Definition:Pole of Complex Function

From ProofWiki
Jump to navigation Jump to search



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:

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


Also see