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Mini Shell
Mini Shell
from sympy.integrals.transforms import (mellin_transform,
inverse_mellin_transform, laplace_transform, inverse_laplace_transform,
fourier_transform, inverse_fourier_transform,
sine_transform, inverse_sine_transform,
cosine_transform, inverse_cosine_transform,
hankel_transform, inverse_hankel_transform,
LaplaceTransform, FourierTransform, SineTransform, CosineTransform,
InverseLaplaceTransform, InverseFourierTransform,
InverseSineTransform, InverseCosineTransform, IntegralTransformError)
from sympy import (
gamma, exp, oo, Heaviside, symbols, Symbol, re, factorial, pi, arg,
cos, S, Abs, And, sin, sqrt, I, log, tan, hyperexpand, meijerg,
EulerGamma, erf, erfc, besselj, bessely, besseli, besselk,
exp_polar, unpolarify, Function, expint, expand_mul, Rational,
gammasimp, trigsimp, atan, sinh, cosh, Ne, periodic_argument, atan2)
from sympy.testing.pytest import XFAIL, slow, skip, raises, warns_deprecated_sympy
from sympy.matrices import Matrix, eye
from sympy.abc import x, s, a, b, c, d
nu, beta, rho = symbols('nu beta rho')
def test_undefined_function():
from sympy import Function, MellinTransform
f = Function('f')
assert mellin_transform(f(x), x, s) == MellinTransform(f(x), x, s)
assert mellin_transform(f(x) + exp(-x), x, s) == \
(MellinTransform(f(x), x, s) + gamma(s), (0, oo), True)
assert laplace_transform(2*f(x), x, s) == 2*LaplaceTransform(f(x), x, s)
# TODO test derivative and other rules when implemented
def test_free_symbols():
from sympy import Function
f = Function('f')
assert mellin_transform(f(x), x, s).free_symbols == {s}
assert mellin_transform(f(x)*a, x, s).free_symbols == {s, a}
def test_as_integral():
from sympy import Function, Integral
f = Function('f')
assert mellin_transform(f(x), x, s).rewrite('Integral') == \
Integral(x**(s - 1)*f(x), (x, 0, oo))
assert fourier_transform(f(x), x, s).rewrite('Integral') == \
Integral(f(x)*exp(-2*I*pi*s*x), (x, -oo, oo))
assert laplace_transform(f(x), x, s).rewrite('Integral') == \
Integral(f(x)*exp(-s*x), (x, 0, oo))
assert str(2*pi*I*inverse_mellin_transform(f(s), s, x, (a, b)).rewrite('Integral')) \
== "Integral(f(s)/x**s, (s, _c - oo*I, _c + oo*I))"
assert str(2*pi*I*inverse_laplace_transform(f(s), s, x).rewrite('Integral')) == \
"Integral(f(s)*exp(s*x), (s, _c - oo*I, _c + oo*I))"
assert inverse_fourier_transform(f(s), s, x).rewrite('Integral') == \
Integral(f(s)*exp(2*I*pi*s*x), (s, -oo, oo))
# NOTE this is stuck in risch because meijerint cannot handle it
@slow
@XFAIL
def test_mellin_transform_fail():
skip("Risch takes forever.")
MT = mellin_transform
bpos = symbols('b', positive=True)
