Definition:Isolated Singularity/Pole/Definition 2

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Definition

Let $U$ be an open subset of a Riemann surface.

Let $z_0 \in U$.

Let $f: U \setminus \set {z_0} \to \C$ be a holomorphic function.


Let $z_0$ be an isolated singularity of $f$.

Then $z_0$ is a pole if and only if $f$ can be written in the form:

$\map f z = \dfrac {\map \phi z} {\paren {z - z_0}^k}$

where:

$\phi$ is analytic at $z_0$
$\map \phi {z_0} \ne 0$
$k \in \Z$ such that $k \ge 1$.


Also see


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