Simply Connected Domains, the Riemann Mapping Theorem and Conformal Mapping

One of the main goals of this chapter is to prove the Riemann Mapping Theorem, that there exists a conformal mapping from any proper simply connected domain onto the disk. It is shown that any non-vanishing analytic function in a simply connected domain has an analytic square root which, together with a normal families argument, is sufficient to prove the Riemann Mapping Theorem. The existence of a conformal map to the disk permits the hyperbolic metric on the disk to be transferred to a general proper simply connected domain. A geometric condition is described under which conformal maps extend continuously to the boundary. This condition together with the Schwarz Reflection Principle is used to classify the conformal types of annuli.