# bneg = symbols('b', negative=True)
expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
# TODO does not work with bneg, argument wrong. Needs changes to matching.
assert MT(expr.subs(b, -bpos), x, s) == \
((-1)**(a + 1)*2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(a + s)
*gamma(1 - a - 2*s)/gamma(1 - s),
(-re(a), -re(a)/2 + S.Half), True)
expr = (sqrt(x + b**2) + b)**a
assert MT(expr.subs(b, -bpos), x, s) == \
(
2**(a + 2*s)*a*bpos**(a + 2*s)*gamma(-a - 2*
s)*gamma(a + s)/gamma(-s + 1),
(-re(a), -re(a)/2), True)
# Test exponent 1:
assert MT(expr.subs({b: -bpos, a: 1}), x, s) == \
(-bpos**(2*s + 1)*gamma(s)*gamma(-s - S.Half)/(2*sqrt(pi)),
(-1, Rational(-1, 2)), True)
def test_mellin_transform():
from sympy import Max, Min
MT = mellin_transform
bpos = symbols('b', positive=True)
# 8.4.2
assert MT(x**nu*Heaviside(x - 1), x, s) == \
(-1/(nu + s), (-oo, -re(nu)), True)
assert MT(x**nu*Heaviside(1 - x), x, s) == \
(1/(nu + s), (-re(nu), oo), True)
assert MT((1 - x)**(beta - 1)*Heaviside(1 - x), x, s) == \
(gamma(beta)*gamma(s)/gamma(beta + s), (0, oo), re(beta) > 0)
assert MT((x - 1)**(beta - 1)*Heaviside(x - 1), x, s) == \
(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
(-oo, -re(beta) + 1), re(beta) > 0)
assert MT((1 + x)**(-rho), x, s) == \
(gamma(s)*gamma(rho - s)/gamma(rho), (0, re(rho)), True)
# TODO also the conditions should be simplified, e.g.
# And(re(rho) - 1 < 0, re(rho) < 1) should just be
# re(rho) < 1
assert MT(abs(1 - x)**(-rho), x, s) == (
2*sin(pi*rho/2)*gamma(1 - rho)*
cos(pi*(rho/2 - s))*gamma(s)*gamma(rho-s)/pi,
(0, re(rho)), And(re(rho) - 1 < 0, re(rho) < 1))
mt = MT((1 - x)**(beta - 1)*Heaviside(1 - x)
+ a*(x - 1)**(beta - 1)*Heaviside(x - 1), x, s)
assert mt[1], mt[2] == ((0, -re(beta) + 1), re(beta) > 0)
assert MT((x**a - b**a)/(x - b), x, s)[0] == \
pi*b**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s)))
assert MT((x**a - bpos**a)/(x - bpos), x, s) == \
(pi*bpos**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s))),
(Max(-re(a), 0), Min(1 - re(a), 1)), True)
expr = (sqrt(x + b**2) + b)**a
assert MT(expr.subs(b, bpos), x, s) == \
(-a*(2*bpos)**(a + 2*s)*gamma(s)*gamma(-a - 2*s)/gamma(-a - s + 1),
(0, -re(a)/2), True)
expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
assert MT(expr.subs(b, bpos), x, s) == \
(2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(s)
*gamma(1 - a - 2*s)/gamma(1 - a - s),
(0, -re(a)/2 + S.Half), True)
# 8.4.2
assert MT(exp(-x), x, s) == (gamma(s), (0, oo), True)
assert MT(exp(-1/x), x, s) == (gamma(-s), (-oo, 0), True)
# 8.4.5
assert MT(log(x)**4*Heaviside(1 - x), x, s) == (24/s**5, (0, oo), True)
assert MT(log(x)**3*Heaviside(x - 1), x, s) == (6/s**4, (-oo, 0), True)
assert MT(log(x + 1), x, s) == (pi/(s*sin(pi*s)), (-1, 0), True)
assert MT(log(1/x + 1), x, s) == (pi/(s*sin(pi*s)), (0, 1), True)
assert MT(log(abs(1 - x)), x, s) == (pi/(s*tan(pi*s)), (-1, 0), True)
assert MT(log(abs(1 - 1/x)), x, s) == (pi/(s*tan(pi*s)), (0, 1), True)
# 8.4.14
assert MT(erf(sqrt(x)), x, s) == \
