Python sympy.besseli函数代码示例

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_airybiprime():
    z = Symbol('z', real=False)
    t = Symbol('t', negative=True)
    p = Symbol('p', positive=True)

    assert isinstance(airybiprime(z), airybiprime)

    assert airybiprime(0) == 3**(S(1)/6)/gamma(S(1)/3)
    assert airybiprime(oo) == oo
    assert airybiprime(-oo) == 0

    assert diff(airybiprime(z), z) == z*airybi(z)

    assert series(airybiprime(z), z, 0, 3) == (
        3**(S(1)/6)/gamma(S(1)/3) + 3**(S(5)/6)*z**2/(6*gamma(S(2)/3)) + O(z**3))

    assert airybiprime(z).rewrite(hyper) == (
        3**(S(5)/6)*z**2*hyper((), (S(5)/3,), z**S(3)/9)/(6*gamma(S(2)/3)) +
        3**(S(1)/6)*hyper((), (S(1)/3,), z**S(3)/9)/gamma(S(1)/3))

    assert isinstance(airybiprime(z).rewrite(besselj), airybiprime)
    assert airyai(t).rewrite(besselj) == (
        sqrt(-t)*(besselj(-S(1)/3, 2*(-t)**(S(3)/2)/3) +
                  besselj(S(1)/3, 2*(-t)**(S(3)/2)/3))/3)
    assert airybiprime(z).rewrite(besseli) == (
        sqrt(3)*(z**2*besseli(S(2)/3, 2*z**(S(3)/2)/3)/(z**(S(3)/2))**(S(2)/3) +
                 (z**(S(3)/2))**(S(2)/3)*besseli(-S(2)/3, 2*z**(S(3)/2)/3))/3)
    assert airybiprime(p).rewrite(besseli) == (
        sqrt(3)*p*(besseli(-S(2)/3, 2*p**(S(3)/2)/3) + besseli(S(2)/3, 2*p**(S(3)/2)/3))/3)

    assert expand_func(airybiprime(2*(3*z**5)**(S(1)/3))) == (
        sqrt(3)*(z**(S(5)/3)/(z**5)**(S(1)/3) - 1)*airyaiprime(2*3**(S(1)/3)*z**(S(5)/3))/2 +
        (z**(S(5)/3)/(z**5)**(S(1)/3) + 1)*airybiprime(2*3**(S(1)/3)*z**(S(5)/3))/2)

def test_airyai():
    z = Symbol('z', real=False)
    t = Symbol('t', negative=True)
    p = Symbol('p', positive=True)

    assert isinstance(airyai(z), airyai)

    assert airyai(0) == 3**(S(1)/3)/(3*gamma(S(2)/3))
    assert airyai(oo) == 0
    assert airyai(-oo) == 0

    assert diff(airyai(z), z) == airyaiprime(z)

    assert series(airyai(z), z, 0, 3) == (
        3**(S(5)/6)*gamma(S(1)/3)/(6*pi) - 3**(S(1)/6)*z*gamma(S(2)/3)/(2*pi) + O(z**3))

    assert airyai(z).rewrite(hyper) == (
        -3**(S(2)/3)*z*hyper((), (S(4)/3,), z**S(3)/9)/(3*gamma(S(1)/3)) +
         3**(S(1)/3)*hyper((), (S(2)/3,), z**S(3)/9)/(3*gamma(S(2)/3)))

    assert isinstance(airyai(z).rewrite(besselj), airyai)
    assert airyai(t).rewrite(besselj) == (
        sqrt(-t)*(besselj(-S(1)/3, 2*(-t)**(S(3)/2)/3) +
                  besselj(S(1)/3, 2*(-t)**(S(3)/2)/3))/3)
    assert airyai(z).rewrite(besseli) == (
        -z*besseli(S(1)/3, 2*z**(S(3)/2)/3)/(3*(z**(S(3)/2))**(S(1)/3)) +
         (z**(S(3)/2))**(S(1)/3)*besseli(-S(1)/3, 2*z**(S(3)/2)/3)/3)
    assert airyai(p).rewrite(besseli) == (
        sqrt(p)*(besseli(-S(1)/3, 2*p**(S(3)/2)/3) -
                 besseli(S(1)/3, 2*p**(S(3)/2)/3))/3)

    assert expand_func(airyai(2*(3*z**5)**(S(1)/3))) == (
        -sqrt(3)*(-1 + (z**5)**(S(1)/3)/z**(S(5)/3))*airybi(2*3**(S(1)/3)*z**(S(5)/3))/6 +
         (1 + (z**5)**(S(1)/3)/z**(S(5)/3))*airyai(2*3**(S(1)/3)*z**(S(5)/3))/2)

