def test_diff():
    assert besselj(n, z).diff(z) == besselj(n - 1, z)/2 - besselj(n + 1, z)/2
    assert bessely(n, z).diff(z) == bessely(n - 1, z)/2 - bessely(n + 1, z)/2
    assert besseli(n, z).diff(z) == besseli(n - 1, z)/2 + besseli(n + 1, z)/2
    assert besselk(n, z).diff(z) == -besselk(n - 1, z)/2 - besselk(n + 1, z)/2
    assert hankel1(n, z).diff(z) == hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2
    assert hankel2(n, z).diff(z) == hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2

def test_rewrite():
    from sympy import polar_lift, exp, I

    assert besselj(n, z).rewrite(jn) == sqrt(2*z/pi)*jn(n - S(1)/2, z)
    assert bessely(n, z).rewrite(yn) == sqrt(2*z/pi)*yn(n - S(1)/2, z)
    assert besseli(n, z).rewrite(besselj) == \
        exp(-I*n*pi/2)*besselj(n, polar_lift(I)*z)
    assert besselj(n, z).rewrite(besseli) == \
        exp(I*n*pi/2)*besseli(n, polar_lift(-I)*z)

    nu = randcplx()

    assert tn(besselj(nu, z), besselj(nu, z).rewrite(besseli), z)
    assert tn(besselj(nu, z), besselj(nu, z).rewrite(bessely), z)

    assert tn(besseli(nu, z), besseli(nu, z).rewrite(besselj), z)
    assert tn(besseli(nu, z), besseli(nu, z).rewrite(bessely), z)

    assert tn(bessely(nu, z), bessely(nu, z).rewrite(besselj), z)
    assert tn(bessely(nu, z), bessely(nu, z).rewrite(besseli), z)

    assert tn(besselk(nu, z), besselk(nu, z).rewrite(besselj), z)
    assert tn(besselk(nu, z), besselk(nu, z).rewrite(besseli), z)
    assert tn(besselk(nu, z), besselk(nu, z).rewrite(bessely), z)

    # check that a rewrite was triggered, when the order is set to a generic
    # symbol 'nu'
    assert yn(nu, z) != yn(nu, z).rewrite(jn)
    assert hn1(nu, z) != hn1(nu, z).rewrite(jn)
    assert hn2(nu, z) != hn2(nu, z).rewrite(jn)
    assert jn(nu, z) != jn(nu, z).rewrite(yn)
    assert hn1(nu, z) != hn1(nu, z).rewrite(yn)
    assert hn2(nu, z) != hn2(nu, z).rewrite(yn)

    # rewriting spherical bessel functions (SBFs) w.r.t. besselj, bessely is
    # not allowed if a generic symbol 'nu' is used as the order of the SBFs
    # to avoid inconsistencies (the order of bessel[jy] is allowed to be
    # complex-valued, whereas SBFs are defined only for integer orders)
    order = nu
    for f in (besselj, bessely):
        assert yn(order, z) == yn(order, z).rewrite(f)
        assert jn(order, z) == jn(order, z).rewrite(f)
        assert hn1(order, z) == hn1(order, z).rewrite(f)
        assert hn2(order, z) == hn2(order, z).rewrite(f)

    # for integral orders rewriting SBFs w.r.t bessel[jy] is allowed
    N = Symbol('n', integer=True)
    ri = randint(-11, 10)
    for order in (ri, N):
        for f in (besselj, bessely):
            assert yn(order, z) != yn(order, z).rewrite(f)
            assert jn(order, z) != jn(order, z).rewrite(f)
            assert hn1(order, z) != hn1(order, z).rewrite(f)
            assert hn2(order, z) != hn2(order, z).rewrite(f)

    for func, refunc in product((yn, jn, hn1, hn2),
                                (jn, yn, besselj, bessely)):
        assert tn(func(ri, z), func(ri, z).rewrite(refunc), z)

