## 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.