I need to solve this sum: $$\sum_{n=-\infty}^\infty\frac{y^2}{[(x-n\pi)^2+y^2]^{3/2}}.$$ Do you have any ideas for how I could do this?
I know that this sum: $$\sum_{n=-\infty}^\infty\frac{y}{(x-n\pi)^2+y^2},$$ can be simplified by using the following identities (see this Wikipedia article): $$\cot(x+iy) = \sum_{n=-\infty}^\infty\frac{1}{x-n\pi + iy},$$ $$\sum_{n=-\infty}^\infty\frac{y}{(x-n\pi)^2+y^2}=\frac{i}{2}\sum_{n=-\infty}^\infty\frac{1}{x-n\pi+iy}-\frac{1}{x-n\pi-iy}.$$ Hence, $$\begin{aligned} \sum_{n=-\infty}^\infty\frac{y}{(x-n\pi)^2+y^2}&=\frac{i}{2}[\cot(x+iy)-\cot(x-iy)] \\ &=\frac{\sinh(2y)}{\cosh(2y)-\cos(2x)}. \end{aligned}$$
Do you know if a similar trick can be used to solve the original sum?
