| 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 | $\exists U,g,n.\ \left(z\in U\right)\land \left(f\ \mathrm{holomorphic\ on}\ U\setminus\{z\}\right)\land \left(g\ \mathrm{holomorphic\ on}\ U\right)\land \left(f(z)=\frac{g(z)}{(z-a)^n}\right)$ |
The natural number $n$ associated with $a$ is called the order of the pole.
Wikipedia: Pole (complex analysis)