympy.simplify.simplify import simplify
|
|
ILT = inverse_laplace_transform
|
|
a, b, c, = symbols('a b c', positive=True)
|
|
t = symbols('t')
|
|
|
|
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((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) +\
|
|
(a*sin(b*t) - b*cos(b*t))*exp(-a*t)*Heaviside(t) == 0
|
|
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 - 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 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).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) == (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)]])
|
|
|
|
def test_inverse_laplace_transform_delta():
|
|
from sympy.functions.special.delta_functions import DiracDelta
|
|
ILT = inverse_laplace_transform
|
|
t = symbols('t')
|
|
assert ILT(2, s, t) == 2*DiracDelta(t)
|
|
assert ILT(2*exp(3*s) - 5*exp(-7*s), s, t) == \
|
|
2*DiracDelta(t + 3) - 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) == DiracDelta(t + 2)
|
|
r = Symbol('r', real=True)
|
|
assert ILT(exp(r*s), s, t) == DiracDelta(t + r)
|
|
|
|
|
|
def test_inverse_laplace_transform_delta_cond():
|
|
from sympy.functions.elementary.complexes import im
|
|
from sympy.functions.special.delta_functions import DiracDelta
|
|
ILT = inverse_laplace_transform
|
|
t = symbols('t')
|
|
r = Symbol('r', real=True)
|
|
assert ILT(exp(r*s), s, t, noconds=False) == (DiracDelta(t + r), True)
|
|
z = Symbol('z')
|
|
assert ILT(exp(z*s), s, t, noconds=False) == \
|
|
(DiracDelta(t + z), Eq(im(z), 0))
|
|
# 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
|
|
# issue 15043
|
|
assert ILT(1/s + exp(r*s)/s, s, t, noconds=False) == (
|
|
Heaviside(t) + Heaviside(r + t), True)
|
|
|
|
def test_fourier_transform():
|
|
from sympy.core.function import (expand, expand_complex, expand_trig)
|
|
from sympy.polys.polytools import factor
|
|
from sympy.simplify.simplify import simplify
|
|
FT = fourier_transform
|
|
IFT = inverse_fourier_transform
|
|
|
|
def simp(x):
|
|
return simplify(expand_trig(expand_complex(expand(x))))
|
|
|
|
def sinc(x):
|
|
return sin(pi*x)/(pi*x)
|
|
k = symbols('k', real=True)
|
|
f = Function("f")
|
|
|
|
# TODO for this to work with real a, need to expand abs(a*x) to abs(a)*abs(x)
|
|
a = symbols('a', positive=True)
|
|
b = symbols('b', positive=True)
|
|
|
|
posk = symbols('posk', positive=True)
|
|
|
|
# Test unevaluated form
|
|
assert fourier_transform(f(x), x, k) == FourierTransform(f(x), x, k)
|
|
assert inverse_fourier_transform(
|
|
f(k), k, x) == InverseFourierTransform(f(k), k, x)
|
|
|
|
# basic examples from wikipedia
|
|
assert simp(FT(Heaviside(1 - abs(2*a*x)), x, k)) == sinc(k/a)/a
|
|
# TODO IFT is a *mess*
|
|
assert simp(FT(Heaviside(1 - abs(a*x))*(1 - abs(a*x)), x, k)) == sinc(k/a)**2/a
|
|
# TODO IFT
|
|
|
|
assert factor(FT(exp(-a*x)*Heaviside(x), x, k), extension=I) == \
|
|
1/(a + 2*pi*I*k)
|
|
# NOTE: the ift comes out in pieces
|
|
assert IFT(1/(a + 2*pi*I*x), x, posk,
|
|
noconds=False) == (exp(-a*posk), True)
|
|
assert IFT(1/(a + 2*pi*I*x), x, -posk,
|
|
noconds=False) == (0, True)
|
|
assert IFT(1/(a + 2*pi*I*x), x, symbols('k', negative=True),
|
|
noconds=False) == (0, True)
|
|
# TODO IFT without factoring comes out as meijer g
|
|
|
|
assert factor(FT(x*exp(-a*x)*Heaviside(x), x, k), extension=I) == \
|
|
1/(a + 2*pi*I*k)**2
|
|
assert FT(exp(-a*x)*sin(b*x)*Heaviside(x), x, k) == \
|
|
b/(b**2 + (a + 2*I*pi*k)**2)
|
|
|
|
assert FT(exp(-a*x**2), x, k) == sqrt(pi)*exp(-pi**2*k**2/a)/sqrt(a)
|
|
assert IFT(sqrt(pi/a)*exp(-(pi*k)**2/a), k, x) == exp(-a*x**2)
|
|
assert FT(exp(-a*abs(x)), x, k) == 2*a/(a**2 + 4*pi**2*k**2)
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# TODO IFT (comes out as meijer G)
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# TODO besselj(n, x), n an integer > 0 actually can be done...
