Riemann Mapping Theorem

Let  be a point in a simply connected region , where  is the complex plane. Then there is a unique analytic function  mapping  one-to-one onto the disk  such that  and . The corollary guarantees that any two simply connected regions except  (the Euclidean plane) can be mapped conformally onto each other.

REFERENCES:

Krantz, S. G. "The Riemann Mapping Theorem." §6.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 86-87, 1999.