This is an exercise problem in GTM11 by Conway (chap. V 2.8). It asks me to show
$$\pi\cot\pi a=\frac{1}{a}+\sum_{n\neq0}\left(\frac{1}{a-n}+\frac{1}{n}\right)$$
by using the fact that
$$\lim_{n\to\infty}\int_{\gamma_n}\frac{\pi\cot\pi z}{z^2-a^2}dz=0$$
where $\gamma_n$ is the square centered at 0 with side length $2n-1$.
Now I have been able to show the integral goes to 0, but I cannot see how this leads to the partial fraction development of $\pi\cot\pi z$. Thanks for any suggestions.