def test_bessel_eval():
    from sympy import I
    assert besselj(-4, z) == besselj(4, z)
    assert besselj(-3, z) == -besselj(3, z)

    assert bessely(-2, z) == bessely(2, z)
    assert bessely(-1, z) == -bessely(1, z)

    assert besselj(0, -z) == besselj(0, z)
    assert besselj(1, -z) == -besselj(1, z)

    assert besseli(0, I*z) == besselj(0, z)
    assert besseli(1, I*z) == I*besselj(1, z)
    assert besselj(3, I*z) == -I*besseli(3, z)

def test_pmint_bessel_products():
    # Note: Derivatives of Bessel functions have many forms.
    # Recurrence relations are needed for comparisons.
    if ON_TRAVIS:
        skip("Too slow for travis.")

    f = x*besselj(nu, x)*bessely(nu, 2*x)
    g = -2*x*besselj(nu, x)*bessely(nu - 1, 2*x)/3 + x*besselj(nu - 1, x)*bessely(nu, 2*x)/3

    assert heurisch(f, x) == g

    f = x*besselj(nu, x)*besselk(nu, 2*x)
    g = -2*x*besselj(nu, x)*besselk(nu - 1, 2*x)/5 - x*besselj(nu - 1, x)*besselk(nu, 2*x)/5

    assert heurisch(f, x) == g

def test_hyper():
    for x in sorted(exparg):
        test("erf", x, N(sp.erf(x)))
    for x in sorted(exparg):
        test("erfc", x, N(sp.erfc(x)))

    gamarg = FiniteSet(*(x+S(1)/12 for x in exparg))
    betarg = ProductSet(gamarg, gamarg)
    for x in sorted(gamarg):
        test("lgamma", x, N(sp.log(abs(sp.gamma(x)))))
    for x in sorted(gamarg):
        test("gamma", x, N(sp.gamma(x)))
    for x, y in sorted(betarg, key=lambda (x, y): (y, x)):
        test("beta", x, y, N(sp.beta(x, y)))

    pgamarg = FiniteSet(S(1)/12, S(1)/3, S(3)/2, 5)
    pgamargp = ProductSet(gamarg & Interval(0, oo, True), pgamarg)
    for a, x in sorted(pgamargp):
        test("pgamma", a, x, N(sp.lowergamma(a, x)))
    for a, x in sorted(pgamargp):
        test("pgammac", a, x, N(sp.uppergamma(a, x)))
    for a, x in sorted(pgamargp):
        test("pgammar", a, x, N(sp.lowergamma(a, x)/sp.gamma(a)))
    for a, x in sorted(pgamargp):
        test("pgammarc", a, x, N(sp.uppergamma(a, x)/sp.gamma(a)))
    for a, x in sorted(pgamargp):
        test("ipgammarc", a, N(sp.uppergamma(a, x)/sp.gamma(a)), x)

    pbetargp = [(a, b, x) for a, b, x in ProductSet(betarg, pgamarg)
                if a > 0 and b > 0 and x < 1]
    pbetargp.sort(key=lambda (a, b, x): (b, a, x))
    for a, b, x in pbetargp:
        test("pbeta", a, b, x, mp.betainc(mpf(a), mpf(b), x2=mpf(x)))
    for a, b, x in pbetargp:
        test("pbetar", a, b, x, mp.betainc(mpf(a), mpf(b), x2=mpf(x),
                                           regularized=True))
    for a, b, x in pbetargp:
        test("ipbetar", a, b, mp.betainc(mpf(a), mpf(b), x2=mpf(x),
                                         regularized=True), x)

    for x in sorted(posarg):
        test("j0", x, N(sp.besselj(0, x)))
    for x in sorted(posarg):
        test("j1", x, N(sp.besselj(1, x)))
    for x in sorted(posarg-FiniteSet(0)):
        test("y0", x, N(sp.bessely(0, x)))
    for x in sorted(posarg-FiniteSet(0)):
        test("y1", x, N(sp.bessely(1, x)))

