Order of a pole

Definition

Let $c$ be a pole of $f$. The order of the pole $c$ is the unique integer $m \ge 1$ with the following property. On some punctured disc $0 < |z - c| < \delta$, $$f(z) = \frac{g(z)}{(z - c)^m},$$ where $g$ is holomorphic on the disc $|z - c| < \delta$ and $g(c) \neq 0$. A pole of order $1$ is a simple pole.

Remarks

The order of the pole of $f$ at $c$ is the order of the zero of $1/f$ at $c$. In the Laurent series $\sum a_n (z - c)^n$ of $f$ at $c$, the order is $m$ if and only if $a_{-m} \neq 0$ and $a_n = 0$ for each $n < -m$. Many authors give a pole of order $m$ the signed order $-m$. Then zeros have positive order and poles have negative order. For example, $\frac{1}{\sin^2 z}$ has a pole of order $2$ at each point $k\pi$ with $k \in \mathbb{Z}$.

Explanations

L1 human

The order of a pole says how fast the function goes to infinity at the point. Near a pole of order $m$, the function behaves like a constant times $1/(z - c)^m$.