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)
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_sympy__integrals__transforms__FourierTransform():
from sympy.integrals.transforms import FourierTransform
assert _test_args(FourierTransform(2, x, y))
