The Cauchy integral formula is the most central result in all of classical function theory. A recent discovery of Kerzman and Stein allows more theorems than ever to be deduced from simple facts about the Cauchy integral. In this book, the Riemann Mapping Theorem is deduced, the Dirichlet and Neumann problems for the Laplace operator are solved, the Poisson kernal is constructed, and the inhomogenous Cauchy-Reimann equations are solved concretely 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. The Cauchy Transform, Potential Theory, and Conformal Mapping is suitable for pure and applied math students taking a beginning graduate-level topics course on aspects of complex analysis. It will also be useful to physicists and engineers interested in a clear exposition on a fundamental topic of complex analysis, methods, and their application.
Introduction. The Improved Cauchy Integral Formula. The Cauchy Transform. The Hardy Space, the Szego Projection, and the Kerzman-Stein Formula. The Kerzman-Stein Operator and Kernel. The Classical Definition of the Hardy Space. The Szegö Kernel Function. The Reimann Mapping Function. A Density Lemma. Solution of the Dirichlet Problem and the Poisson Extension Operator. 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 Dirichelt Problem Again. 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(?). Bibliographic Notes. References.