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# pole

Let $U\subset\mathbb{C}$ be a domain and let $a\in\mathbb{C}$. A function $f\colon U\to\mathbb{C}$ has a *pole* at $a$ if it can be represented by a Laurent series centered about $a$ with only finitely many terms of negative exponent; that is,

$f(z)=\sum_{{k=-n}}^{\infty}c_{k}(z-a)^{k}$ |

in some nonempty deleted neighborhood of $a$, with $c_{{-n}}\neq 0$, for some $n\in\mathbb{N}$. The number $n$ is called the *order* of the pole. A *simple pole* is a pole of order 1.

Defines:

simple pole, simple

Related:

EssentialSingularity

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

30D30*no label found*

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