Definition:Order of Pole/Definition 1

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 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 $\size {n_0} > 0$.

Also see

• Equivalence of Definitions of Order of Pole