The exercise asks to find the closed form of $$\sum_{n=-\infty}^{\infty}\frac{1}{(z+n)^2+a^2}$$where $a$ is a complex number.
I know that $\frac{\pi^2}{\sin^2(\pi z)}=\sum_{n=-\infty}^{\infty}\frac{1}{(z+n)^2}$, but don't how to develop this fact. Is this some kind of a shift of this series?
Thank you!