Python randtest.td函数代码示例

def test_E():
    assert E(z, 0) == z
    assert E(0, m) == 0
    assert E(i*pi/2, m) == i*E(m)
    assert E(z, oo) == zoo
    assert E(z, -oo) == zoo
    assert E(0) == pi/2
    assert E(1) == 1
    assert E(oo) == I*oo
    assert E(-oo) == oo
    assert E(zoo) == zoo

    assert E(-z, m) == -E(z, m)

    assert E(z, m).diff(z) == sqrt(1 - m*sin(z)**2)
    assert E(z, m).diff(m) == (E(z, m) - F(z, m))/(2*m)
    assert E(z).diff(z) == (E(z) - K(z))/(2*z)
    r = randcplx()
    assert td(E(r, m), m)
    assert td(E(z, r), z)
    assert td(E(z), z)

    mi = Symbol('m', real=False)
    assert E(z, mi).conjugate() == E(z.conjugate(), mi.conjugate())
    mr = Symbol('m', real=True, negative=True)
    assert E(z, mr).conjugate() == E(z.conjugate(), mr)

    assert E(z).rewrite(hyper) == (pi/2)*hyper((-S.Half, S.Half), (S.One,), z)
    assert tn(E(z), (pi/2)*hyper((-S.Half, S.Half), (S.One,), z))
    assert E(z).rewrite(meijerg) == \
        -meijerg(((S.Half, S(3)/2), []), ((S.Zero,), (S.Zero,)), -z)/4
    assert tn(E(z), -meijerg(((S.Half, S(3)/2), []), ((S.Zero,), (S.Zero,)), -z)/4)

def test_uppergamma():
    from sympy import meijerg, exp_polar, I, expint
    assert uppergamma(4, 0) == 6
    assert uppergamma(x, y).diff(y) == -y**(x-1)*exp(-y)
    assert td(uppergamma(randcplx(), y), y)
    assert uppergamma(x, y).diff(x) == \
           uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
    assert td(uppergamma(x, randcplx()), x)

    assert uppergamma(S.Half, x) == sqrt(pi)*(1 - erf(sqrt(x)))
    assert not uppergamma(S.Half - 3, x).has(uppergamma)
    assert not uppergamma(S.Half + 3, x).has(uppergamma)
    assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
    assert tn(uppergamma(S.Half + 3, x, evaluate=False),
              uppergamma(S.Half + 3, x), x)
    assert tn(uppergamma(S.Half - 3, x, evaluate=False),
              uppergamma(S.Half - 3, x), x)

    assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)

    assert tn_branch(-3, uppergamma)
    assert tn_branch(-4, uppergamma)
    assert tn_branch(S(1)/3, uppergamma)
    assert tn_branch(pi, uppergamma)
    assert uppergamma(3, exp_polar(4*pi*I)*x) == uppergamma(3, x)
    assert uppergamma(y, exp_polar(5*pi*I)*x) == \
           exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + gamma(y)*(1-exp(4*pi*I*y))
    assert uppergamma(-2, exp_polar(5*pi*I)*x) == \
           uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I

    assert uppergamma(-2, x) == expint(3, x)/x**2
    assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y)

def test_lowergamma():
    from sympy import meijerg, exp_polar, I, expint

    assert lowergamma(x, y).diff(y) == y ** (x - 1) * exp(-y)
    assert td(lowergamma(randcplx(), y), y)
    assert td(lowergamma(x, randcplx()), x)
    assert lowergamma(x, y).diff(x) == gamma(x) * polygamma(0, x) - uppergamma(x, y) * log(y) - meijerg(
        [], [1, 1], [0, 0, x], [], y
    )

    assert lowergamma(S.Half, x) == sqrt(pi) * erf(sqrt(x))
    assert not lowergamma(S.Half - 3, x).has(lowergamma)
    assert not lowergamma(S.Half + 3, x).has(lowergamma)
    assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
    assert tn(lowergamma(S.Half + 3, x, evaluate=False), lowergamma(S.Half + 3, x), x)
    assert tn(lowergamma(S.Half - 3, x, evaluate=False), lowergamma(S.Half - 3, x), x)

