def eval(cls, n, z):
n, z = map(sympify, (n, z))
from sympy import unpolarify
if n.is_integer:
if n.is_nonnegative:
nz = unpolarify(z)
if z != nz:
return polygamma(n, nz)
if n.is_positive:
if z is S.Half:
return (
(-1) ** (n + 1)
* factorial(n)
* (2 ** (n + 1) - 1)
* zeta(n + 1)
)
if n is S.NegativeOne:
return loggamma(z)
else:
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
if n.is_Number:
if n.is_zero:
return S.Infinity
else:
return S.Zero
if n.is_zero:
return S.Infinity
elif z.is_Integer:
if z.is_nonpositive:
return S.ComplexInfinity
else:
if n.is_zero:
return -S.EulerGamma + harmonic(z - 1, 1)
elif n.is_odd:
return (-1) ** (n + 1) * factorial(n) * zeta(n + 1, z)
if n.is_zero:
if z is S.NaN:
return S.NaN
elif z.is_Rational:
p, q = z.as_numer_denom()
# only expand for small denominators to avoid creating long expressions
if q <= 5:
return expand_func(polygamma(S.Zero, z, evaluate=False))
elif z in (S.Infinity, S.NegativeInfinity):
return S.Infinity
else:
t = z.extract_multiplicatively(S.ImaginaryUnit)
if t in (S.Infinity, S.NegativeInfinity):
return S.Infinity
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
def _eval_rewrite_as_harmonic(self, n, z, **kwargs):
if n.is_integer:
if n.is_zero:
return harmonic(z - 1) - S.EulerGamma
else:
return S.NegativeOne**(n + 1) * factorial(n) * (
zeta(n + 1) - harmonic(z - 1, n + 1))
def test_moment_generating_functions():
t = S('t')
geometric_mgf = moment_generating_function(Geometric('g', S.Half))(t)
assert geometric_mgf.diff(t).subs(t, 0) == 2
logarithmic_mgf = moment_generating_function(Logarithmic('l', S.Half))(t)
assert logarithmic_mgf.diff(t).subs(t, 0) == 1 / log(2)
negative_binomial_mgf = moment_generating_function(
NegativeBinomial('n', 5, Rational(1, 3)))(t)
assert negative_binomial_mgf.diff(t).subs(t, 0) == Rational(5, 2)
poisson_mgf = moment_generating_function(Poisson('p', 5))(t)
assert poisson_mgf.diff(t).subs(t, 0) == 5
skellam_mgf = moment_generating_function(Skellam('s', 1, 1))(t)
assert skellam_mgf.diff(t).subs(
t, 2) == (-exp(-2) + exp(2)) * exp(-2 + exp(-2) + exp(2))
yule_simon_mgf = moment_generating_function(YuleSimon('y', 3))(t)
assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == Rational(3, 2)
zeta_mgf = moment_generating_function(Zeta('z', 5))(t)
assert zeta_mgf.diff(t).subs(t, 0) == pi**4 / (90 * zeta(5))
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_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_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 eval_zeta_function(self, f, limits):
"""
Check whether the function matches with the zeta function.
If it matches, then return a `Piecewise` expression because
zeta function does not converge unless `s > 1` and `q > 0`
"""
i, a, b = limits
w, y, z = Wild('w', exclude=[i]), Wild('y', exclude=[i]), Wild('z', exclude=[i])
result = f.match((w * i + y) ** (-z))
if result is not None and b == S.Infinity:
coeff = 1 / result[w] ** result[z]
s = result[z]
q = result[y] / result[w] + a
return Piecewise((coeff * zeta(s, q), And(q > 0, s > 1)), (self, True))
def eval_zeta_function(self, f, limits):
"""
Check whether the function matches with the zeta function.
