Ramanujan Cos/Cosh Identity

Author

Eric W. Weisstein
February 8, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/RamanujanCosCoshIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RamanujanCosCoshIdentity.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Identity

(((2 (∑_n^∞)((cosh(n θ))/(cosh(n π)))+1))^(-2))+(((2 (∑_n^∞)((cos(n θ))/(cosh(n π)))+1))^(-2))==((2 ((Γ((3/4)))^4))/π)

(((1+2Sum[((Cos[n θ])/(Cosh[n π])),{n,∞}]))^(-2))+(((1+2Sum[((Cosh[n θ])/(Cosh[n π])),{n,∞}]))^(-2))

Numerical Verification

N[With[{θ=Random[Complex]},{(1+2*Sum[Cos[k*θ]/Cosh[k*Pi],{k,1000}])^(-2)+(1+2*Sum[Cosh[k*θ]/Cosh[k*Pi],{k,1000}])^(-2),(2/Pi)*Gamma[3/4]^4}]]