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 $f'(a)$ 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.