def test_hyperexpand_bases():
    assert (
        hyperexpand(hyper([2], [a], z))
        == a + z ** (-a + 1) * (-a ** 2 + 3 * a + z * (a - 1) - 2) * exp(z) * lowergamma(a - 1, z) - 1
    )
    # TODO [a+1, a-S.Half], [2*a]
    assert hyperexpand(hyper([1, 2], [3], z)) == -2 / z - 2 * log(exp_polar(-I * pi) * z + 1) / z ** 2
    assert hyperexpand(hyper([S.Half, 2], [S(3) / 2], z)) == -1 / (2 * z - 2) + log((sqrt(z) + 1) / (-sqrt(z) + 1)) / (
        4 * sqrt(z)
    )
    assert hyperexpand(hyper([S(1) / 2, S(1) / 2], [S(5) / 2], z)) == (-3 * z + 3) / 4 / (z * sqrt(-z + 1)) + (
        6 * z - 3
    ) * asin(sqrt(z)) / (4 * z ** (S(3) / 2))
    assert hyperexpand(hyper([1, 2], [S(3) / 2], z)) == -1 / (2 * z - 2) - asin(sqrt(z)) / (
        sqrt(z) * (2 * z - 2) * sqrt(-z + 1)
    )
    assert hyperexpand(hyper([-S.Half - 1, 1, 2], [S.Half, 3], z)) == sqrt(z) * (6 * z / 7 - S(6) / 5) * atanh(
        sqrt(z)
    ) + (-30 * z ** 2 + 32 * z - 6) / 35 / z - 6 * log(-z + 1) / (35 * z ** 2)
    assert hyperexpand(hyper([1 + S.Half, 1, 1], [2, 2], z)) == -4 * log(sqrt(-z + 1) / 2 + S(1) / 2) / z
    # TODO hyperexpand(hyper([a], [2*a + 1], z))
    # TODO [S.Half, a], [S(3)/2, a+1]
    assert hyperexpand(hyper([2], [b, 1], z)) == z ** (-b / 2 + S(1) / 2) * besseli(b - 1, 2 * sqrt(z)) * gamma(
        b
    ) + z ** (-b / 2 + 1) * besseli(b, 2 * sqrt(z)) * gamma(b)

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
    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_branching():
    from sympy import exp_polar, polar_lift, Symbol, I, exp
    assert besselj(polar_lift(k), x) == besselj(k, x)
    assert besseli(polar_lift(k), x) == besseli(k, x)

    n = Symbol('n', integer=True)
    assert besselj(n, exp_polar(2*pi*I)*x) == besselj(n, x)
    assert besselj(n, polar_lift(x)) == besselj(n, x)
    assert besseli(n, exp_polar(2*pi*I)*x) == besseli(n, x)
    assert besseli(n, polar_lift(x)) == besseli(n, x)

    def tn(func, s):
        from random import uniform
        c = uniform(1, 5)
        expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
        eps = 1e-15
        expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
        return abs(expr.n() - expr2.n()).n() < 1e-10

    nu = Symbol('nu')
    assert besselj(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besselj(nu, x)
    assert besseli(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besseli(nu, x)
    assert tn(besselj, 2)
    assert tn(besselj, pi)
    assert tn(besselj, I)
    assert tn(besseli, 2)
    assert tn(besseli, pi)
    assert tn(besseli, I)

def test_bessel():
    from sympy import (besselj, Heaviside, besseli, polar_lift, exp_polar,
                       powdenest)
    assert simplify(integrate(besselj(a, z)*besselj(b, z)/z, (z, 0, oo),
                     meijerg=True, conds='none')) == \
           2*sin(pi*a/2 - pi*b/2)/(pi*(a - b)*(a + b))
    assert simplify(integrate(besselj(a, z)*besselj(a, z)/z, (z, 0, oo),
                     meijerg=True, conds='none')) == 1/(2*a)

    # TODO more orthogonality integrals

    # TODO there is some improvement possible here:
    #  - the result can be simplified to besselj(y, z))
    assert powdenest(simplify(integrate(sin(z*x)*(x**2-1)**(-(y+S(1)/2)),
                              (x, 1, oo), meijerg=True, conds='none')
                              *2/((z/2)**y*sqrt(pi)*gamma(S(1)/2-y))),
                     polar=True) == \
           exp(-I*pi*y/2)*besseli(y, z*exp_polar(I*pi/2))

