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.

Readership

Graduate students as well as experts in theoretical and mathematical physics, differential and integral equations and mathematical analysis.

Reviews

"The exposition is clear and very detailed ... most constructive methods are described in detail while in the proofs of theorems every step is made explicitly. Moreover at the end of each chapter a series of exercises is provided, allowing the reader to check his/her understanding of the preceeding chapter."

-- mededelingen van het wiskundig genootschap

"Suitable for use as an undergraduate or graduate level textbook."

-- Zentralblatt MATH

Table of Contents

• Some properties of analytic and harmonic functions
• Conformal mappings of simply connected domains
• Conformal mappings of multiply connected domains
• Applications of integrals of Cauchy type to boundary value problems
• The Hilbert boundary value problem for analytic functions on multiply connected domains
• Basic boundary value problems for harmonic functions
• Appendix 1. A brief introduction to quasiconformal mappings
• Appendix 2. Some connections between integral equations and boundary value problems