def test_bessel_rand():
    assert td(besselj(randcplx(), z), z)
    assert td(bessely(randcplx(), z), z)
    assert td(besseli(randcplx(), z), z)
    assert td(besselk(randcplx(), z), z)
    assert td(hankel1(randcplx(), z), z)
    assert td(hankel2(randcplx(), z), z)
    assert td(jn(randcplx(), z), z)
    assert td(yn(randcplx(), z), z)

def test_rewrite():
    from sympy import polar_lift, exp, I
    assert besselj(n, z).rewrite(jn) == sqrt(2*z/pi)*jn(n - S(1)/2, z)
    assert bessely(n, z).rewrite(yn) == sqrt(2*z/pi)*yn(n - S(1)/2, z)
    assert besseli(n, z).rewrite(besselj) == \
        exp(-I*n*pi/2)*besselj(n, polar_lift(I)*z)
    assert besselj(n, z).rewrite(besseli) == \
        exp(I*n*pi/2)*besseli(n, polar_lift(-I)*z)
    nu = randcplx()
    assert tn(besselj(nu, z), besselj(nu, z).rewrite(besseli), z)
    assert tn(besseli(nu, z), besseli(nu, z).rewrite(besselj), z)

def test_bessel_eval():
    from sympy import I, Symbol
    n, m, k = Symbol('n', integer=True), Symbol('m'), Symbol('k', integer=True, zero=False)

    for f in [besselj, besseli]:
        assert f(0, 0) == S.One
        assert f(2.1, 0) == S.Zero
        assert f(-3, 0) == S.Zero
        assert f(-10.2, 0) == S.ComplexInfinity
        assert f(1 + 3*I, 0) == S.Zero
        assert f(-3 + I, 0) == S.ComplexInfinity
        assert f(-2*I, 0) == S.NaN
        assert f(n, 0) != S.One and f(n, 0) != S.Zero
        assert f(m, 0) != S.One and f(m, 0) != S.Zero
        assert f(k, 0) == S.Zero

    assert bessely(0, 0) == S.NegativeInfinity
    assert besselk(0, 0) == S.Infinity
    for f in [bessely, besselk]:
        assert f(1 + I, 0) == S.ComplexInfinity
        assert f(I, 0) == S.NaN

    for f in [besselj, bessely]:
        assert f(m, S.Infinity) == S.Zero
        assert f(m, S.NegativeInfinity) == S.Zero

    for f in [besseli, besselk]:
        assert f(m, I*S.Infinity) == S.Zero
        assert f(m, I*S.NegativeInfinity) == S.Zero

    for f in [besseli, besselk]:
        assert f(-4, z) == f(4, z)
        assert f(-3, z) == f(3, z)
        assert f(-n, z) == f(n, z)
        assert f(-m, z) != f(m, z)

    for f in [besselj, bessely]:
        assert f(-4, z) == f(4, z)
        assert f(-3, z) == -f(3, z)
        assert f(-n, z) == (-1)**n*f(n, z)
        assert f(-m, z) != (-1)**m*f(m, z)

    for f in [besselj, besseli]:
        assert f(m, -z) == (-z)**m*z**(-m)*f(m, z)

    assert besseli(2, -z) == besseli(2, z)
    assert besseli(3, -z) == -besseli(3, z)

    assert besselj(0, -z) == besselj(0, z)
    assert besselj(1, -z) == -besselj(1, z)

    assert besseli(0, I*z) == besselj(0, z)
    assert besseli(1, I*z) == I*besselj(1, z)
    assert besselj(3, I*z) == -I*besseli(3, z)

def test_expand():
    assert expand_func(besselj(S(1)/2, z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
    assert expand_func(
        bessely(S(1)/2, z)) == -sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))

def test_rewrite():
    assert besselj(n, z).rewrite(jn) == sqrt(2*z/pi)*jn(n - S(1)/2, z)
    assert bessely(n, z).rewrite(yn) == sqrt(2*z/pi)*yn(n - S(1)/2, z)

def test_mellin_transform_bessel():
    from sympy import Max, Min, hyper, meijerg
    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)

def test_expand():
    from sympy import besselsimp, Symbol, exp, exp_polar, I

    assert expand_func(besselj(S(1)/2, z).rewrite(jn)) == \
        sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
    assert expand_func(bessely(S(1)/2, z).rewrite(yn)) == \
        -sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))

