Definition:Order of Pole/Definition 2

Definition

Let $f: \C \to \C$ be a complex function.

Let $z_0 \in U \subset \C$ be such that $f$ is holomorphic in $U \setminus \set {z_0}$, with a pole at $z_0$:

By definition $z_0$ 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 $k \ge 1$.

The order of the pole at $z_0$ is defined to be $k$.

Also see

• Equivalence of Definitions of Order of Pole

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): singular point (singularity): 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): singular point (singularity): 1.