Finding residues of $\frac{z}{e^z+e^{-z}}$.

Question

I am having trouble finding the residues of

$$\frac{z}{e^z+e^{-z}}.$$

I have found the poles (at $z= ±i(\frac{\pi}{2} +n\pi)$) but I cannot find a way to expand the function into a Laurent series to find the residues.

Answer 1

You don't need such a Laurent expansion, since\begin{align}\require{cancel}\operatorname{res}\left(i\left(\frac\pi2+n\pi\right),\frac z{e^z+e^{-z}}\right)&=\frac{i\left(\frac\pi2+n\pi\right)}{e^z-e^{-z}|_{z=i\left(\frac\pi2+n\pi\right)}}\text{ (because }(e^z+e^{-z})'=e^z-e^{-z}\text)\\&=\frac{\cancel i\left(\frac\pi2+n\pi\right)}{(-1)^n2\cancel i}\\&=(-1)^n\left(\frac\pi4+\frac\pi2n\right).\end{align}