Conformal Mapping

Abstract

In this chapter we will be concerned with conformal maps from domains onto the open unit disk. One of our goals is the celebrated Riemann mapping theorem: Any simply connected domain in the complex plane, except the entire complex plane itself, can be mapped conformally onto the open unit disk. We begin in Section 1 by reviewing and enlarging our repertoire of conformal maps onto the open unit disk, or equivalently, onto the upper half-plane. In Section 2 we state and discuss the Riemann mapping theorem. Before embarking on the proof, we give some applications to the conformal mapping of polygons in Section 3 and to fluid dynamics in Section 4. In Section 5 we develop some prerequisite material concerning compactness of families of analytic functions, which is at a deeper level than the analysis used up to this point. The proof of the Riemann mapping theorem follows in Section 6.KeywordsConnected DomainStraight Line SegmentExtremal FunctionRiemann MappingOpen Unit DiskThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.