Definition:Order of Pole

Theorem
Let $f: \C \to \C$ be a function and $x \in U \subset \C$ such that $f$ is analytic in $U - \{ x \}$, with a pole at $x$.

Then by Existence of Laurent Series there is a series


 * $\displaystyle f(z) = \sum_{n \geq n_0}^\infty a_j \left({z - x}\right)^n$

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

If the pole has order $1$, it is called simple.