Definition:Pole of Complex Function

Definition
Let $U \subset \C$ be an open subset.

Let $f : U \to \C$ be a holomorphic function.

Let $p\in \C$ be an isolated singularity of $f$.

Definition 1
The point $p$ is a pole of $f$ the Laurent expansion of $f$ around $p$ has the form:
 * $f(z) = \displaystyle \sum_{k = -n}^\infty a_k(z-p)^k$

Definition 2
The point $p$ is a pole of $f$ there exists a natural number $m>0$ such that:
 * $\displaystyle\lim_{z \to p}(z-p)^m f(z) \in \C\setminus\{0\}$

Definition 3
The point $p$ is a pole of $f$ the improper limit:
 * $\displaystyle\lim_{z\to p}|f(z)| = \infty$.

Also see

 * Equivalence of Definitions of Pole of Complex Function
 * Definition:Order of Pole of Complex Function
 * Definition:Removeable Singularity