The Cauchy Transform, Potential Theory and Conformal Mapping  book cover
2nd Edition

The Cauchy Transform, Potential Theory and Conformal Mapping

ISBN 9781498727204
Published November 23, 2015 by Chapman and Hall/CRC
221 Pages

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Book Description

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

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Steven R. Bell, PhD, professor, Department of Mathematics, Purdue University, West Lafayette, Indiana, USA, and Fellow of the AMS