Definition:Isolated Singularity/Pole/Definition 2

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 $f$ 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 $k \ge 1$.

Also see

 * Equivalence of Definitions of Poles in the Context of Complex Analysis