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test_zeta_functions.py
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test_zeta_functions.py
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from sympy import (Symbol, zeta, nan, Rational, Float, pi, dirichlet_eta, log,
zoo, expand_func, polylog, lerchphi, S, exp, sqrt, I,
exp_polar, polar_lift, O, stieltjes, Abs, Sum, oo)
from sympy.core.function import ArgumentIndexError
from sympy.functions.combinatorial.numbers import bernoulli, factorial, harmonic
from sympy.utilities.pytest import raises
from sympy.utilities.randtest import (test_derivative_numerically as td,
random_complex_number as randcplx, verify_numerically as tn)
x = Symbol('x')
a = Symbol('a')
b = Symbol('b', negative=True)
z = Symbol('z')
s = Symbol('s')
def test_zeta_eval():
assert zeta(nan) == nan
assert zeta(x, nan) == nan
assert zeta(0) == Rational(-1, 2)
assert zeta(0, x) == Rational(1, 2) - x
assert zeta(0, b) == Rational(1, 2) - b
assert zeta(1) == zoo
assert zeta(1, 2) == zoo
assert zeta(1, -7) == zoo
assert zeta(1, x) == zoo
assert zeta(2, 1) == pi**2/6
assert zeta(2) == pi**2/6
assert zeta(4) == pi**4/90
assert zeta(6) == pi**6/945
assert zeta(2, 2) == pi**2/6 - 1
assert zeta(4, 3) == pi**4/90 - Rational(17, 16)
assert zeta(6, 4) == pi**6/945 - Rational(47449, 46656)
assert zeta(2, -2) == pi**2/6 + Rational(5, 4)
assert zeta(4, -3) == pi**4/90 + Rational(1393, 1296)
assert zeta(6, -4) == pi**6/945 + Rational(3037465, 2985984)
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(119, 12)
assert zeta(-1, -9) == Rational(539, 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, 0, 2) == \
zeta(x, 0) - x*a*zeta(x + 1, 0) + O(a**2)
def test_dirichlet_eta_eval():
assert dirichlet_eta(0) == Rational(1, 2)
assert dirichlet_eta(-1) == Rational(1, 4)
assert dirichlet_eta(1) == log(2)
assert dirichlet_eta(2) == pi**2/12
assert dirichlet_eta(4) == pi**4*Rational(7, 720)
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 tn(dirichlet_eta(x), dirichlet_eta(x).rewrite(zeta), x)
assert tn(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 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():
from sympy import log
assert polylog(s, 0) == 0
assert polylog(s, 1) == zeta(s)
assert polylog(s, -1) == -dirichlet_eta(s)
assert polylog(s, exp_polar(4*I*pi/3)) == polylog(s, exp(4*I*pi/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_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():
from sympy.utilities.randtest import verify_numerically as tn
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 tn(polylog(s, z), polylog(s, z, evaluate=False), z,
a=-3, b=-2, c=S.Half, d=2)
assert tn(polylog(s, z), polylog(s, z, evaluate=False), z,
a=2, b=-2, c=5, d=2)
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(1)/2),
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, S(3)/2), None)
assert myexpand(lerchphi(z, s, S(7)/3), None)
assert myexpand(lerchphi(z, s, -S(1)/3), None)
assert myexpand(lerchphi(z, s, -S(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(2*I*pi/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) == nan
assert stieltjes(0, nan) == nan
assert stieltjes(-1) == S.ComplexInfinity
assert stieltjes(1.5) == S.ComplexInfinity
assert stieltjes(z, 0) == S.ComplexInfinity
assert stieltjes(z, -1) == 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', positive=True, integer=True)
assert zeta(2*n) == (-1)**(n + 1)*2**(2*n - 1)*pi**(2*n)*bernoulli(2*n)/factorial(2*n)
assert zeta(-n) == (-1)**(-n)*bernoulli(n + 1)/(n + 1)
n = Symbol('n')
assert zeta(2*n) == zeta(2*n) # As sign of z (= 2*n) is not determined