def test_cosine_transform():
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 + Rational(1, 2)) * w**(a - 1) * gamma(
(-a + 1) / 2) / gamma(a / 2)
assert inverse_cosine_transform(
2**(-a + Rational(1, 2)) * w**(a - 1) *
gamma(-a / 2 + Rational(1, 2)) / 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(((Rational(1, 2), 0), ()),
((Rational(1, 2), 0, 0),
(Rational(1, 2), )), 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(
((Rational(1, 2), ), ()),
((0, 0), (Rational(1, 2), )), a**2 * w**2 / 4) / (2 * sqrt(pi))
assert inverse_cosine_transform(
sqrt(2) * meijerg(
((Rational(1, 2), ), ()),
((0, 0), (Rational(1, 2), )), a**2 * w**2 / 4) / (2 * sqrt(pi)), w,
t) == 1 / (a * sqrt(1 + t**2 / a**2))
def test_besselsimp():
assert besselsimp(exp(-I*pi*y/2)*besseli(y, z*exp_polar(I*pi/2))) == \
besselj(y, z)
assert besselsimp(exp(-I*pi*a/2)*besseli(a, 2*sqrt(x)*exp_polar(I*pi/2))) == \
besselj(a, 2*sqrt(x))
assert besselsimp(sqrt(2)*sqrt(pi)*root(x, 4)*exp(I*pi/4)*exp(-I*pi*a/2) *
besseli(-Rational(1, 2), sqrt(x)*exp_polar(I*pi/2)) *
besseli(a, sqrt(x)*exp_polar(I*pi/2))/2) == \
besselj(a, sqrt(x)) * cos(sqrt(x))
assert besselsimp(besseli(Rational(-1, 2), z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(a, z*exp_polar(-I*pi/2))) == \
exp(-I*pi*a/2)*besselj(a, z)
assert cosine_transform(1/t*sin(a/t), t, y) == \
sqrt(2)*sqrt(pi)*besselj(0, 2*sqrt(a)*sqrt(y))/2