Definition:Order of Pole

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

Let $x \in U \subset \C$ be such that $f$ is analytic in $U \setminus \set x$, with a pole at $x$.

By Existence of Laurent Series there is a series:

$\ds \map f z = \sum_{n \mathop \ge n_0}^\infty a_j \paren {z - x}^n$

The order of the pole at $x$ is defined to be $\size {n_0} > 0$.

Simple Pole

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

Then $x$ is a simple pole.