A Question About Poles.

Question

I have some questions in my mind bothering me to understand poles.

Let $z_0$ be a pole of order $m$ for $f(z)$. Does that mean:

1- $(m+1)$ is the smallest positive integer such that: $\lim_{z\rightarrow z_0}f(z)(z-z_0)^{m+1}=0$ ?

2- For any $n <m$, we have: $\lim_{z\rightarrow z_0}f(z)(z-z_0)^n=+\infty$ ?

3- Could we have $\lim_{z\rightarrow z_0}f(z)(z-z_0)^m=0$?

Answer 1

HINT:

We can write $f(z)=\dfrac{h(z)}{(z-z_0)^m}$, in a neighborhood of $z_0$, where $h$ is an analytic function. Then, we have

$$h(z)=\sum_{n=0}^{\infty}\dfrac{h^{(n)}(z_0)(z-z_0)^n}{n!}$$

and therefore

$$f(z)=\sum_{n=0}^{\infty}\dfrac{h^{(n)}(z_0)(z-z_0)^{n-m}}{n!}$$