    assert tn_branch(-3, lowergamma)
    assert tn_branch(-4, lowergamma)
    assert tn_branch(S(1) / 3, lowergamma)
    assert tn_branch(pi, lowergamma)
    assert lowergamma(3, exp_polar(4 * pi * I) * x) == lowergamma(3, x)
    assert lowergamma(y, exp_polar(5 * pi * I) * x) == exp(4 * I * pi * y) * lowergamma(y, x * exp_polar(pi * I))
    assert lowergamma(-2, exp_polar(5 * pi * I) * x) == lowergamma(-2, x * exp_polar(I * pi)) + 2 * pi * I

    assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x), conjugate(y))
    assert conjugate(lowergamma(x, 0)) == conjugate(lowergamma(x, 0))
    assert conjugate(lowergamma(x, -oo)) == conjugate(lowergamma(x, -oo))

    assert lowergamma(x, y).rewrite(expint) == -y ** x * expint(-x + 1, y) + gamma(x)
    k = Symbol("k", integer=True)
    assert lowergamma(k, y).rewrite(expint) == -y ** k * expint(-k + 1, y) + gamma(k)
    k = Symbol("k", integer=True, positive=False)
    assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
    assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)

def test_meijer():
    raises(TypeError, lambda: meijerg(1, z))
    raises(TypeError, lambda: meijerg(((1,), (2,)), (3,), (4,), z))

    assert meijerg(((1, 2), (3,)), ((4,), (5,)), z) == \
        meijerg(Tuple(1, 2), Tuple(3), Tuple(4), Tuple(5), z)

    g = meijerg((1, 2), (3, 4, 5), (6, 7, 8, 9), (10, 11, 12, 13, 14), z)
    assert g.an == Tuple(1, 2)
    assert g.ap == Tuple(1, 2, 3, 4, 5)
    assert g.aother == Tuple(3, 4, 5)
    assert g.bm == Tuple(6, 7, 8, 9)
    assert g.bq == Tuple(6, 7, 8, 9, 10, 11, 12, 13, 14)
    assert g.bother == Tuple(10, 11, 12, 13, 14)
    assert g.argument == z
    assert g.nu == 75
    assert g.delta == -1
    assert g.is_commutative is True

    assert meijerg([1, 2], [3], [4], [5], z).delta == S(1)/2

    # just a few checks to make sure that all arguments go where they should
    assert tn(meijerg(Tuple(), Tuple(), Tuple(0), Tuple(), -z), exp(z), z)
    assert tn(sqrt(pi)*meijerg(Tuple(), Tuple(),
                               Tuple(0), Tuple(S(1)/2), z**2/4), cos(z), z)
    assert tn(meijerg(Tuple(1, 1), Tuple(), Tuple(1), Tuple(0), z),
              log(1 + z), z)

    # differentiation
    g = meijerg((randcplx(),), (randcplx() + 2*I,), Tuple(),
                (randcplx(), randcplx()), z)
    assert td(g, z)

    g = meijerg(Tuple(), (randcplx(),), Tuple(),
                (randcplx(), randcplx()), z)
    assert td(g, z)

    g = meijerg(Tuple(), Tuple(), Tuple(randcplx()),
                Tuple(randcplx(), randcplx()), z)
    assert td(g, z)

    a1, a2, b1, b2, c1, c2, d1, d2 = symbols('a1:3, b1:3, c1:3, d1:3')
    assert meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z).diff(z) == \
        (meijerg((a1 - 1, a2), (b1, b2), (c1, c2), (d1, d2), z)
         + (a1 - 1)*meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z))/z

    assert meijerg([z, z], [], [], [], z).diff(z) == \
        Derivative(meijerg([z, z], [], [], [], z), z)

    # meijerg is unbranched wrt parameters
    from sympy import polar_lift as pl
    assert meijerg([pl(a1)], [pl(a2)], [pl(b1)], [pl(b2)], pl(z)) == \
        meijerg([a1], [a2], [b1], [b2], pl(z))