If it matches, then return a `Piecewise` expression because
zeta function does not converge unless `s > 1` and `q > 0`
"""
i, a, b = limits
w, y, z = Wild('w', exclude=[i]), Wild('y', exclude=[i]), Wild('z', exclude=[i])
result = f.match((w * i + y) ** (-z))
if result is not None and b == S.Infinity:
coeff = 1 / result[w] ** result[z]
s = result[z]
q = result[y] / result[w] + a
return Piecewise((coeff * zeta(s, q), And(q > 0, s > 1)), (self, True))
def canonize(cls, n, m=None):
if m is None:
m = S.One
if n == oo:
from sympy.functions.special.zeta_functions import zeta
return zeta(m)
if n.is_Integer and n.is_nonnegative and m.is_Integer:
if n == 0:
return S.Zero
if not m in cls._functions:
@recurrence_memo([0])
def f(n, prev):
return prev[-1] + S.One / n**m
cls._functions[m] = f
return cls._functions[m](int(n))
def _eval_apply(cls, n, m=None):
if m is None:
m = Integer(1)
if n == oo:
from sympy.functions.special.zeta_functions import zeta
return zeta(m)
if isinstance(n, Integer) and n.is_nonnegative and \
isinstance(m, Integer):
if n == 0:
return S.Zero
if not m in cls._functions:
@recurrence_memo([0])
def f(n, prev):
return prev[-1] + S.One / n**m
cls._functions[m] = f
return cls._functions[m](int(n))
def canonize(cls, n, m=None):
if m is None:
m = S.One
if n == oo:
from sympy.functions.special.zeta_functions import zeta
return zeta(m)
if n.is_Integer and n.is_nonnegative and m.is_Integer:
if n == 0:
return S.Zero
if not m in cls._functions:
@recurrence_memo([0])
def f(n, prev):
return prev[-1] + S.One / n**m
cls._functions[m] = f
return cls._functions[m](int(n))
def test_trigamma():
assert trigamma(nan) == nan
assert trigamma(oo) == 0
assert trigamma(1) == pi**2/6
assert trigamma(2) == pi**2/6 - 1
assert trigamma(3) == pi**2/6 - Rational(5, 4)
assert trigamma(x, evaluate=False).rewrite(zeta) == zeta(2, x)
assert trigamma(x, evaluate=False).rewrite(harmonic) == \
trigamma(x).rewrite(polygamma).rewrite(harmonic)
assert trigamma(x,evaluate=False).fdiff() == polygamma(2, x)
assert trigamma(x,evaluate=False).is_real is None
assert trigamma(x,evaluate=False).is_positive is None
assert trigamma(x,evaluate=False).is_negative is None
assert trigamma(x,evaluate=False).rewrite(polygamma) == polygamma(1, x)
def _characteristic_function(self, t):
return polylog(self.s, exp(I * t)) / zeta(self.s)
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_sympy__functions__special__zeta_functions__zeta():
from sympy.functions.special.zeta_functions import zeta
assert _test_args(zeta(101))
def test_zeta():
assert str(zeta(3)) == "zeta(3)"
def test_sympy__functions__special__zeta_functions__zeta():
from sympy.functions.special.zeta_functions import zeta
assert _test_args(zeta(101))
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 _eval_rewrite_as_zeta(self, z, **kwargs):
return zeta(2, z)
def pdf(self, k):
s = self.s
return 1 / (k**s * zeta(s))
def test_polygamma():
assert polygamma(n, nan) is nan
assert polygamma(0, oo) is oo
assert polygamma(0, -oo) is oo
assert polygamma(0, I*oo) is oo
assert polygamma(0, -I*oo) is oo
assert polygamma(1, oo) == 0
assert polygamma(5, oo) == 0
assert polygamma(0, -9) is zoo
assert polygamma(0, -9) is zoo
assert polygamma(0, -1) is zoo
assert polygamma(0, 0) is zoo
assert polygamma(0, 1) == -EulerGamma
assert polygamma(0, 7) == Rational(49, 20) - EulerGamma
assert polygamma(1, 1) == pi**2/6
assert polygamma(1, 2) == pi**2/6 - 1
assert polygamma(1, 3) == pi**2/6 - Rational(5, 4)
assert polygamma(3, 1) == pi**4 / 15
assert polygamma(3, 5) == 6*(Rational(-22369, 20736) + pi**4/90)
assert polygamma(5, 1) == 8 * pi**6 / 63
assert polygamma(1, S.Half) == pi**2 / 2
assert polygamma(2, S.Half) == -14*zeta(3)
assert polygamma(11, S.Half) == 176896*pi**12
def t(m, n):
x = S(m)/n
r = polygamma(0, x)
if r.has(polygamma):
return False
return abs(polygamma(0, x.n()).n() - r.n()).n() < 1e-10
assert t(1, 2)
assert t(3, 2)
assert t(-1, 2)
assert t(1, 4)
assert t(-3, 4)
assert t(1, 3)
assert t(4, 3)
assert t(3, 4)
assert t(2, 3)
assert t(123, 5)
assert polygamma(0, x).rewrite(zeta) == polygamma(0, x)
assert polygamma(1, x).rewrite(zeta) == zeta(2, x)
assert polygamma(2, x).rewrite(zeta) == -2*zeta(3, x)
assert polygamma(I, 2).rewrite(zeta) == polygamma(I, 2)
n1 = Symbol('n1')
n2 = Symbol('n2', real=True)
n3 = Symbol('n3', integer=True)
n4 = Symbol('n4', positive=True)
n5 = Symbol('n5', positive=True, integer=True)
assert polygamma(n1, x).rewrite(zeta) == polygamma(n1, x)
assert polygamma(n2, x).rewrite(zeta) == polygamma(n2, x)
assert polygamma(n3, x).rewrite(zeta) == polygamma(n3, x)
assert polygamma(n4, x).rewrite(zeta) == polygamma(n4, x)
assert polygamma(n5, x).rewrite(zeta) == (-1)**(n5 + 1) * factorial(n5) * zeta(n5 + 1, x)
assert polygamma(3, 7*x).diff(x) == 7*polygamma(4, 7*x)
assert polygamma(0, x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma
assert polygamma(2, x).rewrite(harmonic) == 2*harmonic(x - 1, 3) - 2*zeta(3)
ni = Symbol("n", integer=True)
assert polygamma(ni, x).rewrite(harmonic) == (-1)**(ni + 1)*(-harmonic(x - 1, ni + 1)
+ zeta(ni + 1))*factorial(ni)
# Polygamma of non-negative integer order is unbranched:
k = Symbol('n', integer=True, nonnegative=True)
assert polygamma(k, exp_polar(2*I*pi)*x) == polygamma(k, x)
