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Summation from minus infinity to infinity of 1/(n^4 + a^4)

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Evaluate the summation from minus infinity to infinity of 1/(n^4 + a^4).

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We can evaluate the summation as follows. When summing a meromorphic function f(z) over the zeroes of an analytic function g(z) we can consider the contour integral of f(z) g'(z)/g(z) dz over a counterclockwise circle of radius R. Then the zeroes of g(z) are the poles of the integrand (unless they are canceled by f(z)) and the residue there is just the value of f there. Then in the limit of R to infinity, the integral equals a pi i times the desired summation plus the sum of the residues of f(z) g'(z)/g(z) at the poles of f(z). If the integral tends to zero for R to infinity, then the desired summation is thus minus the sum of the residues of f(z) g'(z)/g(z) at the poles of f(z).

Since the function g(z) = sin(pi z) has its zeroes at the integers, we can sum ...

Solution Summary

A detailed calculation is given for the summation from minus infinity to infinity of 1/(n^4 + a^4).

$2.19