    # integrand
    from sympy.abc import a, b, c, d, s
    assert meijerg([a], [b], [c], [d], z).integrand(s) == \
        z**s*gamma(c - s)*gamma(-a + s + 1)/(gamma(b - s)*gamma(-d + s + 1))

def test_derivatives():
    from sympy import Derivative
    assert zeta(x, a).diff(x) == Derivative(zeta(x, a), x)
    assert zeta(x, a).diff(a) == -x*zeta(x + 1, a)
    assert lerchphi(z, s, a).diff(z) == (lerchphi(z, s-1, a) - a*lerchphi(z, s, a))/z
    assert lerchphi(z, s, a).diff(a) == -s*lerchphi(z, s+1, a)
    assert polylog(s, z).diff(z) == polylog(s - 1, z)/z

    b = randcplx()
    c = randcplx()
    assert td(zeta(b, x), x)
    assert td(polylog(b, z), z)
    assert td(lerchphi(c, b, x), x)
    assert td(lerchphi(x, b, c), x)

def test_uppergamma():
    from sympy import meijerg

    assert uppergamma(4, 0) == 6
    assert uppergamma(x, y).diff(y) == -y ** (x - 1) * exp(-y)
    assert td(uppergamma(randcplx(), y), y)
    assert uppergamma(x, y).diff(x) == uppergamma(x, y) * log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
    assert td(uppergamma(x, randcplx()), x)

    assert uppergamma(S.Half, x) == sqrt(pi) * (1 - erf(sqrt(x)))
    assert not uppergamma(S.Half - 3, x).has(uppergamma)
    assert not uppergamma(S.Half + 3, x).has(uppergamma)
    assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
    assert tn(uppergamma(S.Half + 3, x, evaluate=False), uppergamma(S.Half + 3, x), x)
    assert tn(uppergamma(S.Half - 3, x, evaluate=False), uppergamma(S.Half - 3, x), x)

def test_lowergamma():
    from sympy import meijerg, exp_polar, I
    assert lowergamma(x, y).diff(y) == y**(x-1)*exp(-y)
    assert td(lowergamma(randcplx(), y), y)
    assert lowergamma(x, y).diff(x) == \
           gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
           + meijerg([], [1, 1], [0, 0, x], [], y)

    assert lowergamma(S.Half, x) == sqrt(pi)*erf(sqrt(x))
    assert not lowergamma(S.Half - 3, x).has(lowergamma)
    assert not lowergamma(S.Half + 3, x).has(lowergamma)
    assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
    assert tn(lowergamma(S.Half + 3, x, evaluate=False),
              lowergamma(S.Half + 3, x), x)
    assert tn(lowergamma(S.Half - 3, x, evaluate=False),
              lowergamma(S.Half - 3, x), x)

    assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)

    assert tn_branch(-3, lowergamma)
    assert tn_branch(-4, lowergamma)
    assert tn_branch(S(1)/3, lowergamma)
    assert tn_branch(pi, lowergamma)
    assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x)
    assert lowergamma(y, exp_polar(5*pi*I)*x) == \
           exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
    assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
           lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I

def test_K():
    assert K(0) == pi / 2
    assert K(S(1) / 2) == 8 * pi ** (S(3) / 2) / gamma(-S(1) / 4) ** 2
    assert K(1) == zoo
    assert K(-1) == gamma(S(1) / 4) ** 2 / (4 * sqrt(2 * pi))
    assert K(oo) == 0
    assert K(-oo) == 0
    assert K(I * oo) == 0
    assert K(-I * oo) == 0
    assert K(zoo) == 0

    assert K(z).diff(z) == (E(z) - (1 - z) * K(z)) / (2 * z * (1 - z))
    assert td(K(z), z)

    zi = Symbol("z", real=False)
    assert K(zi).conjugate() == K(zi.conjugate())
    zr = Symbol("z", real=True, negative=True)
    assert K(zr).conjugate() == K(zr)

