sympy/functions/special/zeta_functions.py

   1
   2 from sympy.core import SingleValuedFunction, S, Basic, pi
   3
   4 ###############################################################################
   5 ###################### HURWITZ GENERALIZED ZETA FUNCTION ######################
   6 ###############################################################################
   7
   8 class zeta(SingleValuedFunction):
   9
  10     nofargs = (1, 2)
  11
  12     @classmethod
  13     def canonize(cls, z, a=S.One):
  14         z, a = map(Basic.sympify, (z, a))
  15
  16         if isinstance(a, Basic.Number):
  17             if isinstance(a, Basic.NaN):
  18                 return S.NaN
  19             elif isinstance(a, Basic.Zero):
  20                 return cls(z)
  21
  22         if isinstance(z, Basic.Number):
  23             if isinstance(z, Basic.NaN):
  24                 return S.NaN
  25             elif isinstance(z, Basic.Infinity):
  26                 return S.One
  27             elif isinstance(z, Basic.Zero):
  28                 if a.is_negative:
  29                     return S.Half - a - 1
  30                 else:
  31                     return S.Half - a
  32             elif isinstance(z, Basic.One):
  33                 return S.ComplexInfinity
  34             elif isinstance(z, Basic.Integer):
  35                 if isinstance(a, Basic.Integer):
  36                     if z.is_negative:
  37                         zeta = (-1)**z * Basic.bernoulli(-z+1)/(-z+1)
  38                     elif z.is_even:
  39                         B, F = Basic.bernoulli(z), Basic.Factorial(z)
  40                         zeta = 2**(z-1) * abs(B) * pi**z / F
  41
  42                     if a.is_negative:
  43                         return zeta + Basic.harmonic(abs(a), z)
  44                     else:
  45                         return zeta - Basic.harmonic(a-1, z)
  46
  47
  48 class dirichlet_eta(SingleValuedFunction):
  49     """
  50     Dirichlet eta function
  51     """
  52     nofargs = 1
  53
  54     @classmethod
  55     def canonize(cls, s):
  56         if s == 1:
  57             return Basic.log(2)
  58         else:
  59             return (1-2**(1-s)) * zeta(s)