(-gamma(s + S.Half)/(sqrt(pi)*s), (Rational(-1, 2), 0), True)
def test_mellin_transform2():
MT = mellin_transform
# TODO we cannot currently do these (needs summation of 3F2(-1))
# this also implies that they cannot be written as a single g-function
# (although this is possible)
mt = MT(log(x)/(x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)**2/(x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)/(x + 1)**2, x, s)
assert mt[1:] == ((0, 2), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
@slow
def test_mellin_transform_bessel():
from sympy import Max
MT = mellin_transform
# 8.4.19
assert MT(besselj(a, 2*sqrt(x)), x, s) == \
(gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, Rational(3, 4)), True)
assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(-2*s + S.Half)*gamma(a/2 + s + S.Half)/(
gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), (
-re(a)/2 - S.Half, Rational(1, 4)), True)
assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(a/2 + s)*gamma(-2*s + S.Half)/(
gamma(-a/2 - s + S.Half)*gamma(a - 2*s + 1)), (
-re(a)/2, Rational(1, 4)), True)
assert MT(besselj(a, sqrt(x))**2, x, s) == \
(gamma(a + s)*gamma(S.Half - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
(-re(a), S.Half), True)
assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
(gamma(s)*gamma(S.Half - s)
/ (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
(0, S.Half), True)
# NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as
# I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large)
assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(gamma(1 - s)*gamma(a + s - S.Half)
/ (sqrt(pi)*gamma(Rational(3, 2) - s)*gamma(a - s + S.Half)),
(S.Half - re(a), S.Half), True)
assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \
(4**s*gamma(1 - 2*s)*gamma((a + b)/2 + s)
/ (gamma(1 - s + (b - a)/2)*gamma(1 - s + (a - b)/2)
*gamma( 1 - s + (a + b)/2)),
(-(re(a) + re(b))/2, S.Half), True)
assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
((Max(re(a), -re(a)), S.Half), True)
# Section 8.4.20
assert MT(bessely(a, 2*sqrt(x)), x, s) == \
(-cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)/pi,
(Max(-re(a)/2, re(a)/2), Rational(3, 4)), True)
assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*sin(pi*(a/2 - s))*gamma(S.Half - 2*s)
* gamma((1 - a)/2 + s)*gamma((1 + a)/2 + s)
/ (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)),
(Max(-(re(a) + 1)/2, (re(a) - 1)/2), Rational(1, 4)), True)
assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)*gamma(S.Half - 2*s)
/ (sqrt(pi)*gamma(S.Half - s - a/2)*gamma(S.Half - s + a/2)),
(Max(-re(a)/2, re(a)/2), Rational(1, 4)), True)
assert MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-cos(pi*s)*gamma(s)*gamma(a + s)*gamma(S.Half - s)
/ (pi**S('3/2')*gamma(1 + a - s)),
(Max(-re(a), 0), S.Half), True)
assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \
(-4**s*cos(pi*(a/2 - b/2 + s))*gamma(1 - 2*s)
* 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)),
(Max((-re(a) + re(b))/2, (-re(a) - re(b))/2), S.Half), True)
