Even-valued zeta functions evaluated using Parseval's theorem

zeta.py

# Evaluating zeta functions using Parseval sums
# Amar
# version 1, Aug 2013

from sympy import *
from sympy import init_printing
init_printing() 

n = symbols('n',integer=True)
s = symbols('s',rational=True)
t = symbols('t')

def a0(s):
    # DC component of fourier series
    return ((2*pi)**s)/(s+1)

def b(n,s):
    return integrate(t**s * E**(I*n*t),(t,0,2*pi))/(2*pi)

def a(n,s):
    # fourier components of the function t**s, defined recursively
    if s!=0 and s > Rational(1,2): return (s)/(I*n) * ( a0(s-1) - a(n,s-1) )
    elif s==Rational(1,2): return b(n,Rational(1,2))
    else: return 0

def lhs(n,s):
    # fourier side of Parseval sum. Factor of 2 is because only single-sided fourier series is considered
    return  apart( 2 * Abs(a(n,s))**2, n ) #,full=True )  

def rhs(s):
    # time side of Parseval sum
    return a0(2*s) - a0(s)**2

def equation(n,s):
    return lhs(n,s), rhs(s)

pprint(equation(n,1))          # gives the series sum for zeta(2)
pprint(equation(n,2))          # gives the series sum for zeta(4)