# 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.

Title pole Pole 2013-03-22 12:05:56 2013-03-22 12:05:56 djao (24) djao (24) 8 djao (24) Definition msc 30D30 EssentialSingularity simple pole simple