Riemann mapping theorem
Let $U$ be a simply connected open proper subset^{} of $\u2102$, and let $a\in U$. There is a unique analytic function^{} $f:U\to \u2102$ such that

1.
$f(a)=0$, and ${f}^{\prime}(a)$ is real and positive;

2.
$f$ is injective^{};

3.
$$.
Remark. As a consequence of this theorem, any two simply connected regions, none of which is the whole plane, are conformally equivalent.
Title  Riemann mapping theorem^{} 

Canonical name  RiemannMappingTheorem 
Date of creation  20130322 13:15:03 
Last modified on  20130322 13:15:03 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  5 
Author  Koro (127) 
Entry type  Theorem 
Classification  msc 30A99 
Related topic  ConformalRadius 