The Cauchy Transform, Potential Theory and Conformal Mapping explores the most central result in all of classical function theory, the Cauchy integral formula, in a new and novel way based on an advance made by Kerzman and Stein in 1976.
The book provides a fast track to understanding the Riemann Mapping Theorem. The Dirichlet and Neumann problems for the Laplace operator are solved, the Poisson kernel is constructed, and the inhomogenous Cauchy-Reimann equations are solved concretely and efficiently using formulas stemming from the Kerzman-Stein result.
These explicit formulas yield new numerical methods for computing the classical objects of potential theory and conformal mapping, and the book provides succinct, complete explanations of these methods.
Four new chapters have been added to this second edition: two on quadrature domains and another two on complexity of the objects of complex analysis and improved Riemann mapping theorems.
The book is suitable for pure and applied math students taking a beginning graduate-level topics course on aspects of complex analysis as well as physicists and engineers interested in a clear exposition on a fundamental topic of complex analysis, methods, and their application.
Table of Contents
The Improved Cauchy Integral Formula
The Cauchy Transform
The Hardy Space, the Szegö Projection, and the Kerzman-Stein Formula
The Kerzman-Stein Operator and Kernel
The Classical Definition of the Hardy Space
The Szegö Kernel Function
The Riemann Mapping Function
A Density Lemma and Consequences
Solution of the Dirichlet Problem in Simply Connected Domains
The Case of Real Analytic Boundary
The Transformation Law for the Szegö Kernel under Conformal Mappings
The Ahlfors Map of a Multiply Connected Domain
The Dirichlet Problem in Multiply Connected Domains
The Bergman Space
Proper Holomorphic Mappings and the Bergman Projection
The Solid Cauchy Transform
The Classical Neumann Problem
Harmonic Measure and the Szegö Kernel
The Neumann Problem in Multiply Connected Domains
The Dirichlet Problem Again
Area Quadrature Domains
Arc Length Quadrature Domains
The Hilbert Transform
The Bergman Kernel and the Szegö Kernel
Pseudo-Local Property of the Cauchy Transform and Consequences
Zeroes of the Szegö Kernel
The Kerzman-Stein Integral Equation
Local Boundary Behavior of Holomorphic Mappings
The Dual Space of A∞(Ω)
The Green’s Function and the Bergman Kernel
Zeroes of the Bergman Kernel
Complexity in Complex Analysis
Area Quadrature Domains and the Double
The Cauchy-Kovalevski Theorem for the Cauchy-Riemann Operator
Steven R. Bell, PhD, professor, Department of Mathematics, Purdue University, West Lafayette, Indiana, USA, and Fellow of the AMS