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,
    cos, S, Abs, And, Or, sin, sqrt, I, log, tan, hyperexpand, meijerg,
    EulerGamma, erf, erfc, besselj, bessely, besseli, besselk,
    exp_polar, polar_lift, unpolarify, Function, expint, expand_mul,
    combsimp, trigsimp, atan, sinh, cosh, Ne, periodic_argument, atan2, Abs)
from sympy.utilities.pytest import XFAIL, slow, skip, raises
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 == set([s])
    assert mellin_transform(f(x)*a, x, s).free_symbols == set([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(inverse_mellin_transform(f(s), s, x, (a, b)).rewrite('Integral')) \
        == "Integral(x**(-s)*f(s), (s, _c - oo*I, _c + oo*I))"
    assert str(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(1)/2), 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(1)/2)/(2*sqrt(pi)),
            (-1, -S(1)/2), True)


@slow
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
    assert MT(abs(1 - x)**(-rho), x, s) == (
        cos(pi*(rho/2 - s))*gamma(s)*gamma(rho - s)/(cos(pi*rho/2)*gamma(rho)),
        (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), True)

    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(1)/2), 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)

    # 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)

    # 8.4.14
    assert MT(erf(sqrt(x)), x, s) == \
        (-gamma(s + S(1)/2)/(sqrt(pi)*s), (-S(1)/2, 0), True)


@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, S(3)/4), True)
    assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
        (2**a*gamma(-2*s + S(1)/2)*gamma(a/2 + s + S(1)/2)/(
        gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), (
        -re(a)/2 - S(1)/2, S(1)/4), True)
    assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
        (2**a*gamma(a/2 + s)*gamma(-2*s + S(1)/2)/(
        gamma(-a/2 - s + S(1)/2)*gamma(a - 2*s + 1)), (
        -re(a)/2, S(1)/4), True)
    assert MT(besselj(a, sqrt(x))**2, x, s) == \
        (gamma(a + s)*gamma(S(1)/2 - s)
         / (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
            (-re(a), S(1)/2), True)
    assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
        (gamma(s)*gamma(S(1)/2 - s)
         / (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
            (0, S(1)/2), 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(1)/2)
         / (sqrt(pi)*gamma(S(3)/2 - s)*gamma(a - s + S(1)/2)),
            (S(1)/2 - re(a), S(1)/2), 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(1)/2), True)
    assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
        ((Max(re(a), -re(a)), S(1)/2), 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), S(3)/4), True)
    assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
        (-4**s*sin(pi*(a/2 - s))*gamma(S(1)/2 - 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), S(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(1)/2 - 2*s)
         / (sqrt(pi)*gamma(S(1)/2 - s - a/2)*gamma(S(1)/2 - s + a/2)),
            (Max(-re(a)/2, re(a)/2), S(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(1)/2 - s)
         / (pi**S('3/2')*gamma(1 + a - s)),
            (Max(-re(a), 0), S(1)/2), 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(1)/2), 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(1)/2), 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(1)/2)/(2*sqrt(pi)*gamma(a - s + 1)),
        (Max(-re(a), 0), S(1)/2), 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(1)/2), True)

    # TODO products of besselk are a mess

    mt = MT(exp(-x/2)*besselk(a, x/2), x, s)
    mt0 = combsimp((trigsimp(combsimp(mt[0].expand(func=True)))))
    assert mt0 == 2*pi**(S(3)/2)*cos(pi*s)*gamma(-s + S(1)/2)/(
        (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(1)/2)/(
        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(1)/2)), (0, 1), True)
    assert inverse_mellin_transform(
        -4**s*sqrt(pi)*gamma(s)/(2*s*gamma(-s + S(1)/2)),
        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*polar_lift(-1), 1, a), 0, S(0) < re(a))
    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, finite=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, (-S(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(1)/2)/(sqrt(pi)*s), s, x, (-S(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, S(3)/4))) \
        == besselj(a, 2*sqrt(x))
    assert simplify(IMT(2**a*gamma(S(1)/2 - 2*s)*gamma(s + (a + 1)/2)
                      / (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
                      s, x, (-(re(a) + 1)/2, S(1)/4))) == \
        sin(sqrt(x))*besselj(a, sqrt(x))
    assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(S(1)/2 - 2*s)
                      / (gamma(S(1)/2 - s - a/2)*gamma(1 - 2*s + a)),
                      s, x, (-re(a)/2, S(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(1)/2 - s)
                      / (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
                      s, x, (-re(a), S(1)/2))) == \
        besselj(a, sqrt(x))**2
    assert simplify(IMT(gamma(s)*gamma(S(1)/2 - s)
                      / (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
                      s, x, (0, S(1)/2))) == \
        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(1)/2))) == \
        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(1)/2))) == \
                    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(1)/2)) == sqrt(x)/(x + 1)


