Handbook of Complex Analysis

547 Pages·2022·8.85 MB·english

by  Steven G. Krantz

Handbook of Complex
Analysis

Handbook of Complex
Analysis
Edited by
Steven G. Krantz
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.
Contents
Preface vii
Editor ix
List of Contributors xi
1 Something about Poisson and Dirichlet 1
Steven R. Bell and Luis Reyna de la Torre
2 The Cauchy-Leray Operator for Convex Domains 19
David Barrett and Michael Bolt
3 Fractional Linear Maps and Some Applications. An
“Augenblick”. 41
Joseph A. Cima
4 Biholomorphic Transformations 61
Buma Fridman and Daowei Ma
5 Positivity in the ∂¯-Neumann Problem 89
Siqi Fu
6 Symmetry and Art 133
Emily Gullerud and James S. Walker
7 A Glimpse into Invariant Distances in Complex Analysis 171
Marek Jarnicki and Peter Pflug
8 Variations on the (Eternal) Theme of Analytic Continuation 229
Dmitry Khavinson
9 Complex Convexity 245
Christer Oscar Kiselman
10 Reproducing Kernels in Complex Analysis 379
Steven G. Krantz
v
vi Contents
11 The Green’s Function Method for the Riemann Mapping
Theorem 427
Bingyuan Liu
12 PolynomialTraceIdentitiesinSL(2,C),QuaternionAlgebras,
and Two-Generator Kleinian Groups 433
T. H. Marshall and Gaven Martin
13 Boundary Value Problems on Klein Surfaces 487
Vicentiu Radulescu and Monica Rosiu
Index 521
Preface
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 ap-
plicationstophysics,engineering,cosmology,andotheraspectsoftechnology.
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 sys-
tems, 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 dif-
ferential 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 re-
searchers. More than a compilation of recent results, this book offers a step-
ping stone for students to gain entry into the professional life of complex
analysis.Theessayspresentedhereareallaccessibletograduatestudentsbut
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.
Asnoted,thisbookwillserveasareferenceandasourceofinspirationfor
maturemathematicians—bothspecialistsincomplexanalysisandotherswho
want to become acquainted with current modes of thought and investigation.
And it will help the neophyte to become inured in the subject matter.
Thechaptersinthisvolumeareauthoredbyleadingexpertsinthesubject
area,alsogiftedexpositors.Theyarecarefullycraftedpresentationsofdiverse
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 con-
strued 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.
vii

Editor
Steven G. Krantz is a professor of mathematics at Washington Univer-
sity 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.
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