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from sympy.concrete.summations import Sum
from sympy.core.function import expand_func
from sympy.core.numbers import (Float, I, Rational, nan, oo, pi, zoo)
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.elementary.complexes import (Abs, polar_lift)
from sympy.functions.elementary.exponential import (exp, exp_polar, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.special.zeta_functions import (dirichlet_eta, lerchphi, polylog, riemann_xi, stieltjes, zeta)
from sympy.series.order import O
from sympy.core.function import ArgumentIndexError
from sympy.functions.combinatorial.numbers import bernoulli, factorial, genocchi, harmonic
from sympy.testing.pytest import raises
from sympy.core.random import (test_derivative_numerically as td,
random_complex_number as randcplx, verify_numerically)
x = Symbol('x')
a = Symbol('a')
b = Symbol('b', negative=True)
z = Symbol('z')
s = Symbol('s')
def test_zeta_eval():
assert zeta(nan) is nan
assert zeta(x, nan) is nan
assert zeta(0) == Rational(-1, 2)
assert zeta(0, x) == S.Half - x
assert zeta(0, b) == S.Half - b
assert zeta(1) is zoo
assert zeta(1, 2) is zoo
assert zeta(1, -7) is zoo
assert zeta(1, x) is zoo
assert zeta(2, 1) == pi**2/6
assert zeta(3, 1) == zeta(3)
assert zeta(2) == pi**2/6
assert zeta(4) == pi**4/90
assert zeta(6) == pi**6/945
assert zeta(4, 3) == pi**4/90 - Rational(17, 16)
assert zeta(7, 4) == zeta(7) - Rational(282251, 279936)
assert zeta(S.Half, 2).func == zeta
assert expand_func(zeta(S.Half, 2)) == zeta(S.Half) - 1
assert zeta(x, 3).func == zeta
assert expand_func(zeta(x, 3)) == zeta(x) - 1 - 1/2**x
assert zeta(2, 0) is nan
assert zeta(3, -1) is nan
assert zeta(4, -2) is nan
assert zeta(oo) == 1
assert zeta(-1) == Rational(-1, 12)
assert zeta(-2) == 0
assert zeta(-3) == Rational(1, 120)
assert zeta(-4) == 0
assert zeta(-5) == Rational(-1, 252)
assert zeta(-1, 3) == Rational(-37, 12)
assert zeta(-1, 7) == Rational(-253, 12)
assert zeta(-1, -4) == Rational(-121, 12)
assert zeta(-1, -9) == Rational(-541, 12)
assert zeta(-4, 3) == -17
assert zeta(-4, -8) == 8772
assert zeta(0, 1) == Rational(-1, 2)
assert zeta(0, -1) == Rational(3, 2)
assert zeta(0, 2) == Rational(-3, 2)
assert zeta(0, -2) == Rational(5, 2)
assert zeta(
3).evalf(20).epsilon_eq(Float("1.2020569031595942854", 20), 1e-19)
def test_zeta_series():
assert zeta(x, a).series(a, z, 2) == \
zeta(x, z) - x*(a-z)*zeta(x+1, z) + O((a-z)**2, (a, z))
def test_dirichlet_eta_eval():
assert dirichlet_eta(0) == S.Half
assert dirichlet_eta(-1) == Rational(1, 4)
assert dirichlet_eta(1) == log(2)
assert dirichlet_eta(1, S.Half).simplify() == pi/2
assert dirichlet_eta(1, 2) == 1 - log(2)
assert dirichlet_eta(2) == pi**2/12
assert dirichlet_eta(4) == pi**4*Rational(7, 720)
assert str(dirichlet_eta(I).evalf(n=10)) == '0.5325931818 + 0.2293848577*I'
assert str(dirichlet_eta(I, I).evalf(n=10)) == '3.462349253 + 0.220285771*I'
def test_riemann_xi_eval():
assert riemann_xi(2) == pi/6
assert riemann_xi(0) == Rational(1, 2)
assert riemann_xi(1) == Rational(1, 2)
assert riemann_xi(3).rewrite(zeta) == 3*zeta(3)/(2*pi)
assert riemann_xi(4) == pi**2/15
def test_rewriting():
from sympy.functions.elementary.piecewise import Piecewise
assert isinstance(dirichlet_eta(x).rewrite(zeta), Piecewise)
assert isinstance(dirichlet_eta(x).rewrite(genocchi), Piecewise)
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(dirichlet_eta(x), dirichlet_eta(x).rewrite(genocchi), 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 myexpand(func, target):
expanded = expand_func(func)
if target is not None:
return expanded == target
if expanded == func: # it didn't expand
return False