    # XXX: teach sin/cos to work around arguments like
    # x*exp_polar(I*pi*n/2).  Then change besselsimp -> expand_func
    assert besselsimp(besselj(S(1)/2, z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
    assert besselsimp(besselj(S(-1)/2, z)) == sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
    assert besselsimp(besselj(S(5)/2, z)) == \
        -sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**(S(5)/2))
    assert besselsimp(besselj(-S(5)/2, z)) == \
        -sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**(S(5)/2))

    assert besselsimp(bessely(S(1)/2, z)) == \
        -(sqrt(2)*cos(z))/(sqrt(pi)*sqrt(z))
    assert besselsimp(bessely(S(-1)/2, z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
    assert besselsimp(bessely(S(5)/2, z)) == \
        sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**(S(5)/2))
    assert besselsimp(bessely(S(-5)/2, z)) == \
        -sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**(S(5)/2))

    assert besselsimp(besseli(S(1)/2, z)) == sqrt(2)*sinh(z)/(sqrt(pi)*sqrt(z))
    assert besselsimp(besseli(S(-1)/2, z)) == \
        sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
    assert besselsimp(besseli(S(5)/2, z)) == \
        sqrt(2)*(z**2*sinh(z) - 3*z*cosh(z) + 3*sinh(z))/(sqrt(pi)*z**(S(5)/2))
    assert besselsimp(besseli(S(-5)/2, z)) == \
        sqrt(2)*(z**2*cosh(z) - 3*z*sinh(z) + 3*cosh(z))/(sqrt(pi)*z**(S(5)/2))

    assert besselsimp(besselk(S(1)/2, z)) == \
        besselsimp(besselk(S(-1)/2, z)) == sqrt(pi)*exp(-z)/(sqrt(2)*sqrt(z))
    assert besselsimp(besselk(S(5)/2, z)) == \
        besselsimp(besselk(S(-5)/2, z)) == \
        sqrt(2)*sqrt(pi)*(z**2 + 3*z + 3)*exp(-z)/(2*z**(S(5)/2))

    def check(eq, ans):
        return tn(eq, ans) and eq == ans

    rn = randcplx(a=1, b=0, d=0, c=2)

    for besselx in [besselj, bessely, besseli, besselk]:
        ri = S(2*randint(-11, 10) + 1) / 2  # half integer in [-21/2, 21/2]
        assert tn(besselsimp(besselx(ri, z)), besselx(ri, z))

    assert check(expand_func(besseli(rn, x)),
                 besseli(rn - 2, x) - 2*(rn - 1)*besseli(rn - 1, x)/x)
    assert check(expand_func(besseli(-rn, x)),
                 besseli(-rn + 2, x) + 2*(-rn + 1)*besseli(-rn + 1, x)/x)

    assert check(expand_func(besselj(rn, x)),
                 -besselj(rn - 2, x) + 2*(rn - 1)*besselj(rn - 1, x)/x)
    assert check(expand_func(besselj(-rn, x)),
                 -besselj(-rn + 2, x) + 2*(-rn + 1)*besselj(-rn + 1, x)/x)

    assert check(expand_func(besselk(rn, x)),
                 besselk(rn - 2, x) + 2*(rn - 1)*besselk(rn - 1, x)/x)
    assert check(expand_func(besselk(-rn, x)),
                 besselk(-rn + 2, x) - 2*(-rn + 1)*besselk(-rn + 1, x)/x)

