Finding residues of $ 1/(\exp{z^2}-1)$

Question

i was asked to find the poles and the residues of

$f(z) = \frac{1}{\exp{(z^2)}-1}$

So i found the poles at $z=0$ and $z = \sqrt{2 n \pi i}, n \in \mathbb{N}$

I wanted to find the residues, but it's not going well. First of all when the task ask me to find residues is it at the two poles or just wherever? I just assumed it was at $a=0$ so I could use:

$ Res(f,a) = c_{-1} = 1/(2\pi i) \int f(z) dz$, where we integrate on the circumference of a ball with radius 1 and center 0.

However, this does not work for me. I tried writing the integral with polar coordinates but i think it's the wrong way, because the integral becomes too hard.

Would love your input

Answer 1

Just use the fact that, if $z_0$ is a square root of $2\pi i n$, then$$\operatorname{res}_{z=z_0}\bigl(f(z)\bigr)=\frac1{2z_0e^{{z_0}^2}}=\frac1{2z_0},$$unless, of course, $n=0$. If $n=0$, then $z_0=0$ and, since $f$ is an even function, its residue at $0$ is $0$.