def test_cosine_transform():
from sympy import sinh, cosh, 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)*(-sinh(a*w) + cosh(a*w))/(2*a)
assert cosine_transform(t**(-a), t, w) == 2**(-a + S(1)/2)*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(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) == -sinh(a*t) + cosh(a*t)
assert cosine_transform(exp(-a*sqrt(t))*cos(a*sqrt(t)), t, w) == a*(-sinh(a**2/(2*w)) + cosh(a**2/(2*w)))/(2*w**(S(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))/(2*sqrt(pi))
assert inverse_cosine_transform(sqrt(2)*meijerg(((S(1)/2, 0), ()), ((S(1)/2, 0, 0), (S(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(((S(1)/2,), ()), ((0, 0), (S(1)/2,)), a**2*w**2/4)/(2*sqrt(pi))
assert inverse_cosine_transform(sqrt(2)*meijerg(((S(1)/2,), ()), ((0, 0), (S(1)/2,)), a**2*w**2/4)/(2*sqrt(pi)), w, t) == 1/(t*sqrt(a**2/t**2 + 1))
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_cosine_transform():
from sympy import sinh, cosh, 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) * (-sinh(a * w) + cosh(a * w)) / (2 * a)
assert cosine_transform(t**(-a), t,
w) == 2**(-a + S(1) / 2) * 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(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) == -sinh(a * t) + cosh(a * t)
assert cosine_transform(
exp(-a * sqrt(t)) * cos(a * sqrt(t)), t,
w) == a * (-sinh(a**2 / (2 * w)) + cosh(a**2 /
(2 * w))) / (2 * w**(S(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)) / (2 * sqrt(pi))
assert inverse_cosine_transform(
sqrt(2) * meijerg(
((S(1) / 2, 0), ()),
((S(1) / 2, 0, 0), (S(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(
((S(1) / 2, ), ()),
((0, 0), (S(1) / 2, )), a**2 * w**2 / 4) / (2 * sqrt(pi))
assert inverse_cosine_transform(
sqrt(2) * meijerg(
((S(1) / 2, ), ()),
((0, 0), (S(1) / 2, )), a**2 * w**2 / 4) / (2 * sqrt(pi)), w,
t) == 1 / (t * sqrt(a**2 / t**2 + 1))
