Definition:Pole of Complex Function

Definition
Let $U \subset \C$ be an open subset of the complex plane.

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

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:
 * $\map f z = \ds \sum_{k \mathop = -n}^\infty a_k \paren {z - p}^k$

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

Definition 3
The point $p$ is a pole of $f$ the improper limit:
 * $\ds \lim_{z \mathop \to p} \size {\map f z} = \infty$

Also see

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