Editing Conformal map (section)

==In two dimensions==
If <math>U</math> is an [[open set|open subset]] of the complex plane <math>\mathbb{C}</math>, then a [[function (mathematics)|function]] <math>f:U\to\mathbb{C}</math> is conformal [[if and only if]] it is [[holomorphic function|holomorphic]] and its [[derivative]] is everywhere non-zero on <math>U</math>. If <math>f</math> is [[antiholomorphic function|antiholomorphic]] ([[complex conjugate|conjugate]] to a holomorphic function), it preserves angles but reverses their orientation.

In the literature, there is another definition of conformal: a mapping <math>f</math> which is one-to-one and holomorphic on an open set in the plane. The open mapping theorem forces the inverse function (defined on the image of <math>f</math>) to be holomorphic. Thus, under this definition, a map is conformal [[if and only if]] it is biholomorphic. The two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative. In fact, we have the following relation, the [[inverse function theorem]]:
::<math>(f^{-1}(z_0))'=\frac{1}{f'(z_0)}</math>
where <math>z_0 \in \mathbb{C}</math>. However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic.<ref>Richard M. Timoney (2004), [https://www.maths.tcd.ie/~richardt/414/414-ch7.pdf Riemann mapping theorem] from [[Trinity College Dublin]]</ref>

The [[Riemann mapping theorem]], one of the profound results of [[complex analysis]], states that any non-empty open [[simply connected]] proper subset of <math>\mathbb{C}</math> admits a [[bijection|bijective]] conformal map to the open [[unit disk]] in <math>\mathbb{C}</math>. Informally, this means that any blob can be transformed into a perfect circle by some conformal map.

=== Global conformal maps on the Riemann sphere ===

A map of the [[Riemann sphere]] [[surjection|onto]] itself is conformal if and only if it is a [[Möbius transformation]].

The complex conjugate of a Möbius transformation preserves angles, but reverses the orientation. For example, [[Inversive geometry#Circle inversion|circle inversions]].

===Conformality with respect to three types of angles===
In plane geometry there are three types of angles that may be preserved in a conformal map.<ref>{{wikibooks-inline|Geometry/Unified Angles}}</ref> Each is hosted by its own real algebra, ordinary [[complex number]]s, [[split-complex number]]s, and [[dual number]]s. The conformal maps are described by [[linear fractional transformation#Conformal property|linear fractional transformations]] in each case.<ref>Tsurusaburo Takasu (1941) [https://projecteuclid.org/euclid.pja/1195578674 Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie, 2], [[Japan Academy|Proceedings of the Imperial Academy]] 17(8): 330–8, link from [[Project Euclid]], {{mr|id=14282}}</ref>