I would like to write the sum over a series as a complex integral using residual theorem. I don't know if it is possible or not but I tried something and I don't know if I wrote something true...
Given the series $\sum_{n=1} ^{\infty}{e^{-n\nu}}$ how can someone write it as a complex integral? I know it looks seely because everyone knows what the sum equals to but I have reasons to write it that way.
Using the function $\pi coth(\pi z)$ one can sample all the points in the sum, but it's not just from 1 to $\infty$, so by adding heaviside function: $\theta(z+z^*)$ we can sample just the upper plane.
Now, naivly I can say that $\sum_{n=1} ^{\infty}{e^{-n\nu}}= \sum{Res(\pi coth(\pi z) \theta(z+z*)e^{-\nu z},z=z_{poles})}$
$= \sum_{poles} \int \frac{1}{2\pi i} \pi coth(\pi z) \theta(z+z*)e^{-\nu z} dz$ with a half circle contour. But I have two problems, first- the theorem is for a final number of poles and what I wrote has infinit number of poles, and I'm not sure if it's legit to add the heaviside...
Maybe someone has a different approach? maybe it's not possible?
Thank you