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