Conformal mapping: Difference between revisions
(New page: ==Definition== Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math>. A map <math>f: U \to \mathbb{C}</math> (or to a subset of <math>\mathbb{C}</math> is termed a '''confor...) |
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* <math>f</math> is a [[holomorphic function]] and <math>|f'(z)| \ne 0</math> for all <math>z \in U</math> | * <math>f</math> is a [[holomorphic function]] and <math>|f'(z)| \ne 0</math> for all <math>z \in U</math> | ||
* <math>f</math> maps smooth curves to smooth curves, and preserves both the magnitude and orientation of angle between curves | * <math>f</math> maps smooth curves to smooth curves, and preserves both the magnitude and orientation of angle between curves at their intersection. | ||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 20:52, 20 April 2008
Definition
Suppose is an open subset of . A map (or to a subset of is termed a conformal mapping if it satisfies the following equivalent conditions:
- is a holomorphic function and for all
- maps smooth curves to smooth curves, and preserves both the magnitude and orientation of angle between curves at their intersection.
Relation with other properties
Weaker properties
- Isogonal mapping: This is a smooth mapping that preserves the magnitude of angles, but not necessarily the orientation. An isogonal mapping could be conformal or anticonformal