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| + | ===== 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]] | ||
