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
- Results about poles in the context of Complex Analysis can be found here.
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.
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): pole (in complex analysis)