# NOTE bessely(a, sqrt(x))**2 and bessely(a, sqrt(x))*bessely(b, sqrt(x))
# are a mess (no matter what way you look at it ...)
assert MT(bessely(a, sqrt(x))**2, x, s)[1:] == \
((Max(-re(a), 0, re(a)), S.Half), True)
# Section 8.4.22
# TODO we can't do any of these (delicate cancellation)
# Section 8.4.23
assert MT(besselk(a, 2*sqrt(x)), x, s) == \
(gamma(
s - a/2)*gamma(s + a/2)/2, (Max(-re(a)/2, re(a)/2), oo), True)
assert MT(besselj(a, 2*sqrt(2*sqrt(x)))*besselk(
a, 2*sqrt(2*sqrt(x))), x, s) == (4**(-s)*gamma(2*s)*
gamma(a/2 + s)/(2*gamma(a/2 - s + 1)), (Max(0, -re(a)/2), oo), True)
# TODO bessely(a, x)*besselk(a, x) is a mess
assert MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
(gamma(s)*gamma(
a + s)*gamma(-s + S.Half)/(2*sqrt(pi)*gamma(a - s + 1)),
(Max(-re(a), 0), S.Half), True)
assert MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
(2**(2*s - 1)*gamma(-2*s + 1)*gamma(-a/2 + b/2 + s)* \
gamma(a/2 + b/2 + s)/(gamma(-a/2 + b/2 - s + 1)* \
gamma(a/2 + b/2 - s + 1)), (Max(-re(a)/2 - re(b)/2, \
re(a)/2 - re(b)/2), S.Half), True)
# TODO products of besselk are a mess
mt = MT(exp(-x/2)*besselk(a, x/2), x, s)
mt0 = gammasimp(trigsimp(gammasimp(mt[0].expand(func=True))))
assert mt0 == 2*pi**Rational(3, 2)*cos(pi*s)*gamma(-s + S.Half)/(
(cos(2*pi*a) - cos(2*pi*s))*gamma(-a - s + 1)*gamma(a - s + 1))
assert mt[1:] == ((Max(-re(a), re(a)), oo), True)
# TODO exp(x/2)*besselk(a, x/2) [etc] cannot currently be done
# TODO various strange products of special orders
@slow
def test_expint():
from sympy import E1, expint, Max, re, lerchphi, Symbol, simplify, Si, Ci, Ei
aneg = Symbol('a', negative=True)
u = Symbol('u', polar=True)
assert mellin_transform(E1(x), x, s) == (gamma(s)/s, (0, oo), True)
assert inverse_mellin_transform(gamma(s)/s, s, x,
(0, oo)).rewrite(expint).expand() == E1(x)
assert mellin_transform(expint(a, x), x, s) == \
(gamma(s)/(a + s - 1), (Max(1 - re(a), 0), oo), True)
# XXX IMT has hickups with complicated strips ...
assert simplify(unpolarify(
inverse_mellin_transform(gamma(s)/(aneg + s - 1), s, x,
(1 - aneg, oo)).rewrite(expint).expand(func=True))) == \
expint(aneg, x)
assert mellin_transform(Si(x), x, s) == \
(-2**s*sqrt(pi)*gamma(s/2 + S.Half)/(
2*s*gamma(-s/2 + 1)), (-1, 0), True)
assert inverse_mellin_transform(-2**s*sqrt(pi)*gamma((s + 1)/2)
/(2*s*gamma(-s/2 + 1)), s, x, (-1, 0)) \
== Si(x)
assert mellin_transform(Ci(sqrt(x)), x, s) == \
(-2**(2*s - 1)*sqrt(pi)*gamma(s)/(s*gamma(-s + S.Half)), (0, 1), True)
assert inverse_mellin_transform(
-4**s*sqrt(pi)*gamma(s)/(2*s*gamma(-s + S.Half)),
s, u, (0, 1)).expand() == Ci(sqrt(u))
# 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) == \
(lerchphi(s*exp_polar(I*pi), 1, a), 0, re(a) > S.Zero)
assert laplace_transform(expint(1, x), x, s) == (log(s + 1)/s, 0, True)
assert laplace_transform(expint(2, x), x, s) == \
((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).rewrite(expint) == \
Heaviside(x)*E1(x)
assert inverse_laplace_transform((s - log(s + 1))/s**2, s,
x).rewrite(expint).expand() == \
(expint(2, x)*Heaviside(x)).rewrite(Ei).rewrite(expint).expand()
@slow
def test_inverse_mellin_transform():
from sympy import (sin, simplify, Max, Min, expand,
powsimp, exp_polar, cos, cot)
IMT = inverse_mellin_transform
assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