    assert K(z).rewrite(hyper) == (pi / 2) * hyper((S.Half, S.Half), (S.One,), z)
    assert tn(K(z), (pi / 2) * hyper((S.Half, S.Half), (S.One,), z))
    assert K(z).rewrite(meijerg) == meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z) / 2
    assert tn(K(z), meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z) / 2)

    assert K(z).series(
        z
    ) == pi / 2 + pi * z / 8 + 9 * pi * z ** 2 / 128 + 25 * pi * z ** 3 / 512 + 1225 * pi * z ** 4 / 32768 + 3969 * pi * z ** 5 / 131072 + O(
        z ** 6
    )

def test_hyper():
    raises(TypeError, lambda: hyper(1, 2, z))

    assert hyper((1, 2), (1,), z) == hyper(Tuple(1, 2), Tuple(1), z)

    h = hyper((1, 2), (3, 4, 5), z)
    assert h.ap == Tuple(1, 2)
    assert h.bq == Tuple(3, 4, 5)
    assert h.argument == z
    assert h.is_commutative is True

    # just a few checks to make sure that all arguments go where they should
    assert tn(hyper(Tuple(), Tuple(), z), exp(z), z)
    assert tn(z*hyper((1, 1), Tuple(2), -z), log(1 + z), z)

    # differentiation
    h = hyper(
        (randcplx(), randcplx(), randcplx()), (randcplx(), randcplx()), z)
    assert td(h, z)

    a1, a2, b1, b2, b3 = symbols('a1:3, b1:4')
    assert hyper((a1, a2), (b1, b2, b3), z).diff(z) == \
        a1*a2/(b1*b2*b3) * hyper((a1 + 1, a2 + 1), (b1 + 1, b2 + 1, b3 + 1), z)

    # differentiation wrt parameters is not supported
    assert hyper([z], [], z).diff(z) == Derivative(hyper([z], [], z), z)

    # hyper is unbranched wrt parameters
    from sympy import polar_lift
    assert hyper([polar_lift(z)], [polar_lift(k)], polar_lift(x)) == \
        hyper([z], [k], polar_lift(x))

def test_hyper():
    raises(TypeError, 'hyper(1, 2, z)')

    assert hyper((1, 2),(1,), z) == hyper(Tuple(1, 2), Tuple(1), z)

    h = hyper((1, 2), (3, 4, 5), z)
    assert h.ap == Tuple(1, 2)
    assert h.bq == Tuple(3, 4, 5)
    assert h.argument == z
    assert h.is_commutative is True

    # just a few checks to make sure that all arguments go where they should
    assert tn(hyper(Tuple(), Tuple(), z), exp(z), z)
    assert tn(z*hyper((1, 1), Tuple(2), -z), log(1 + z), z)

    # differentiation
    h = hyper((randcplx(), randcplx(), randcplx()), (randcplx(), randcplx()), z)
    assert td(h, z)

    a1, a2, b1, b2, b3 = symbols('a1:3, b1:4')
    assert hyper((a1, a2), (b1, b2, b3), z).diff(z) == \
             a1*a2/(b1*b2*b3) * hyper((a1+1, a2+1), (b1+1, b2+1, b3+1), z)

    # differentiation wrt parameters is not supported
    raises(NotImplementedError, 'hyper((z,), (), z).diff(z)')

def test_K():
    assert K(0) == pi/2
    assert K(S(1)/2) == 8*pi**(S(3)/2)/gamma(-S(1)/4)**2
    assert K(1) == zoo
    assert K(-1) == gamma(S(1)/4)**2/(4*sqrt(2*pi))
    assert K(oo) == 0
    assert K(-oo) == 0
    assert K(I*oo) == 0
    assert K(-I*oo) == 0
    assert K(zoo) == 0

    assert K(z).diff(z) == (E(z) - (1 - z)*K(z))/(2*z*(1 - z))
    assert td(K(z), z)

    zi = Symbol('z', real=False)
    assert K(zi).conjugate() == K(zi.conjugate())
    zr = Symbol('z', real=True, negative=True)
    assert K(zr).conjugate() == K(zr)