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# TODO are there other common transforms (no distributions!)?
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def test_sine_transform():
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t = symbols("t")
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w = symbols("w")
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a = symbols("a")
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f = Function("f")
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# Test unevaluated form
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assert sine_transform(f(t), t, w) == SineTransform(f(t), t, w)
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assert inverse_sine_transform(
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f(w), w, t) == InverseSineTransform(f(w), w, t)
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assert sine_transform(1/sqrt(t), t, w) == 1/sqrt(w)
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assert inverse_sine_transform(1/sqrt(w), w, t) == 1/sqrt(t)
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assert sine_transform((1/sqrt(t))**3, t, w) == 2*sqrt(w)
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assert sine_transform(t**(-a), t, w) == 2**(
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-a + S.Half)*w**(a - 1)*gamma(-a/2 + 1)/gamma((a + 1)/2)
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assert inverse_sine_transform(2**(-a + S(
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1)/2)*w**(a - 1)*gamma(-a/2 + 1)/gamma(a/2 + S.Half), w, t) == t**(-a)
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assert sine_transform(
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exp(-a*t), t, w) == sqrt(2)*w/(sqrt(pi)*(a**2 + w**2))
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assert inverse_sine_transform(
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sqrt(2)*w/(sqrt(pi)*(a**2 + w**2)), w, t) == exp(-a*t)
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assert sine_transform(
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log(t)/t, t, w) == sqrt(2)*sqrt(pi)*-(log(w**2) + 2*EulerGamma)/4
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assert sine_transform(
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t*exp(-a*t**2), t, w) == sqrt(2)*w*exp(-w**2/(4*a))/(4*a**Rational(3, 2))
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assert inverse_sine_transform(
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sqrt(2)*w*exp(-w**2/(4*a))/(4*a**Rational(3, 2)), w, t) == t*exp(-a*t**2)
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def test_cosine_transform():
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from sympy.functions.special.error_functions import (Ci, Si)
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t = symbols("t")
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w = symbols("w")
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a = symbols("a")
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f = Function("f")
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# Test unevaluated form
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assert cosine_transform(f(t), t, w) == CosineTransform(f(t), t, w)
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assert inverse_cosine_transform(
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f(w), w, t) == InverseCosineTransform(f(w), w, t)
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assert cosine_transform(1/sqrt(t), t, w) == 1/sqrt(w)
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assert inverse_cosine_transform(1/sqrt(w), w, t) == 1/sqrt(t)
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assert cosine_transform(1/(
|
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a**2 + t**2), t, w) == sqrt(2)*sqrt(pi)*exp(-a*w)/(2*a)
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assert cosine_transform(t**(
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|
-a), t, w) == 2**(-a + S.Half)*w**(a - 1)*gamma((-a + 1)/2)/gamma(a/2)
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assert inverse_cosine_transform(2**(-a + S(
|
|
1)/2)*w**(a - 1)*gamma(-a/2 + S.Half)/gamma(a/2), w, t) == t**(-a)
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|
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assert cosine_transform(
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exp(-a*t), t, w) == sqrt(2)*a/(sqrt(pi)*(a**2 + w**2))
