Editing Riemann mapping theorem (section)

{{Short description|Mathematical theorem}}
{{Complex analysis sidebar}}
In [[complex analysis]], the '''Riemann mapping theorem''' states that if <math>U</math> is a [[non-empty]] [[simply connected space|simply connected]] [[open set|open subset]] of the [[complex plane|complex number plane]] <math>\mathbb{C}</math> which is not all of <math>\mathbb{C}</math>, then there exists a [[biholomorphy|biholomorphic]] mapping <math>f</math> (i.e. a [[bijective function|bijective]] [[holomorphic function|holomorphic]] mapping whose inverse is also holomorphic) from <math>U</math> onto the [[open unit disk]] 
:<math>D = \{z\in \mathbb{C} : |z| < 1\}.</math>

This mapping is known as a '''Riemann mapping'''.<ref>The existence of f is equivalent to the existence of a [[Green’s function]].</ref>

Intuitively, the condition that <math>U</math> be simply connected means that <math>U</math> does not contain any “holes”. The fact that <math>f</math> is biholomorphic implies that it is a [[conformal map]] and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it.

[[Henri Poincaré]] proved that the map <math>f</math> is unique up to rotation and recentering: if <math>z_0</math> is an element of <math>U</math> and <math>\phi</math> is an arbitrary angle, then there exists precisely one ''f'' as above such that <math>f(z_0)=0</math> and such that the [[Complex number#Complex plane|argument]] of the derivative of <math>f</math> at the point <math>z_0</math> is equal to <math>\phi</math>. This is an easy consequence of the [[Schwarz lemma]].

As a corollary of the theorem, any two simply connected open subsets of the [[Riemann sphere]] which both lack at least two points of the sphere can be conformally mapped into each other.