I need to prove
$$\sum_{n=1}^{+\infty}\frac{1}{n^2-\alpha^2}=\frac{1}{2\alpha^2}-\frac{\pi}{2\alpha\tan(\alpha\pi)},$$
with $\alpha$ a non integer complex.
I know that I have to use the Parseval's Indentity relation but I don't know whose function's expansion makes this possible.