def test_rewriting():
assert dirichlet_eta(x).rewrite(zeta) == (1 - 2**(1 - x))*zeta(x)
assert zeta(x).rewrite(dirichlet_eta) == dirichlet_eta(x)/(1 - 2**(1 - x))
assert zeta(x).rewrite(dirichlet_eta, a=2) == zeta(x)
assert verify_numerically(dirichlet_eta(x), dirichlet_eta(x).rewrite(zeta), x)
assert verify_numerically(zeta(x), zeta(x).rewrite(dirichlet_eta), x)
assert zeta(x, a).rewrite(lerchphi) == lerchphi(1, x, a)
assert polylog(s, z).rewrite(lerchphi) == lerchphi(z, s, 1)*z
assert lerchphi(1, x, a).rewrite(zeta) == zeta(x, a)
assert z*lerchphi(z, s, 1).rewrite(polylog) == polylog(s, z)
def test_derivatives():
from sympy.core.function 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)
raises(ArgumentIndexError, lambda: lerchphi(c, b, x).fdiff(2))
raises(ArgumentIndexError, lambda: lerchphi(c, b, x).fdiff(4))
raises(ArgumentIndexError, lambda: polylog(b, z).fdiff(1))
raises(ArgumentIndexError, lambda: polylog(b, z).fdiff(3))
def test_expint():
from sympy.functions.elementary.miscellaneous import Max
from sympy.functions.special.error_functions import (Ci, E1, Ei, Si)
from sympy.functions.special.zeta_functions import lerchphi
from sympy.simplify.simplify import simplify
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.Half)/(
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.Half)), (0, 1), True)
assert inverse_mellin_transform(
-4**s*sqrt(pi)*gamma(s)/(2*s*gamma(-s + S.Half)),
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*exp_polar(I*pi), 1, a), 0, re(a) > S.Zero)
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()
def test_lerchphi():
from sympy.functions.special.zeta_functions import (lerchphi, polylog)
from sympy.simplify.gammasimp import gammasimp
assert hyperexpand(hyper([1, a], [a + 1], z)/a) == lerchphi(z, 1, a)
assert hyperexpand(
hyper([1, a, a], [a + 1, a + 1], z)/a**2) == lerchphi(z, 2, a)
assert hyperexpand(hyper([1, a, a, a], [a + 1, a + 1, a + 1], z)/a**3) == \
lerchphi(z, 3, a)
assert hyperexpand(hyper([1] + [a]*10, [a + 1]*10, z)/a**10) == \
lerchphi(z, 10, a)
assert gammasimp(hyperexpand(meijerg([0, 1 - a], [], [0],
[-a], exp_polar(-I*pi)*z))) == lerchphi(z, 1, a)
assert gammasimp(hyperexpand(meijerg([0, 1 - a, 1 - a], [], [0],
[-a, -a], exp_polar(-I*pi)*z))) == lerchphi(z, 2, a)
assert gammasimp(hyperexpand(meijerg([0, 1 - a, 1 - a, 1 - a], [], [0],
[-a, -a, -a], exp_polar(-I*pi)*z))) == lerchphi(z, 3, a)
assert hyperexpand(z*hyper([1, 1], [2], z)) == -log(1 + -z)
assert hyperexpand(z*hyper([1, 1, 1], [2, 2], z)) == polylog(2, z)
assert hyperexpand(z*hyper([1, 1, 1, 1], [2, 2, 2], z)) == polylog(3, z)
assert hyperexpand(hyper([1, a, 1 + S.Half], [a + 1, S.Half], z)) == \
-2*a/(z - 1) + (-2*a**2 + a)*lerchphi(z, 1, a)
# Now numerical tests. These make sure reductions etc are carried out
# correctly
# a rational function (polylog at negative integer order)
assert can_do([2, 2, 2], [1, 1])
# NOTE these contain log(1-x) etc ... better make sure we have |z| < 1
# reduction of order for polylog
assert can_do([1, 1, 1, b + 5], [2, 2, b], div=10)
# reduction of order for lerchphi
# XXX lerchphi in mpmath is flaky
assert can_do(
[1, a, a, a, b + 5], [a + 1, a + 1, a + 1, b], numerical=False)
# test a bug
from sympy.functions.elementary.complexes import Abs
assert hyperexpand(hyper([S.Half, S.Half, S.Half, 1],
[Rational(3, 2), Rational(3, 2), Rational(3, 2)], Rational(1, 4))) == \
Abs(-polylog(3, exp_polar(I*pi)/2) + polylog(3, S.Half))
def test_print_lerchphi():
assert mpp.doprint(lerchphi(1, 2, 3)) == '<mrow><mi>Φ</mi><mfenced><mn>1</mn><mn>2</mn><mn>3</mn></mfenced></mrow>'
def test_lerchphi_expansion():
assert myexpand(lerchphi(1, s, a), zeta(s, a))
assert myexpand(lerchphi(z, s, 1), polylog(s, z)/z)
# direct summation
assert myexpand(lerchphi(z, -1, a), a/(1 - z) + z/(1 - z)**2)
assert myexpand(lerchphi(z, -3, a), None)
# polylog reduction
assert myexpand(lerchphi(z, s, S.Half),
2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z)
- polylog(s, polar_lift(-1)*sqrt(z))/sqrt(z)))
assert myexpand(lerchphi(z, s, 2), -1/z + polylog(s, z)/z**2)
assert myexpand(lerchphi(z, s, Rational(3, 2)), None)
assert myexpand(lerchphi(z, s, Rational(7, 3)), None)
assert myexpand(lerchphi(z, s, Rational(-1, 3)), None)
assert myexpand(lerchphi(z, s, Rational(-5, 2)), None)
# hurwitz zeta reduction
assert myexpand(lerchphi(-1, s, a),
2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, (a + 1)/2))
assert myexpand(lerchphi(I, s, a), None)
assert myexpand(lerchphi(-I, s, a), None)
assert myexpand(lerchphi(exp(I*pi*Rational(2, 5)), s, a), None)