# Riemann mapping theorem

Let $U$ be a simply connected open proper subset of $\mathbb{C}$, and let $a\in U$. There is a unique analytic function $f:U\rightarrow\mathbb{C}$ such that

1. 1.

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

2. 2.

$f$ is injective;

3. 3.

$f(U)=\{z\in\mathbb{C}:|z|<1\}$.

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 RiemannMappingTheorem 2013-03-22 13:15:03 2013-03-22 13:15:03 Koro (127) Koro (127) 5 Koro (127) Theorem msc 30A99 ConformalRadius