    assert K(z).rewrite(hyper) == \
        (pi/2)*hyper((S.Half, S.Half), (S.One,), z)
    assert tn(K(z), (pi/2)*hyper((S.Half, S.Half), (S.One,), z))
    assert K(z).rewrite(meijerg) == \
        meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z)/2
    assert tn(K(z), meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -z)/2)

def test_meijer():
    raises(TypeError, 'meijerg(1, z)')
    raises(TypeError, 'meijerg(((1,), (2,)), (3,), (4,), z)')

    assert meijerg(((1, 2), (3,)), ((4,), (5,)), z) == \
           meijerg(Tuple(1, 2), Tuple(3), Tuple(4), Tuple(5), z)

    g = meijerg((1, 2), (3, 4, 5), (6, 7, 8, 9), (10, 11, 12, 13, 14), z)
    assert g.an == Tuple(1, 2)
    assert g.ap == Tuple(1, 2, 3, 4, 5)
    assert g.aother == Tuple(3, 4, 5)
    assert g.bm == Tuple(6, 7, 8, 9)
    assert g.bq == Tuple(6, 7, 8, 9, 10, 11, 12, 13, 14)
    assert g.bother == Tuple(10, 11, 12, 13, 14)
    assert g.argument == z
    assert g.nu == 75
    assert g.delta == -1
    assert g.is_commutative is True

    assert meijerg([1, 2], [3], [4], [5], z).delta == S(1)/2

    # just a few checks to make sure that all arguments go where they should
    assert tn(meijerg(Tuple(), Tuple(), Tuple(0), Tuple(), -z), exp(z), z)
    assert tn(sqrt(pi)*meijerg(Tuple(), Tuple(),
                               Tuple(0), Tuple(S(1)/2), z**2/4), cos(z), z)
    assert tn(meijerg(Tuple(1, 1),Tuple(), Tuple(1), Tuple(0), z),
              log(1 + z), z)

    # differentiation
    g = meijerg((randcplx(),), (randcplx() + 2*I,), Tuple(),
                (randcplx(), randcplx()), z)
    assert td(g, z)

    g = meijerg(Tuple(), (randcplx(),), Tuple(),
                (randcplx(), randcplx()), z)
    assert td(g, z)

    g = meijerg(Tuple(), Tuple(), Tuple(randcplx()),
                Tuple(randcplx(), randcplx()), z)
    assert td(g, z)

    a1, a2, b1, b2, c1, c2, d1, d2 = symbols('a1:3, b1:3, c1:3, d1:3')
    assert meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z).diff(z) == \
        (meijerg((a1-1, a2), (b1, b2), (c1, c2), (d1, d2), z) \
         + (a1 - 1)*meijerg((a1, a2), (b1, b2), (c1, c2), (d1, d2), z))/z

    raises(NotImplementedError, 'meijerg((z,), (), (), (), z).diff(z)')

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_P():
    assert P(0, z, m) == F(z, m)
    assert P(1, z, m) == F(z, m) + (sqrt(1 - m * sin(z) ** 2) * tan(z) - E(z, m)) / (1 - m)
    assert P(n, i * pi / 2, m) == i * P(n, m)
    assert P(n, z, 0) == atanh(sqrt(n - 1) * tan(z)) / sqrt(n - 1)
    assert P(n, z, n) == F(z, n) - P(1, z, n) + tan(z) / sqrt(1 - n * sin(z) ** 2)
    assert P(oo, z, m) == 0
    assert P(-oo, z, m) == 0
    assert P(n, z, oo) == 0
    assert P(n, z, -oo) == 0
    assert P(0, m) == K(m)
    assert P(1, m) == zoo
    assert P(n, 0) == pi / (2 * sqrt(1 - n))
    assert P(2, 1) == -oo
    assert P(-1, 1) == oo
    assert P(n, n) == E(n) / (1 - n)

    assert P(n, -z, m) == -P(n, z, m)

