# Definition:Isolated Singularity/Pole

## Definition

Let $U$ be an open subset of a Riemann surface.

Let $z_0 \in U$.

Let $f: U \setminus \set {z_0} \to \C$ be a holomorphic function.

### Definition 1

Let $z_0$ be an isolated singularity of $f$.

Then $z_0$ is a pole if and only if:

$\ds \lim_{z \mathop \to z_0} \cmod {\map f z} \to \infty$

### Definition 2

Let $z_0$ be an isolated singularity of $f$.

Then $z_0$ 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 $k \ge 1$.

### Order of Pole

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 $\size {n_0} > 0$.

### Simple Pole

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

Let the order of the pole at $z_0$ be $1$.

Then $z_0$ is a simple pole.

## Examples

### Pole (Complex Analysis)/Examples/Example: $\frac 1 {\paren {z - 3}^2 \paren {z + 1} }$

Let $f$ be the complex function:

$\forall z \in \C \setminus \set {-1, 3}: \map f z = \dfrac 1 {\paren {z - 3}^2 \paren {z + 1} }$

Then $f$ has:

a pole of order $2$ at $z = 3$
a simple pole at $z = -1$.