Alternating sine series with a quadratic denominator (${\sum }_{n\in \mathbb{Z}}(-1{)}^{n}n\sin (nx)/(a^2+n^2)=-\pi \sinh (ax)/\sinh (a\pi )$)

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.