    ni, mi = Symbol("n", real=False), Symbol("m", real=False)
    assert P(ni, z, mi).conjugate() == P(ni.conjugate(), z.conjugate(), mi.conjugate())
    nr, mr = Symbol("n", real=True, negative=True), Symbol("m", real=True, negative=True)
    assert P(nr, z, mr).conjugate() == P(nr, z.conjugate(), mr)
    assert P(n, m).conjugate() == P(n.conjugate(), m.conjugate())

    assert P(n, z, m).diff(n) == (
        E(z, m)
        + (m - n) * F(z, m) / n
        + (n ** 2 - m) * P(n, z, m) / n
        - n * sqrt(1 - m * sin(z) ** 2) * sin(2 * z) / (2 * (1 - n * sin(z) ** 2))
    ) / (2 * (m - n) * (n - 1))
    assert P(n, z, m).diff(z) == 1 / (sqrt(1 - m * sin(z) ** 2) * (1 - n * sin(z) ** 2))
    assert P(n, z, m).diff(m) == (
        E(z, m) / (m - 1) + P(n, z, m) - m * sin(2 * z) / (2 * (m - 1) * sqrt(1 - m * sin(z) ** 2))
    ) / (2 * (n - m))
    assert P(n, m).diff(n) == (E(m) + (m - n) * K(m) / n + (n ** 2 - m) * P(n, m) / n) / (2 * (m - n) * (n - 1))
    assert P(n, m).diff(m) == (E(m) / (m - 1) + P(n, m)) / (2 * (n - m))
    rx, ry = randcplx(), randcplx()
    assert td(P(n, rx, ry), n)
    assert td(P(rx, z, ry), z)
    assert td(P(rx, ry, m), m)

    assert P(n, z, m).series(z) == z + z ** 3 * (m / 6 + n / 3) + z ** 5 * (
        3 * m ** 2 / 40 + m * n / 10 - m / 30 + n ** 2 / 5 - n / 15
    ) + O(z ** 6)

def test_lowergamma():
    from sympy import meijerg, exp_polar, I, expint
    assert lowergamma(x, 0) == 0
    assert lowergamma(x, y).diff(y) == y**(x - 1)*exp(-y)
    assert td(lowergamma(randcplx(), y), y)
    assert td(lowergamma(x, randcplx()), x)
    assert lowergamma(x, y).diff(x) == \
        gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
        - meijerg([], [1, 1], [0, 0, x], [], y)

    assert lowergamma(S.Half, x) == sqrt(pi)*erf(sqrt(x))
    assert not lowergamma(S.Half - 3, x).has(lowergamma)
    assert not lowergamma(S.Half + 3, x).has(lowergamma)
    assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
    assert tn(lowergamma(S.Half + 3, x, evaluate=False),
              lowergamma(S.Half + 3, x), x)
    assert tn(lowergamma(S.Half - 3, x, evaluate=False),
              lowergamma(S.Half - 3, x), x)

    assert tn_branch(-3, lowergamma)
    assert tn_branch(-4, lowergamma)
    assert tn_branch(S(1)/3, lowergamma)
    assert tn_branch(pi, lowergamma)
    assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x)
    assert lowergamma(y, exp_polar(5*pi*I)*x) == \
        exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
    assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
        lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I

    assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x), conjugate(y))
    assert conjugate(lowergamma(x, 0)) == conjugate(lowergamma(x, 0))
    assert conjugate(lowergamma(x, -oo)) == conjugate(lowergamma(x, -oo))

    assert lowergamma(
        x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x)
    k = Symbol('k', integer=True)
    assert lowergamma(
        k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k)
    k = Symbol('k', integer=True, positive=False)
    assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
    assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)

    assert lowergamma(70, 6) == factorial(69) - 69035724522603011058660187038367026272747334489677105069435923032634389419656200387949342530805432320 * exp(-6)
    assert (lowergamma(S(77) / 2, 6) - lowergamma(S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
    assert (lowergamma(-S(77) / 2, 6) - lowergamma(-S(77) / 2, 6, evaluate=False)).evalf() < 1e-16