# but negative integers are branched!
k = Symbol('n', integer=True)
assert polygamma(k, exp_polar(2*I*pi)*x).args == (k, exp_polar(2*I*pi)*x)
# Polygamma of order -1 is loggamma:
assert polygamma(-1, x) == loggamma(x)
# But smaller orders are iterated integrals and don't have a special name
assert polygamma(-2, x).func is polygamma
# Test a bug
assert polygamma(0, -x).expand(func=True) == polygamma(0, -x)
assert polygamma(2, 2.5).is_positive == False
assert polygamma(2, -2.5).is_positive == False
assert polygamma(3, 2.5).is_positive == True
assert polygamma(3, -2.5).is_positive is True
assert polygamma(-2, -2.5).is_positive is None
assert polygamma(-3, -2.5).is_positive is None
assert polygamma(2, 2.5).is_negative == True
assert polygamma(3, 2.5).is_negative == False
assert polygamma(3, -2.5).is_negative == False
assert polygamma(2, -2.5).is_negative is True
assert polygamma(-2, -2.5).is_negative is None
assert polygamma(-3, -2.5).is_negative is None
assert polygamma(I, 2).is_positive is None
assert polygamma(I, 3).is_negative is None
# issue 17350
assert polygamma(pi, 3).evalf() == polygamma(pi, 3)
assert (I*polygamma(I, pi)).as_real_imag() == \
(-im(polygamma(I, pi)), re(polygamma(I, pi)))
assert (tanh(polygamma(I, 1))).rewrite(exp) == \
(exp(polygamma(I, 1)) - exp(-polygamma(I, 1)))/(exp(polygamma(I, 1)) + exp(-polygamma(I, 1)))
assert (I / polygamma(I, 4)).rewrite(exp) == \
I*sqrt(re(polygamma(I, 4))**2 + im(polygamma(I, 4))**2)\
/((re(polygamma(I, 4)) + I*im(polygamma(I, 4)))*Abs(polygamma(I, 4)))
assert unchanged(polygamma, 2.3, 1.0)
# issue 12569
assert unchanged(im, polygamma(0, I))
assert polygamma(Symbol('a', positive=True), Symbol('b', positive=True)).is_real is True
assert polygamma(0, I).is_real is None
def test_zeta():
s = S(5)
x = Zeta('x', s)
assert E(x) == zeta(s - 1) / zeta(s)
assert simplify(
variance(x)) == (zeta(s) * zeta(s - 2) - zeta(s - 1)**2) / zeta(s)**2
def test_gruntz_eval_special_fail():
# TODO zeta function series
assert gruntz(
exp((log(2) + 1) * x) * (zeta(x + exp(-x)) - zeta(x)), x,
oo) == -log(2)
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(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 _eval_rewrite_as_zeta(self, n, z, **kwargs):
if n.is_integer:
if (n - S.One).is_nonnegative:
return (-1)**(n + 1) * factorial(n) * zeta(n + 1, z)
def _moment_generating_function(self, t):
return polylog(self.s, exp(t)) / zeta(self.s)
def _eval_rewrite_as_zeta(self, n, z, **kwargs):
if n >= S.One:
return (-1)**(n + 1) * factorial(n) * zeta(n + 1, z)
else:
return self
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