===== Pole of a complex function =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $\mathcal O$ ... open subset of $\mathbb C$ |
| @#55CCEE: context     | @#55CCEE: $f:\mathcal O\to \mathbb C$ |

| @#FFBB00: definiendum | @#FFBB00: $a\in\mathrm{it}$ |

| @#DDDDDD: range       | @#DDDDDD: $U$ ... open subset of $\mathbb O$ |
| @#DDDDDD: range       | @#DDDDDD: $g:\mathcal O\to \mathbb C$ |
| @#DDDDDD: range       | @#DDDDDD: $n\in\mathbb N, n>0$ |

| @#55EE55: postulate   | @#55EE55: $\exists U,g,n.\ \left(z\in U\right)\land \left(f\ \mathrm{holomorphic\ on}\ U\setminus\{z\}\right)\land \left(g\ \mathrm{holomorphic\ on}\ U\right)\land \left(f(z)=\frac{g(z)}{(z-a)^n}\right)$ |

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

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Pole_%28complex_analysis%29|Pole (complex analysis)]]
==== Parents ====
=== Context ===
[[ℂ valued function]]