Upper half-plane: Difference between revisions
(New page: {{particular simply connected domain}} ==Definition== The upper half-plane is defined as the set of complex numbers with strictly positive imaginary part, i.e.: <math>\mathcal{H} := \{ ...) |
m (3 revisions) |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 9: | Line 9: | ||
==Riemann mapping== | ==Riemann mapping== | ||
The upper half-plane admits a particularly easy Riemann mapping; in fact, one coming from a [[fractional linear transformation]]: | The upper half-plane admits a particularly easy [[Riemann mapping]] to the [[open unit disk]]; in fact, one coming from a [[fractional linear transformation]]: | ||
<math>z \mapsto \frac{z - i}{z + i}</math> | <math>z \mapsto \frac{z - i}{z + i}</math> | ||
| Line 15: | Line 15: | ||
==Relation with other domains== | ==Relation with other domains== | ||
* [ | * [[Right half-plane]]: The right half-plane maps bijectively to the upper half-plane via a rotation map <math>z \mapsto iz</math> | ||
* [[Slit plane]]: The slit plane maps bijectively to the upper half-plane via the map <math>z \mapsto i\sqrt{z}</math> | * [[Slit plane]]: The slit plane maps bijectively to the upper half-plane via the map <math>z \mapsto i\sqrt{z}</math> | ||
Latest revision as of 19:19, 18 May 2008
This article defines a particular simply connected domain in , the complex numbers
View a complete list of particular simply connected domains
Definition
The upper half-plane is defined as the set of complex numbers with strictly positive imaginary part, i.e.:
Riemann mapping
The upper half-plane admits a particularly easy Riemann mapping to the open unit disk; in fact, one coming from a fractional linear transformation:
Relation with other domains
- Right half-plane: The right half-plane maps bijectively to the upper half-plane via a rotation map
- Slit plane: The slit plane maps bijectively to the upper half-plane via the map
