# Definition:Order of Pole

## Definition

Let $f: \C \to \C$ be a complex function.

### Definition 1

Let $z_0 \in U \subset \C$ be such that $f$ is holomorphic in $U \setminus \set {z_0}$, with a pole at $z_0$.

By Existence of Laurent Series there is a series:

$\ds \map f z = \sum_{n \mathop \ge n_0}^\infty a_j \paren {z - z_0}^n$

The order of the pole at $z_0$ is defined to be $\size {n_0} > 0$.

### Definition 2

Let $z_0 \in U \subset \C$ be such that $f$ is holomorphic in $U \setminus \set {z_0}$, with a pole at $z_0$:

By definition $z_0$ is a pole if and only if $f$ can be written in the form:

$\map f z = \dfrac {\map \phi z} {\paren {z - z_0}^k}$

where:

$\phi$ is analytic at $z_0$
$\map \phi {z_0} \ne 0$
$k \in \Z$ such that $k \ge 1$.

The order of the pole at $z_0$ is defined to be $k$.

## Also see

• Results about the order of a pole can be found here.