Jordan's lemma

Jordan's lemma controls exponential contour integrals on large semicircles and makes their arc contributions vanish.

For real $a\ne 0$ and $|x|<\pi$, integrate $z\sin (xz)/((a^2+z^2)\sin \pi z)$ around a thin positively oriented strip containing the integers. Its integer residues give the series. Closing the two strip sides outward instead gives minus the sum of the residues at $z=\pm ia$, each $\sinh (ax)/(2\sinh (a\pi ))$. The factor $|x|<\pi$ ensures exponential decay on those closures. The bilateral series is interpreted by symmetric partial sums; at $a=0$ its continuous extension is $-x$ with the zero-index summand defined to be zero.