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.

$f(a)=0$ , and $f^{\prime}(a)$ is real and positive;
2.

$f$ is injective;
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
Canonical name	RiemannMappingTheorem
Date of creation	2013-03-22 13:15:03
Last modified on	2013-03-22 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