Theorem 10.4 (Riemann Mapping Theorem). If D is any simply connected domain in the plane (other than the entire plane itself), then there exists a one-to-one conformal mapping that maps D onto the unit disk .
Proof of Theorem 10.4 is mentioned in the book.
The theorem is attributed to Riemann and can be found in the book by Lars V. Ahlfors Complex Analysis (New York: McGraw-Hill Book Co.) Chapter 6, 1966.
Complex Analysis for Mathematics and Engineering