Riemann's Mapping Theorem

Abstract

Riemann's Mapping Theorem We gave in Section 3.3c some examples of conformal mappings f: z → w = f(z) mapping a region D “of simple shape” in the z -plane into the unit disk in the w -plane. Given an arbitrary region D , in general it will be impossible to find a conformal mapping f which maps D onto the unit disk U by the composition of suitably chosen known functions. However, we have the following theorem concerning the existence of conformal mappings which map D onto U . Riemann's Mapping Theorem . Let D be a region in the complex plane ℂ, z 0 a point in D , and U = { w:|w| w -plane. If D is simply connected and D ≠ ℂ, then there exists exactly one conformal mapping f:z → w = f(z) from D onto U that satisfies f ( z 0 ) = 0 and f ′( z 0 ) > 0. Obviously for a conformal mapping f between D and U to exist, it is necessary that D be simply connected and D ≠ ℂ. Since, by Liouville's Theorem (Theorem 1.24), a function that is holomorphic and bounded on ℂ is a constant, there cannot exist a conformal mapping from ℂ onto U . If a conformal mapping f from D onto U exists, then f is a one-to-one continuous mapping from D onto U and its inverse mapping f −1 is continuous. Therefore, D has to be simply connected by the simple connectedness of U and the definition of simple connectedness (Definition 4.4). This section is devoted to proving Riemann's Theorem. We first give an outline of the proof.