By using the theorem below, I found the sum ($a\not\in\Bbb Z$): $$\sum_{n=-\infty}^\infty \frac1{(n+a)^2}=-\pi Res\left[\frac{\cot\pi z}{(z+a)^2}\right]_{z=-a}=-\pi\frac d{dz}(\cot\pi z)(-a)=\frac{\pi^2}{\sin^2(\pi a)}.$$
Theorem: Let $f(z)$ be a meromorphic function having a finite number of poles: $a_1,a_2,...,a_m$ which do not coincide with any of $z=0,\pm1,\pm2,...$. If there exists a squence of contours $\{C_n\}$ which shrink to the point at infinity and $$\lim_{n\to\infty}\int_{C_n}f(z)\cot\pi z\,dz=0,\tag1$$ then $$\sum_{n=-\infty}^\infty f(n)=-\pi\sum_{k=1}^m Res[f(z)\cot\pi z]_{z=z_m}. $$
In my calculation, I couldn't check the condition $(1)$. Can any body explain the condition $(1)$ and how to check it in the above example?
Thanks.
