Using Residues to Sum a Series
Letbe an even function analytic onexcept for poles at the pointsnone of which is an integer, and possibly at 0, and letbe the square contour with vertices atSuppose also that the functionis such that
With this theorem we can find the sum of a wide range of series exactly.
The functionis even and analytic onapart from a pole of order 2 at
The functionhas a pole of order 3 atThe residue atis given by
Iflies on the contourthensofor
Hence by the Estimation Theoremas