The Cauchy Transform, Potential Theory and Conformal Mapping

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:
Introduction. 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 A8(O). 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.