# Definition:Pole of Complex Function

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## 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 $f$ if and only if 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 $f$ if and only if 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 $f$ if and only if the improper limit:

- $\ds \lim_{z \mathop \to p} \size {\map f z} = \infty$