    assert check(expand_func(bessely(rn, x)),
                 -bessely(rn - 2, x) + 2*(rn - 1)*bessely(rn - 1, x)/x)
    assert check(expand_func(bessely(-rn, x)),
                 -bessely(-rn + 2, x) + 2*(-rn + 1)*bessely(-rn + 1, x)/x)

    n = Symbol('n', integer=True, positive=True)

    assert expand_func(besseli(n + 2, z)) == \
        besseli(n, z) + (-2*n - 2)*(-2*n*besseli(n, z)/z + besseli(n - 1, z))/z
    assert expand_func(besselj(n + 2, z)) == \
        -besselj(n, z) + (2*n + 2)*(2*n*besselj(n, z)/z - besselj(n - 1, z))/z
    assert expand_func(besselk(n + 2, z)) == \
        besselk(n, z) + (2*n + 2)*(2*n*besselk(n, z)/z + besselk(n - 1, z))/z
    assert expand_func(bessely(n + 2, z)) == \
        -bessely(n, z) + (2*n + 2)*(2*n*bessely(n, z)/z - bessely(n - 1, z))/z

    assert expand_func(besseli(n + S(1)/2, z).rewrite(jn)) == \
        (sqrt(2)*sqrt(z)*exp(-I*pi*(n + S(1)/2)/2) *
         exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi))
    assert expand_func(besselj(n + S(1)/2, z).rewrite(jn)) == \
        sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)

    r = Symbol('r', real=True)
    p = Symbol('p', positive=True)
    i = Symbol('i', integer=True)

    for besselx in [besselj, bessely, besseli, besselk]:
        assert besselx(i, p).is_real
        assert besselx(i, x).is_real is None
        assert besselx(x, z).is_real is None

    for besselx in [besselj, besseli]:
        assert besselx(i, r).is_real
    for besselx in [bessely, besselk]:
        assert besselx(i, r).is_real is None

def test_expand():
    from sympy import besselsimp, Symbol, exp, exp_polar, I

    assert expand_func(besselj(S(1)/2, z).rewrite(jn)) == \
        sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
    assert expand_func(bessely(S(1)/2, z).rewrite(yn)) == \
        -sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))

    # XXX: teach sin/cos to work around arguments like
    # x*exp_polar(I*pi*n/2).  Then change besselsimp -> expand_func
    assert besselsimp(besseli(S(1)/2, z)) == sqrt(2)*sinh(z)/(sqrt(pi)*sqrt(z))

    def check(eq, ans):
        return tn(eq, ans) and eq == ans

    rn = randcplx(a=1, b=0, d=0, c=2)

    assert check(expand_func(besseli(rn, x)), \
        besseli(rn - 2, x) - 2*(rn - 1)*besseli(rn - 1, x)/x)
    assert check(expand_func(besseli(-rn, x)), \
        besseli(-rn + 2, x) + 2*(-rn + 1)*besseli(-rn + 1, x)/x)

    assert check(expand_func(besselj(rn, x)), \
        -besselj(rn - 2, x) + 2*(rn - 1)*besselj(rn - 1, x)/x)
    assert check(expand_func(besselj(-rn, x)), \
        -besselj(-rn + 2, x) + 2*(-rn + 1)*besselj(-rn + 1, x)/x)

    assert check(expand_func(besselk(rn, x)), \
        besselk(rn - 2, x) + 2*(rn - 1)*besselk(rn - 1, x)/x)
    assert check(expand_func(besselk(-rn, x)), \
        besselk(-rn + 2, x) - 2*(-rn + 1)*besselk(-rn + 1, x)/x)

    assert check(expand_func(bessely(rn, x)), \
        -bessely(rn - 2, x) + 2*(rn - 1)*bessely(rn - 1, x)/x)
    assert check(expand_func(bessely(-rn, x)), \
        -bessely(-rn + 2, x) + 2*(-rn + 1)*bessely(-rn + 1, x)/x)

