I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m x}{T_x}\right) \cos \left(\frac{2 \pi n y}{T_y}\right) \sin \left(\frac{\pi m l_x}{T_x}\right) \sin \left(\frac{\pi n l_y}{T_y}\right) \left(1-\exp \left(-\pi h \sqrt{\left(\frac{m}{T_x}\right){}^2+\left(\frac{n}{T_y}\right){}^2}\right) \cosh \left(2 \pi z \sqrt{\left(\frac{m}{T_x}\right){}^2+\left(\frac{n}{T_y}\right){}^2}\right)\right)}{\pi ^2 \left(n T_x^2 \left(\left(\frac{m}{T_x}\right){}^2+\left(\frac{n}{T_y}\right){}^2\right)\right)}$
At the first step, the term without the exponentials $\exp \left(-\pi h \sqrt{\left(\frac{m}{T_x}\right){}^2+\left(\frac{n}{T_y}\right){}^2}\right) \cosh \left(2 \pi z \sqrt{\left(\frac{m}{T_x}\right){}^2+\left(\frac{n}{T_y}\right){}^2}\right)$ is not so difficult to evaluate by using residue calculus. In this case h>2z so Jordan's lemma is valid.
The major difficulties arise when dealing with the square roots at the exponent. We have a branch cuts that make things more difficult. At this point I failed to find a nice contour of integration.
Although the sum looks quite symmetric, I wasn't able to separate the variables and transform the double sum to a product of sums.
I assume, finding the following sum as a function of n $\sum _{m=1}^{\infty } \frac{m \cos \left(\frac{2 \pi m x}{T_x}\right) \sin \left(\frac{\pi m l_x}{T_x}\right) \exp \left(-\pi (h-2 z) \sqrt{\left(\frac{m}{T_x}\right){}^2+\left(\frac{n}{T_y}\right){}^2}\right)}{n \left(\left(\frac{m}{T_x}\right){}^2+\left(\frac{n}{T_y}\right){}^2\right)}$ will be instrumental in calculating the whole sum.
If following the approach with contour integration, the following integral will be of interest: $\oint \text{d$\xi $}\left(\frac{\xi \cos (2 \pi \xi x) \sin \left(\pi \xi l_x\right) \cot \left(\pi \xi T_x\right) \exp \left(-\pi (h-2 z) \sqrt{\xi ^2+\left(\frac{n}{T_y}\right){}^2}\right)}{n \left(\xi ^2+\left(\frac{n}{T_y}\right){}^2\right)}\right)$
I would appreciate any advise on how to tackle this problem.