Finding order of poles of complex functions
Complex function $\frac{1}{1-cos z}$ has order 2 poles when $z= 2n\pi, n\in \mathbb Z$. I can find the values of $z$ on poles but how to find the order of the poles?
Solution 1:
The order of a pole of $f$ at a point $a$ is the order of $a$ as a zero of $\frac1f$. And, if $n\in\Bbb Z$ and if $f(z)=1-\cos z$, then$$f'(2\pi n)=\sin(2\pi n)=0\quad\text{and}\quad f''(2\pi n)=\cos(2\pi n)=(-1)^n\ne0.$$So, $2\pi n$ is a zero of order $2$ of $f$, and therefore a pole of order $2$ of $\frac1{1-\cos z}$.