sequences and series - Using the residue theorem to evaluate $sum_limits{n=-infty}^{infty} frac{e^{in alpha}}{(n-beta)^{2}+gamma^{2}}$

I would like to know how to sum up to following series (from the Gradshteyn-Ryzhik tables):

$$\sum_{n=-\infty}^\infty\frac{e^{in\alpha}}{(n-\beta)^2+\gamma^2}=\frac{\pi}{\gamma}\frac{e^{i\beta(\alpha-2\pi)}\sinh(\gamma\alpha)+e^{i\beta\alpha}\sinh[\gamma(2\pi-\alpha)]}{\cosh(2\pi\gamma)-\cos(2\pi\beta)}$$

with $0\leq\alpha\leq2\pi$. In the special case of $\alpha=0$, we have $$\sum_{n=-\infty}^\infty\frac1{(n-\beta)^2+\gamma^2}=\frac{\pi}{\gamma}\frac{\sinh[ 2\pi\gamma]}{\cosh(2\pi\gamma)-\cos(2\pi\beta)}$$ and I now that I can use the function $$\frac{\cot(\pi z)}{(z-\beta)^2+\gamma^2}$$ to sum up this series via the residue theorem.
In more detail, the singularities are $\beta\pm {\rm i}\gamma$ and $z_n=n$, $n\in\mathbb{N}$.
If I sum the corresponding residues, I get what the above result (for $\alpha=0$).

I am not sure, however:

1) How to choose the contour in order to have a correct argumentation?

2) What to do with the general case $\alpha\neq 0$?