@slow
def test_laplace_transform():
    from sympy import fresnels, fresnelc
    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), True)

    # 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),
    ]

    # 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) == (
        (sin(s**2/(2*pi))*fresnelc(s/pi)/s - cos(s**2/(2*pi))*fresnels(s/pi)/s
        + sqrt(2)*cos(s**2/(2*pi) + pi/4)/(2*s), 0, True))

    assert LT(Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]]), t, s) ==\
        Matrix([
            [(1/(s - 1), 1, True), ((s + 1)**(-2), 0, True)],
            [((s + 1)**(-2), 0, True), (1/(s - 1), 1, True)]
        ])


def test_issue_8368_7173():
    LT = laplace_transform
    # hyperbolic
    assert LT(sinh(x), x, s) == (1/(s**2 - 1), 1, True)
    assert LT(cosh(x), x, s) == (s/(s**2 - 1), 1, True)
    assert LT(sinh(x + 3), x, s) == (
        (-s + (s + 1)*exp(6) + 1)*exp(-3)/(s - 1)/(s + 1)/2, 1, True)
    assert LT(sinh(x)*cosh(x), x, s) == (1/(s**2 - 4), 2, Ne(s/2, 1))

    # 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)


def test_inverse_laplace_transform():
    from sympy import sinh, cosh, besselj, besseli, simplify, factor_terms
    ILT = inverse_laplace_transform
    a, b, c, = symbols('a b c', positive=True, finite=True)
    t = symbols('t')

    def simp_hyp(expr):
        return factor_terms(expand_mul(expr)).rewrite(sin)

    # just test inverses of all of the above
    assert ILT(1/s, s, t) == Heaviside(t)
    assert ILT(1/s**2, s, t) == t*Heaviside(t)
    assert ILT(1/s**5, s, t) == t**4*Heaviside(t)/24
    assert ILT(exp(-a*s)/s, s, t) == Heaviside(t - a)
    assert ILT(exp(-a*s)/(s + b), s, t) == exp(b*(a - t))*Heaviside(-a + t)
    assert ILT(a/(s**2 + a**2), s, t) == sin(a*t)*Heaviside(t)
    assert ILT(s/(s**2 + a**2), s, t) == cos(a*t)*Heaviside(t)
    # TODO is there a way around simp_hyp?
    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)
    assert ILT(a/((s + b)**2 + a**2), s, t) == exp(-b*t)*sin(a*t)*Heaviside(t)
    assert ILT(
        (s + b)/((s + b)**2 + a**2), s, t) == exp(-b*t)*cos(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_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) == sqrt(w)*gamma(S(1)/4)/(2*gamma(S(5)/4))

    assert sine_transform(t**(-a), t, w) == 2**(
        -a + S(1)/2)*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(1)/2), 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**(S(3)/2))
    assert inverse_sine_transform(
        sqrt(2)*w*exp(-w**2/(4*a))/(4*a**(S(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(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) == exp(-a*t)

    assert cosine_transform(exp(-a*sqrt(t))*cos(a*sqrt(
        t)), t, w) == a*exp(-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))/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_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 + S(3)/2)/sqrt(pi)
    assert inverse_hankel_transform(
        2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(3)/2)*gamma(
        nu + S(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():
    assert laplace_transform(sinh(a*x)*cosh(a*x), x, s) == \
        (a/(s**2 - 4*a**2), 0,
        And(Or(Abs(periodic_argument(exp_polar(I*pi)*polar_lift(a), oo)) <
        pi/2, Abs(periodic_argument(exp_polar(I*pi)*polar_lift(a), oo)) <=
        pi/2), Or(Abs(periodic_argument(a, oo)) < pi/2,
        Abs(periodic_argument(a, oo)) <= pi/2)))


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 == ((exp(t*(exp(I*atan2(0, -4*a*c + b**2)/2) -
                  exp(-I*atan2(0, -4*a*c + b**2)/2))*
                  sqrt(Abs(4*a*c - b**2))/(4*a))*exp(t*cos(atan2(0, -4*a*c + b**2)/2)
                  *sqrt(Abs(4*a*c - b**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))