    # Werner Rosenheinrich
    # SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS

    assert integrate(x*besselj(0, x), x, meijerg=True) == x*besselj(1, x)
    assert integrate(x*besseli(0, x), x, meijerg=True) == x*besseli(1, x)
    # TODO can do higher powers, but come out as high order ... should they be
    #      reduced to order 0, 1?
    assert integrate(besselj(1, x), x, meijerg=True) == -besselj(0, x)
    assert integrate(besselj(1, x)**2/x, x, meijerg=True) == \
           -(besselj(0, x)**2 + besselj(1, x)**2)/2
    # TODO more besseli when tables are extended or recursive mellin works
    assert integrate(besselj(0, x)**2/x**2, x, meijerg=True) == \
           -2*x*besselj(0, x)**2 - 2*x*besselj(1, x)**2 \
           + 2*besselj(0, x)*besselj(1, x) - besselj(0, x)**2/x
    assert integrate(besselj(0, x)*besselj(1, x), x, meijerg=True) == \
           -besselj(0, x)**2/2
    assert integrate(x**2*besselj(0, x)*besselj(1, x), x, meijerg=True) == \
           x**2*besselj(1, x)**2/2
    assert integrate(besselj(0, x)*besselj(1, x)/x, x, meijerg=True) == \
           (x*besselj(0, x)**2 + x*besselj(1, x)**2 - \
            besselj(0, x)*besselj(1, x))
    # TODO how does besselj(0, a*x)*besselj(0, b*x) work?
    # TODO how does besselj(0, x)**2*besselj(1, x)**2 work?
    # TODO sin(x)*besselj(0, x) etc come out a mess
    # TODO can x*log(x)*besselj(0, x) be done?
    # TODO how does besselj(1, x)*besselj(0, x+a) work?
    # TODO more indefinite integrals when struve functions etc are implemented

    # test a substitution
    assert integrate(besselj(1, x**2)*x, x, meijerg=True) == \
        -besselj(0, x**2)/2

def test_chi_noncentral():
    k = Symbol("k", integer=True)
    l = Symbol("l")

    X = ChiNoncentral("x", k, l)
    assert density(X)(x) == (x**k*l*(x*l)**(-k/2)*
                          exp(-x**2/2 - l**2/2)*besseli(k/2 - 1, x*l))

def test_meijerg_eval():
    from sympy import besseli, exp_polar
    from sympy.abc import l
    a = randcplx()
    arg = x*exp_polar(k*pi*I)
    expr1 = pi*meijerg([[], [(a + 1)/2]], [[a/2], [-a/2, (a + 1)/2]], arg**2/4)
    expr2 = besseli(a, arg)

    # Test that the two expressions agree for all arguments.
    for x_ in [0.5, 1.5]:
        for k_ in [0.0, 0.1, 0.3, 0.5, 0.8, 1, 5.751, 15.3]:
            assert abs((expr1 - expr2).n(subs={x: x_, k: k_})) < 1e-10
            assert abs((expr1 - expr2).n(subs={x: x_, k: -k_})) < 1e-10

    # Test continuity independently
    eps = 1e-13
    expr2 = expr1.subs(k, l)
    for x_ in [0.5, 1.5]:
        for k_ in [0.5, S(1)/3, 0.25, 0.75, S(2)/3, 1.0, 1.5]:
            assert abs((expr1 - expr2).n(
                       subs={x: x_, k: k_ + eps, l: k_ - eps})) < 1e-10
            assert abs((expr1 - expr2).n(
                       subs={x: x_, k: -k_ + eps, l: -k_ - eps})) < 1e-10

    expr = (meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(-I*pi)/4)
            + meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(I*pi)/4)) \
        /(2*sqrt(pi))
    assert (expr - pi/exp(1)).n(chop=True) == 0

def test_bessel():
    from sympy import besselj, besseli

    assert simplify(integrate(besselj(a, z) * besselj(b, z) / z, (z, 0, oo), meijerg=True, conds="none")) == 2 * sin(
        pi * (a / 2 - b / 2)
    ) / (pi * (a - b) * (a + b))
    assert simplify(integrate(besselj(a, z) * besselj(a, z) / z, (z, 0, oo), meijerg=True, conds="none")) == 1 / (2 * a)

    # TODO more orthogonality integrals

    assert simplify(
        integrate(sin(z * x) * (x ** 2 - 1) ** (-(y + S(1) / 2)), (x, 1, oo), meijerg=True, conds="none")
        * 2
        / ((z / 2) ** y * sqrt(pi) * gamma(S(1) / 2 - y))
    ) == besselj(y, z)

