State Jordan's Lemma.

Consider the integral

$$I=\oint_{C} d z \frac{z \sin (x z)}{\left(a^{2}+z^{2}\right) \sin \pi z}$$

for real $x$ and $a$. The rectangular contour $C$ runs from $+\infty+i \epsilon$ to $-\infty+i \epsilon$, to $-\infty-i \epsilon$, to $+\infty-i \epsilon$ and back to $+\infty+i \epsilon$, where $\epsilon$ is infinitesimal and positive. Perform the integral in two ways to show that

$$\sum_{n=-\infty}^{\infty}(-1)^{n} \frac{n \sin n x}{a^{2}+n^{2}}=-\pi \frac{\sinh a x}{\sinh a \pi}$$

for $|x|<\pi$.