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 import expand, logcombine, powsimp
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 fresnels, fresnelc, DiracDelta
LT = laplace_transform
a, b, c, = symbols('a b c', positive=True)
t = symbols('t')
w = Symbol("w")
f = Function("f")
# Test unevaluated form
assert laplace_transform(f(t), t, w) == LaplaceTransform(f(t), t, w)
assert inverse_laplace_transform(
f(w), w, t, plane=0) == InverseLaplaceTransform(f(w), w, t, 0)
# test a bug
spos = symbols('s', positive=True)
assert LT(exp(t), t, spos)[:2] == (1/(spos - 1), 1)
# 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(t**a, t, s) == (s**(-a - 1)*gamma(a + 1), 0, True)
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(1 - exp(-a*t), t, s) == (a/(s*(a + s)), 0, True)
assert LT((exp(2*t) - 1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \
== exp(-b)/(s**2 - 1)
assert LT(exp(t), t, s)[:2] == (1/(s - 1), 1)
assert LT(exp(2*t), t, s)[:2] == (1/(s - 2), 2)
assert LT(exp(a*t), t, s)[:2] == (1/(s - a), a)
assert LT(log(t/a), t, s) == ((log(a*s) + EulerGamma)/s/-1, 0, True)
assert LT(erf(t), t, s) == (erfc(s/2)*exp(s**2/4)/s, 0, True)
assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True)
assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True)
# TODO would be nice to have these come out better
assert LT(exp(-a*t)*sin(b*t), t, s) == (b/(b**2 + (a + s)**2), -a, True)
assert LT(exp(-a*t)*cos(b*t), t, s) == \
((a + s)/(b**2 + (a + s)**2), -a, True)
assert LT(besselj(0, t), t, s) == (1/sqrt(1 + s**2), 0, True)
assert LT(besselj(1, t), t, s) == (1 - 1/sqrt(1 + 1/s**2), 0, True)
# TODO general order works, but is a *mess*
# TODO besseli also works, but is an even greater mess
# test a bug in conditions processing
# TODO the auxiliary condition should be recognised/simplified
assert LT(exp(t)*cos(t), t, s)[:-1] in [
((s - 1)/(s**2 - 2*s + 2), -oo),
((s - 1)/((s - 1)**2 + 1), -oo),
]
# 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) == (-a/(a + s) + 1, 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, -oo, 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))
# What is this testing:
Ne(1/s, 1) & (0 < cos(Abs(periodic_argument(s, oo)))*Abs(s) - 1)
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, s > 1), ((s + 1)**(-2), 0, True)],
[((s + 1)**(-2), 0, True), (1/(s - 1), 1, s > 1)]])
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, s > 1)
# 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_issue_8368_7173():
LT = laplace_transform
# hyperbolic
assert LT(sinh(x), x, s) == (1/(s**2 - 1), 1, s > 1)
assert LT(cosh(x), x, s) == (s/(s**2 - 1), 1, s > 1)
assert LT(sinh(x + 3), x, s) == (
(-s + (s + 1)*exp(6) + 1)*exp(-3)/(s - 1)/(s + 1)/2, 1, s > 1)
assert LT(sinh(x)*cosh(x), x, s) == (
1/(s**2 - 4), 2, s > 2)
# trig (make sure they are not being rewritten in terms of exp)
assert LT(cos(x + 3), x, s) == ((s*cos(3) - sin(3))/(s**2 + 1), 0, True)
@slow
def test_inverse_laplace_transform():
from sympy import sinh, cosh, besselj, besseli, simplify, factor_terms,\
DiracDelta
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 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 import DiracDelta, Eq, im, Heaviside
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 import simplify, expand, expand_complex, factor, expand_trig
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)
