ABSTRACT
This chapter discusses the idea of conformal mapping on the unit disc. On a practical level, one can often study holomorphic functions on a rather complicated open set by first mapping that open set to some simpler open set, then transferring the holomorphic functions. The chapter reviews the construction of Mobius transformations and summarizes the properties of some important linear fractional transformations. The Riemann mapping theorem is provided and the concept of homeomorphism is discussed. The Riemann mapping theorem asserts in effect that any planar domain that has the topology of the unit disc also has the conformal structure of the unit disc. The classification of planar domains up to biholomorphic equivalence is a part of the theory of Riemann surfaces. The chapter presents a graphical compendium of commonly used conformal mappings.