    n = Symbol('n', integer=True, positive=True)

    assert expand_func(besseli(n + 2, z)) == \
        besseli(n, z) + (-2*n - 2)*(-2*n*besseli(n, z)/z + besseli(n - 1, z))/z
    assert expand_func(besselj(n + 2, z)) == \
        -besselj(n, z) + (2*n + 2)*(2*n*besselj(n, z)/z - besselj(n - 1, z))/z
    assert expand_func(besselk(n + 2, z)) == \
        besselk(n, z) + (2*n + 2)*(2*n*besselk(n, z)/z + besselk(n - 1, z))/z
    assert expand_func(bessely(n + 2, z)) == \
        -bessely(n, z) + (2*n + 2)*(2*n*bessely(n, z)/z - bessely(n - 1, z))/z

    assert expand_func(besseli(n + S(1)/2, z).rewrite(jn)) == \
        sqrt(2)*sqrt(z)*exp(-I*pi*(n + S(1)/2)/2)* \
        exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi)
    assert expand_func(besselj(n + S(1)/2, z).rewrite(jn)) == \
        sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)

#.........这里部分代码省略.........
    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)

    # 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(S(1)/2 - 2*s)*gamma((a+1)/2 + s) \
                / (gamma(1 - s- a/2)*gamma(1 + a - 2*s)),
            (-(re(a) + 1)/2, S(1)/4), True)
    assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
           (2**a*gamma(a/2 + s)*gamma(S(1)/2 - 2*s)
                / (gamma(S(1)/2 - s - a/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 - pi*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 - pi*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 - pi*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 - pi*b/2 + pi*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)
    assert MT(bessely(a, sqrt(x))**2, x, s) == \
           ((2*cos(pi*a - pi*s)*gamma(s)*gamma(-a + s)*gamma(-s + 1)*gamma(a - s + 1) \
             + pi*gamma(-s + S(1)/2)*gamma(s + S(1)/2)) \
            *gamma(a + s)/(pi**(S(3)/2)*gamma(-s + 1)*gamma(s + S(1)/2)*gamma(a - s + 1)),
            (Max(-re(a), 0, re(a)), S(1)/2), True)
    # TODO bessely(a, sqrt(x))*bessely(b, sqrt(x)) is a mess
    #      (no matter what way you look at it ...)

    # 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)
    # TODO this result needs expansion of F(a, b; c; 1) using gauss-summation
    assert MT(exp(-x/2)*besselk(a, x/2), x, s)[1:] == \
           ((Max(-re(a), re(a)), oo), True)

def test_mellin_transform_bessel():
    from sympy import Max, Min, hyper, meijerg
    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(S(1)/2 - 2*s)*gamma((a+1)/2 + s) \
                / (gamma(1 - s- a/2)*gamma(1 + a - 2*s)),
            (-(re(a) + 1)/2, S(1)/4), True)
    # TODO why does this 2**(a+2)/4 not cancel?
    assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
           (2**(a+2)*gamma(a/2 + s)*gamma(S(1)/2 - 2*s)
                / (gamma(S(1)/2 - s - a/2)*gamma(a - 2*s + 1)) / 4,
            (-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) == \
           (2**(2*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 - pi*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) == \
           (-2**(2*s)*sin(pi*a/2 - pi*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) == \
           (-2**(2*s)*cos(pi*a/2 - pi*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) == \
           (-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*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)/gamma(a/2 - s + 1)/2,
            (Max(-re(a)/2, 0), 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

    # TODO this can be simplified considerably (although I have no idea how)
    mt = MT(exp(-x/2)*besselk(a, x/2), x, s)
    assert not mt[0].has(meijerg, hyper)
    assert mt[1:] == ((Max(-re(a), re(a)), oo), True)