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assert inverse_cosine_transform(
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sqrt(2)*a/(sqrt(pi)*(a**2 + w**2)), w, t) == exp(-a*t)
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assert cosine_transform(exp(-a*sqrt(t))*cos(a*sqrt(
|
|
t)), t, w) == a*exp(-a**2/(2*w))/(2*w**Rational(3, 2))
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assert cosine_transform(1/(a + t), t, w) == sqrt(2)*(
|
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(-2*Si(a*w) + pi)*sin(a*w)/2 - cos(a*w)*Ci(a*w))/sqrt(pi)
|
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assert inverse_cosine_transform(sqrt(2)*meijerg(((S.Half, 0), ()), (
|
|
(S.Half, 0, 0), (S.Half,)), a**2*w**2/4)/(2*pi), w, t) == 1/(a + t)
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|
|
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assert cosine_transform(1/sqrt(a**2 + t**2), t, w) == sqrt(2)*meijerg(
|
|
((S.Half,), ()), ((0, 0), (S.Half,)), a**2*w**2/4)/(2*sqrt(pi))
|
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assert inverse_cosine_transform(sqrt(2)*meijerg(((S.Half,), ()), ((0, 0), (S.Half,)), a**2*w**2/4)/(2*sqrt(pi)), w, t) == 1/(t*sqrt(a**2/t**2 + 1))
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|
|
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def test_hankel_transform():
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|
r = Symbol("r")
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|
k = Symbol("k")
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|
nu = Symbol("nu")
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|
m = Symbol("m")
|
|
a = symbols("a")
|
|
|
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assert hankel_transform(1/r, r, k, 0) == 1/k
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assert inverse_hankel_transform(1/k, k, r, 0) == 1/r
|
|
|
|
assert hankel_transform(
|
|
1/r**m, r, k, 0) == 2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2)
|
|
assert inverse_hankel_transform(
|
|
2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2), k, r, 0) == r**(-m)
|
|
|
|
assert hankel_transform(1/r**m, r, k, nu) == (
|
|
2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2))
|
|
assert inverse_hankel_transform(2**(-m + 1)*k**(
|
|
m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2), k, r, nu) == r**(-m)
|
|
|
|
assert hankel_transform(r**nu*exp(-a*r), r, k, nu) == \
|
|
2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(
|
|
3)/2)*gamma(nu + Rational(3, 2))/sqrt(pi)
|
|
assert inverse_hankel_transform(
|
|
2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - Rational(3, 2))*gamma(
|
|
nu + Rational(3, 2))/sqrt(pi), k, r, nu) == r**nu*exp(-a*r)
|
|
|
|
|
|
def test_issue_7181():
|
|
assert mellin_transform(1/(1 - x), x, s) != None
|
|
|
|
|
|
def test_issue_8882():
|
|
# This is the original test.
|
|
# from sympy import diff, Integral, integrate
|
|
# r = Symbol('r')
|
|
# psi = 1/r*sin(r)*exp(-(a0*r))
|
|
# h = -1/2*diff(psi, r, r) - 1/r*psi
|
|
# f = 4*pi*psi*h*r**2
|
|
# assert integrate(f, (r, -oo, 3), meijerg=True).has(Integral) == True
|
|
|
|
# To save time, only the critical part is included.
|
|
F = -a**(-s + 1)*(4 + 1/a**2)**(-s/2)*sqrt(1/a**2)*exp(-s*I*pi)* \
|
|
sin(s*atan(sqrt(1/a**2)/2))*gamma(s)
|
|
raises(IntegralTransformError, lambda:
|
|
inverse_mellin_transform(F, s, x, (-1, oo),
|
|
**{'as_meijerg': True, 'needeval': True}))
|
|
|
|
|
|
def test_issue_8514():
|
|
from sympy.simplify.simplify import simplify
|
|
a, b, c, = symbols('a b c', positive=True)
|
|
t = symbols('t', positive=True)
|
|
ft = simplify(inverse_laplace_transform(1/(a*s**2+b*s+c),s, t))
|
|
assert ft == (I*exp(t*cos(atan2(0, -4*a*c + b**2)/2)*sqrt(Abs(4*a*c -
|
|
b**2))/a)*sin(t*sin(atan2(0, -4*a*c + b**2)/2)*sqrt(Abs(
|
|
4*a*c - b**2))/(2*a)) + exp(t*cos(atan2(0, -4*a*c + b**2)
|
|
/2)*sqrt(Abs(4*a*c - b**2))/a)*cos(t*sin(atan2(0, -4*a*c
|
|
+ b**2)/2)*sqrt(Abs(4*a*c - b**2))/(2*a)) + I*sin(t*sin(
|
|
atan2(0, -4*a*c + b**2)/2)*sqrt(Abs(4*a*c - b**2))/(2*a))
|
|
- cos(t*sin(atan2(0, -4*a*c + b**2)/2)*sqrt(Abs(4*a*c -
|
|
b**2))/(2*a)))*exp(-t*(b + cos(atan2(0, -4*a*c + b**2)/2)
|
|
*sqrt(Abs(4*a*c - b**2)))/(2*a))/sqrt(-4*a*c + b**2)
|
|
|
|
|
|
def test_issue_12591():
|
|
x, y = symbols("x y", real=True)
|
|
assert fourier_transform(exp(x), x, y) == FourierTransform(exp(x), x, y)
|
|
|
|
|
|
def test_issue_14692():
|
|
b = Symbol('b', negative=True)
|
|
assert laplace_transform(1/(I*x - b), x, s) == \
|
|
(-I*exp(I*b*s)*expint(1, b*s*exp_polar(I*pi/2)), 0, True)
|