# 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.

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

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

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