The Cauchy Transform, Potential Theory and Conformal Mapping

by Steven R. Bell

Overview

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 f

Product Details

ISBN-13:	9781040213049
Publisher:	CRC Press
Publication date:	11/04/2015
Sold by:	Barnes & Noble
Format:	eBook
Pages:	221
File size:	2 MB

About the Author

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

Table of Contents

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 Szego 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 Szego 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 Szego 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 Szego Kernel. Pseudo-Local Property of the Cauchy Transform and Consequences. Zeroes of the Szego 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.

Editorial Reviews

A text for a second course in complex analysis, applying the well known Cauchy integral to nonholomorphic functions. Among new tricks the old dog can do are deducing the Riemann Mapping Theorem, solving the Dirichlet and Neumann problems for the Laplace operator, constructing the Poisson kernel, and solving the inhomogeneous Cauchy-Riemann equations. Assumes a previous one- semester course. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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