    # Werner Rosenheinrich
    # SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS

    assert integrate(x * besselj(0, x), x, meijerg=True) == x * besselj(1, x)
    assert integrate(x * besseli(0, x), x, meijerg=True) == x * besseli(1, x)
    # TODO can do higher powers, but come out as high order ... should they be
    #      reduced to order 0, 1?
    assert integrate(besselj(1, x), x, meijerg=True) == -besselj(0, x)
    assert integrate(besselj(1, x) ** 2 / x, x, meijerg=True) == -(besselj(0, x) ** 2 + besselj(1, x) ** 2) / 2
    # TODO more besseli when tables are extended or recursive mellin works
    assert (
        integrate(besselj(0, x) ** 2 / x ** 2, x, meijerg=True)
        == -2 * x * besselj(0, x) ** 2
        - 2 * x * besselj(1, x) ** 2
        + 2 * besselj(0, x) * besselj(1, x)
        - besselj(0, x) ** 2 / x
    )
    assert integrate(besselj(0, x) * besselj(1, x), x, meijerg=True) == -besselj(0, x) ** 2 / 2
    assert integrate(x ** 2 * besselj(0, x) * besselj(1, x), x, meijerg=True) == x ** 2 * besselj(1, x) ** 2 / 2
    assert integrate(besselj(0, x) * besselj(1, x) / x, x, meijerg=True) == (
        x * besselj(0, x) ** 2 + x * besselj(1, x) ** 2 - besselj(0, x) * besselj(1, x)
    )
    # TODO how does besselj(0, a*x)*besselj(0, b*x) work?
    # TODO how does besselj(0, x)**2*besselj(1, x)**2 work?
    # TODO sin(x)*besselj(0, x) etc come out a mess
    # TODO can x*log(x)*besselj(0, x) be done?
    # TODO how does besselj(1, x)*besselj(0, x+a) work?
    # TODO more indefinite integrals when struve functions etc are implemented

    # test a substitution
    assert integrate(besselj(1, x ** 2) * x, x, meijerg=True) == -besselj(0, x ** 2) / 2

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(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 hn1(order, z) == hn1(order, z).rewrite(f)
        assert hn2(order, z) == hn2(order, z).rewrite(f)

    assert jn(order, z).rewrite(besselj) == sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(order + S(1)/2, z)/2
    assert jn(order, z).rewrite(bessely) == (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-order - S(1)/2, z)/2

    # 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, 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_hyperexpand_bases():
    assert hyperexpand(hyper([2], [a], z)) == \
  a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z)*lowergamma(a - 1, z) - 1
    # TODO [a+1, a-S.Half], [2*a]
    assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2
    assert hyperexpand(hyper([S.Half, 2], [S(3)/2], z)) == \
      -1/(2*z - 2) + log((z**(S(1)/2) + 1)/(-z**(S(1)/2) + 1))/(4*z**(S(1)/2))
    assert hyperexpand(hyper([S(1)/2, S(1)/2], [S(5)/2], z)) == \
               (-3*z + 3)/(4*z*(-z + 1)**(S(1)/2)) \
               + (6*z - 3)*asin(z**(S(1)/2))/(4*z**(S(3)/2))
    assert hyperexpand(hyper([1, 2], [S(3)/2], z)) == -1/(2*z - 2) \
            - asin(z**(S(1)/2))/(z**(S(1)/2)*(2*z - 2)*(-z + 1)**(S(1)/2))
    assert hyperexpand(hyper([-S.Half - 1, 1, 2], [S.Half, 3], z)) == \
             z**(S(1)/2)*(6*z/7 - S(6)/5)*atanh(z**(S(1)/2)) \
           + (-30*z**2 + 32*z - 6)/(35*z) - 6*log(-z + 1)/(35*z**2)
    assert hyperexpand(hyper([1+S.Half, 1, 1], [2, 2], z)) == \
           -4*log((-z + 1)**(S(1)/2)/2 + S(1)/2)/z
    # TODO hyperexpand(hyper([a], [2*a + 1], z))
    # TODO [S.Half, a], [S(3)/2, a+1]
    assert hyperexpand(hyper([2], [b, 1], z)) == \
             z**(-b/2 + S(1)/2)*besseli(b - 1, 2*z**(S(1)/2))*gamma(b) \
           + z**(-b/2 + 1)*besseli(b, 2*z**(S(1)/2))*gamma(b)