# check to see that the expanded and original evaluate to the same value
subs = {}
for a in func.free_symbols:
subs[a] = randcplx()
return abs(func.subs(subs).n()
- expanded.replace(exp_polar, exp).subs(subs).n()) < 1e-10
def test_polylog_expansion():
assert polylog(s, 0) == 0
assert polylog(s, 1) == zeta(s)
assert polylog(s, -1) == -dirichlet_eta(s)
assert polylog(s, exp_polar(I*pi*Rational(4, 3))) == polylog(s, exp(I*pi*Rational(4, 3)))
assert polylog(s, exp_polar(I*pi)/3) == polylog(s, exp(I*pi)/3)
assert myexpand(polylog(1, z), -log(1 - z))
assert myexpand(polylog(0, z), z/(1 - z))
assert myexpand(polylog(-1, z), z/(1 - z)**2)
assert ((1-z)**3 * expand_func(polylog(-2, z))).simplify() == z*(1 + z)
assert myexpand(polylog(-5, z), None)
def test_polylog_series():
assert polylog(1, z).series(z, n=5) == z + z**2/2 + z**3/3 + z**4/4 + O(z**5)
assert polylog(1, sqrt(z)).series(z, n=3) == z/2 + z**2/4 + sqrt(z)\
+ z**(S(3)/2)/3 + z**(S(5)/2)/5 + O(z**3)
# https://github.com/sympy/sympy/issues/9497
assert polylog(S(3)/2, -z).series(z, 0, 5) == -z + sqrt(2)*z**2/4\
- sqrt(3)*z**3/9 + z**4/8 + O(z**5)
def test_issue_8404():
i = Symbol('i', integer=True)
assert Abs(Sum(1/(3*i + 1)**2, (i, 0, S.Infinity)).doit().n(4)
- 1.122) < 0.001
def test_polylog_values():
assert polylog(2, 2) == pi**2/4 - I*pi*log(2)
assert polylog(2, S.Half) == pi**2/12 - log(2)**2/2
for z in [S.Half, 2, (sqrt(5)-1)/2, -(sqrt(5)-1)/2, -(sqrt(5)+1)/2, (3-sqrt(5))/2]:
assert Abs(polylog(2, z).evalf() - polylog(2, z, evaluate=False).evalf()) < 1e-15
z = Symbol("z")
for s in [-1, 0]:
for _ in range(10):
assert verify_numerically(polylog(s, z), polylog(s, z, evaluate=False),
z, a=-3, b=-2, c=S.Half, d=2)
assert verify_numerically(polylog(s, z), polylog(s, z, evaluate=False),
z, a=2, b=-2, c=5, d=2)
from sympy.integrals.integrals import Integral
assert polylog(0, Integral(1, (x, 0, 1))) == -S.Half
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)
def test_stieltjes():
assert isinstance(stieltjes(x), stieltjes)
assert isinstance(stieltjes(x, a), stieltjes)
# Zero'th constant EulerGamma
assert stieltjes(0) == S.EulerGamma
assert stieltjes(0, 1) == S.EulerGamma
# Not defined
assert stieltjes(nan) is nan
assert stieltjes(0, nan) is nan
assert stieltjes(-1) is S.ComplexInfinity
assert stieltjes(1.5) is S.ComplexInfinity
assert stieltjes(z, 0) is S.ComplexInfinity
assert stieltjes(z, -1) is S.ComplexInfinity
def test_stieltjes_evalf():
assert abs(stieltjes(0).evalf() - 0.577215664) < 1E-9
assert abs(stieltjes(0, 0.5).evalf() - 1.963510026) < 1E-9
assert abs(stieltjes(1, 2).evalf() + 0.072815845) < 1E-9
def test_issue_10475():
a = Symbol('a', extended_real=True)
b = Symbol('b', extended_positive=True)
s = Symbol('s', zero=False)
assert zeta(2 + I).is_finite
assert zeta(1).is_finite is False
assert zeta(x).is_finite is None
assert zeta(x + I).is_finite is None
assert zeta(a).is_finite is None
assert zeta(b).is_finite is None
assert zeta(-b).is_finite is True
assert zeta(b**2 - 2*b + 1).is_finite is None
assert zeta(a + I).is_finite is True
assert zeta(b + 1).is_finite is True
assert zeta(s + 1).is_finite is True
def test_issue_14177():
n = Symbol('n', nonnegative=True, integer=True)
assert zeta(-n).rewrite(bernoulli) == bernoulli(n+1) / (-n-1)
assert zeta(-n, a).rewrite(bernoulli) == bernoulli(n+1, a) / (-n-1)
z2n = -(2*I*pi)**(2*n)*bernoulli(2*n) / (2*factorial(2*n))
assert zeta(2*n).rewrite(bernoulli) == z2n
assert expand_func(zeta(s, n+1)) == zeta(s) - harmonic(n, s)
assert expand_func(zeta(-b, -n)) is nan
assert expand_func(zeta(-b, n)) == zeta(-b, n)
n = Symbol('n')
assert zeta(2*n) == zeta(2*n) # As sign of z (= 2*n) is not determined
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