pole_of_a_complex

Pole of a complex function

Set

context	$\mathcal O$ … open subset of $\mathbb C$
context	$f:\mathcal O\to \mathbb C$

definiendum

$a\in\mathrm{it}$

range	$U$ … open subset of $\mathbb O$
range	$g:\mathcal O\to \mathbb C$
range	$n\in\mathbb N, n>0$

postulate

$\exists U,g,n.\ \left(z\in U\right)\land \left(f\ \mathrm{holomorphic\ on}\ U\setminus\{z\}\right)\land \left(g\ \mathrm{holomorphic\ on}\ U\right)\land \left(f(z)=\frac{g(z)}{(z-a)^n}\right)$

Discussion

The natural number $n$ associated with $a$ is called the order of the pole.

Reference

Wikipedia: Pole (complex analysis)

Parents

Context

ℂ valued function