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pole_of_a_complex_function [2014/03/21 11:11] (current)
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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]]
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