Conformal Mappings and Boundary Value Problems

This book presents a systematic exposition of the theory of conformal mappings, boundary value problems for analytic and harmonic functions, and the relationship between the two subjects. It is suitable for use as an undergraduate or graduate level textbook, and exercises are included. The first three chapters recount existence and uniqueness theorems of conformal mappings from simply and multiply connected domains to standard domains, some properties of analytic functions, harmonic functions and schlicht meromorphic functions, and representations of conformal mappings. In the remaining three chapters, the basic boundary value problems for analytic and harmonic functions are discussed in detail, including some new methods and results obtained by the author. For example, the Riemann-Hilbert boundary value problem with piecewise continuous coefficients in a multiply connected domain is covered in chapter five, and some irregular oblique derivative problems are treated in chapter six.