Show that $\sum_{n=-\infty}^{\infty}\frac{e^{-i \omega_n t}}{\omega^2_n+k^2}=\frac{1}{2 k T}\left[\frac{2}{e^{k/T}-1}\cosh{k t}+e^{-k t}\right]$

Question

I'm trying to evaluate sum $$S=\sum_{n=-\infty}^{\infty}\frac{e^{-i \omega_n t}}{\omega^2_n+k^2}$$ where $\omega_n=2 \pi T n$, $n \in \mathbb{Z}$

The similar sum without exponent can be evaluated by residue trick. Using the same trick we get $$S=-\sum_i\operatorname*{Res}_{z=i}\frac{1}{T(e^{z/T}-1)}\frac{e^{-z t}}{(-z^2+k^2)}$$ where $i=\{k,-k,\infty\}$

However in infiniy resudie is not equal to zero, and i don't know how to evaluate it.