2nd Edition
The Cauchy Transform, Potential Theory and Conformal Mapping
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
Biography
Steven R. Bell, PhD, professor, Department of Mathematics, Purdue University, West Lafayette, Indiana, USA, and Fellow of the AMS
