Steven G. Krantz (editor) - Handbook of Complex Analysis

1. Steven R. Bell and Luis Reyna de la Torre

2. David Barrett and Michael Bolt

3. Joseph A. Cima

4. Buma Fridman and Daowei Ma

5. Siqi Fu

6. Emily J. Gullerud and James S. Walker

7. Marek Jarnicki and Peter Pflug

8. Dmitri Khavinson

9. Christer Oscar Kiselman

10. Steven G. Krantz

11. Bingyuan Liu

12. T. H.Marshall and Gaven Martin

13. Vicentiu Radulescu and Monica Rosiu

First edition published 2022

by CRC Press

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ISBN: 9781138064041 (hbk)

ISBN: 9781032202105 (pbk)

ISBN: 9781315160658 (ebk)

DOI: 10.1201/9781315160658

Typeset in font CMR10

by KnowledgeWorks Global Ltd.

Despite being nearly 500 years old, with work dating back to Cardano, Euler, Gauss, Cauchy, Riemann, and many others, the subject of complex analysis is still today a vital and active part of the mathematical sciences. In addition to all the exciting theoretical work being done today, there are important applications to physics, engineering, cosmology, and other aspects of technology. Many of the world's most distinguished and accomplished mathematicians conduct research in complex analysis. Several recent Fields Medalists study complex analysis.

Although a venerable subject, complex analysis continues to grow and prosper. New directions of development in the subject include dynamical systems, quasiconformal mappings, harmonic measure, automorphism groups, and the list can go on at some length. One of the sources of strength for the subject is its interaction with diverse parts of mathematics, including differential geometry, partial differential equations, functional analysis, algebra, combinatorics, and many other aspects of the subject.

This Handbook of Complex Analysis presents contributed chapters by several distinguished mathematicians, including a new generation of researchers. More than a compilation of recent results, this book offers a stepping stone for students to gain entry into the professional life of complex analysis. The essays presented here are all accessible to graduate students but will also be of considerable interest to the seasoned mathematician. Classes and seminars, of course, play a role in the maturation process that we are describing. But more is needed for the unilateral study. This handbook will play such a role.

As noted, this book will serve as a reference and a source of inspiration for mature mathematiciansboth specialists in complex analysis and others who want to become acquainted with current modes of thought and investigation. And it will help the neophyte to become inured in the subject matter.

The chapters in this volume are authored by leading experts in the subject area, also gifted expositors. They are carefully crafted presentations of diverse aspects of the field, formulated for a broad and diverse audience. The editor intends this volume to be a touchstone for current ideas in the broadly construed subject area of complex analysis. It should enrich the literature and point to some new directions. The point here is not to present an epitaph for complex analysis but rather to provide an entree to a whole new life. We anticipate that the reader of this volume will be eager to explore other parts of complex analysis literature and to begin to play an active role in complex analysis research life.

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 130 books and more than 250 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a PhD from Princeton University.

David Barrett

University of Michigan

Steven R. Bell

Purdue University

Michael Bolt

Calvin Univeristy

Joseph A. Cima

University of North Carolina

Buma Fridman

Wichita State University

Siqi Fu

Rutgers University

Emily J. Gullerud

University of Minnesota

Dmitri Khavinson

University of South Florida

Christer Kiselman

Uppsala University

Steven G. Krantz

Washington University

Bingyuan Liu

University of California Riverside

Daowei Ma

Wichita State University

T. H. Marshall

Massey University

Gaven Martin

Massey University

Vicentiu Radulescu

University of Craiova

Luis Reyna de la Torre

IUPUI

James S. Walker

University of Wisconsin-Eau Claire

Steven R. Bell and Luis Reyna de la Torre

DOI: 10.1201/9781315160658-1

CONTENTS

For reasons that have always been mysterious to the first author, he has often found himself thinking about the Dirichlet problem in the plane and has fought urges to look for explicit formulas for the Poisson kernels associated to various kinds of multiply connected domains, especially quadrature domains. He has obsessed about solutions to the Dirichlet problem with rational boundary data (see [] for an expository treatment of some of these results.) After thinking about the Dirichlet problem his entire adult life, he can solve it a different way every day of the week. He recently looked up his mathematical lineage at the Math Genealogy Project and found a possible explanation for his obsession. He is a direct mathematical descendent of both Poisson and Dirichlet. These problems are in his blood!

The authors worked together on a summer research project at Purdue University in 2018 to find a particularly elegant and simple way to approach these problems. We want to demonstrate here how Poisson and Dirichlet might solve their famous problems today if they had lived another 200 years and developed a major lazy streak. We assume that our reader has seen a traditional approach to this subject in a course on complex analysis and so will appreciate the novelty and smooth sailing of the line of reasoning here, but just in case the reader hasnt, we have tried to present the material in a way that can be understood assuming only a background in basic analysis.