Definition:Order of Pole

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

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

By Existence of Laurent Series there is a series:
 * $\displaystyle f \left({z}\right) = \sum_{n \mathop \ge n_0}^\infty a_j \left({z - x}\right)^n$

The order of the pole at $x$ is defined to be $n_0 < 0$.