Definition:Order of Pole/Definition 2

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

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 $f$ can be written in the form:
 * $\map f z = \dfrac {\map \phi z} {\paren {z - z_0}^k}$

where:
 * $\phi$ is analytic at $z_0$
 * $\map \phi {z_0} \ne 0$

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

Also see

 * Equivalence of Definitions of Order of Pole