Riemann mapping theorem

Let $U$ be a simply connected open proper subset of $ℂ$, and let$a\in U$. There is a unique analytic function $f:U\to ℂ$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.