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2008-05-18T19:13:04Z

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Vipul: New page: {{generic simply connected domain}} ==Definition== A '''generalized half-plane''' is a domain $U \subset \mathbb{C}$ with the following properties: * $0 \notin U$
2008-04-27T20:30:05Z

New page: {{generic simply connected domain}} ==Definition== A '''generalized half-plane''' is a domain <math>U \subset \mathbb{C}</math> with the following properties: * <math>0 \notin U</ma...

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{{generic simply connected domain}}

==Definition==

A '''generalized half-plane''' is a [[domain]] <math>U \subset \mathbb{C}</math> with the following properties:

* <math>0 \notin U</math>
* For any <math>z \in \mathbb{C}</math>, either exactly one of <math>z, \overline{z}</math> belongs to <math>U</math>, and the other does not belong to <math>\overline{U}</math>, ''or'' both <math>z, \overline{z}</math> belong <math>\overline{U} \setminus U</math> (here <math>\overline{U}</math> denotes the closure of <math>U</math> in <math>\mathbb{C}</math>).

Generalized half-planes arise as the image of [[generalized slit plane]]s under a [[holomorphic squareroot]]. Conversely, the image of a generalized half-plane under the square map is a generalized slit plane.