Category:Definitions/Poles (Complex Analysis)

This category contains definitions related to poles in the context of Complex Analysis.
Related results can be found in Category:Poles (Complex Analysis).

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.

Definition 1

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

Then $z_0$ is a pole if and only if:

$\ds \lim_{z \mathop \to z_0} \cmod {\map f z} \to \infty$

Definition 2

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 $k \ge 1$.