# TODO IFT (comes out as meijer G)
# TODO besselj(n, x), n an integer > 0 actually can be done...
# TODO are there other common transforms (no distributions!)?
def test_sine_transform():
from sympy import EulerGamma
t = symbols("t")
w = symbols("w")
a = symbols("a")
f = Function("f")
# Test unevaluated form
assert sine_transform(f(t), t, w) == SineTransform(f(t), t, w)
assert inverse_sine_transform(
f(w), w, t) == InverseSineTransform(f(w), w, t)
assert sine_transform(1/sqrt(t), t, w) == 1/sqrt(w)
assert inverse_sine_transform(1/sqrt(w), w, t) == 1/sqrt(t)
assert sine_transform((1/sqrt(t))**3, t, w) == 2*sqrt(w)
assert sine_transform(t**(-a), t, w) == 2**(
-a + S.Half)*w**(a - 1)*gamma(-a/2 + 1)/gamma((a + 1)/2)
assert inverse_sine_transform(2**(-a + S(
1)/2)*w**(a - 1)*gamma(-a/2 + 1)/gamma(a/2 + S.Half), w, t) == t**(-a)
assert sine_transform(
exp(-a*t), t, w) == sqrt(2)*w/(sqrt(pi)*(a**2 + w**2))
assert inverse_sine_transform(
sqrt(2)*w/(sqrt(pi)*(a**2 + w**2)), w, t) == exp(-a*t)
assert sine_transform(
log(t)/t, t, w) == -sqrt(2)*sqrt(pi)*(log(w**2) + 2*EulerGamma)/4
assert sine_transform(
t*exp(-a*t**2), t, w) == sqrt(2)*w*exp(-w**2/(4*a))/(4*a**Rational(3, 2))
assert inverse_sine_transform(
sqrt(2)*w*exp(-w**2/(4*a))/(4*a**Rational(3, 2)), w, t) == t*exp(-a*t**2)
def test_cosine_transform():
from sympy import Si, Ci
t = symbols("t")
w = symbols("w")
a = symbols("a")
f = Function("f")
# Test unevaluated form
assert cosine_transform(f(t), t, w) == CosineTransform(f(t), t, w)
assert inverse_cosine_transform(
f(w), w, t) == InverseCosineTransform(f(w), w, t)
assert cosine_transform(1/sqrt(t), t, w) == 1/sqrt(w)
assert inverse_cosine_transform(1/sqrt(w), w, t) == 1/sqrt(t)
assert cosine_transform(1/(
a**2 + t**2), t, w) == sqrt(2)*sqrt(pi)*exp(-a*w)/(2*a)
assert cosine_transform(t**(
-a), t, w) == 2**(-a + S.Half)*w**(a - 1)*gamma((-a + 1)/2)/gamma(a/2)
assert inverse_cosine_transform(2**(-a + S(
1)/2)*w**(a - 1)*gamma(-a/2 + S.Half)/gamma(a/2), w, t) == t**(-a)
assert cosine_transform(
exp(-a*t), t, w) == sqrt(2)*a/(sqrt(pi)*(a**2 + w**2))
assert inverse_cosine_transform(
sqrt(2)*a/(sqrt(pi)*(a**2 + w**2)), w, t) == exp(-a*t)
assert cosine_transform(exp(-a*sqrt(t))*cos(a*sqrt(
t)), t, w) == a*exp(-a**2/(2*w))/(2*w**Rational(3, 2))
assert cosine_transform(1/(a + t), t, w) == sqrt(2)*(
(-2*Si(a*w) + pi)*sin(a*w)/2 - cos(a*w)*Ci(a*w))/sqrt(pi)
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)
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))
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))
def test_hankel_transform():
from sympy import gamma, sqrt, exp
r = Symbol("r")
k = Symbol("k")
nu = Symbol("nu")
m = Symbol("m")
a = symbols("a")
assert hankel_transform(1/r, r, k, 0) == 1/k
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_7173():
from sympy import cse
x0, x1, x2, x3 = symbols('x:4')
ans = laplace_transform(sinh(a*x)*cosh(a*x), x, s)
r, e = cse(ans)
assert r == [
(x0, arg(a)),
(x1, Abs(x0)),
(x2, pi/2),
(x3, Abs(x0 + pi))]
assert e == [
a/(-4*a**2 + s**2),
0,
((x1 <= x2) | (x1 < x2)) & ((x3 <= x2) | (x3 < x2))]
def test_issue_8514():
from sympy 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)
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