Advanced Complex Analysis - A Comprehensive Course in Analysis, Part 2B

339 Pages·2015·4.542 MB·English

by  Barry Simon

Advanced Complex Analysis
A Comprehensive Course in Analysis, Part 2B
Barry Simon
Advanced Complex Analysis
A Comprehensive Course in Analysis, Part 2B

Advanced Complex Analysis
A Comprehensive Course in Analysis, Part 2B
Barry Simon
Providence, Rhode Island
2010 Mathematics Subject Classification. Primary 30-01, 33-01, 34-01, 11-01;
Secondary 30C55, 30D35, 33C05, 60J67.
For additional informationand updates on this book, visit
www.ams.org/bookpages/simon
Library of Congress Cataloging-in-Publication Data
Simon,Barry,1946–
Advancedcomplexanalysis/BarrySimon.
pagescm. —(Acomprehensivecourseinanalysis;part2B)
Includesbibliographicalreferencesandindexes.
ISBN978-1-4704-1101-5(alk.paper)
1.Mathematicalanalysis—Textbooks. I.Title.
QA300.S526 2015
515—dc23
2015015258
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To the memory of Cherie Galvez
extraordinary secretary, talented helper, caring person
and to the memory of my mentors,
Ed Nelson (1932-2014) and Arthur Wightman (1922-2013)
who not only taught me Mathematics
but taught me how to be a mathematician

Contents
Preface to the Series ix
Preface to Part 2 xv
Chapter 12. Riemannian Metrics and Complex Analysis 1
§12.1. Conformal Metrics and Curvature 3
§12.2. The Poincar´e Metric 6
§12.3. The Ahlfors–Schwarz Lemma 14
§12.4. Robinson’s Proof of Picard’s Theorems 16
§12.5. The Bergman Kernel and Metric 18
§12.6. The Bergman Projection and Painlev´e’s Conformal
Mapping Theorem 27
Chapter 13. Some Topics in Analytic Number Theory 37
§13.1. Jacobi’s Two- and Four-Square Theorems 46
§13.2. Dirichlet Series 56
§13.3. The Riemann Zeta and Dirichlet L-Function 72
§13.4. Dirichlet’s Prime Progression Theorem 80
§13.5. The Prime Number Theorem 87
Chapter 14. Ordinary Differential Equations in the Complex
Domain 95
§14.1. Monodromy and Linear ODEs 99
§14.2. Monodromy in Punctured Disks 101
§14.3. ODEs in Punctured Disks 106
vii
viii Contents
§14.4. Hypergeometric Functions 116
§14.5. Bessel and Airy Functions 139
§14.6. Nonlinear ODEs: Some Remarks 150
§14.7. Integral Representation 152
Chapter 15. Asymptotic Methods 161
§15.1. Asymptotic Series 163
§15.2. Laplace’s Method: Gaussian Approximation and
Watson’s Lemma 171
§15.3. The Method of Stationary Phase 183
§15.4. The Method of Steepest Descent 194
§15.5. The WKB Approximation 213
Chapter 16. Univalent Functions and Loewner Evolution 231
§16.1. Fundamentals of Univalent Function Theory 233
§16.2. Slit Domains and Loewner Evolution 241
§16.3. SLE: A First Glimpse 251
Chapter 17. Nevanlinna Theory 257
§17.1. The First Main Theorem of Nevanlinna Theory 262
§17.2. Cartan’s Identity 268
§17.3. The Second Main Theorem and Its Consequences 271
§17.4. Ahlfors’ Proof of the SMT 278
Bibliography 285
Symbol Index 309
Subject Index 311
Author Index 315
Index of Capsule Biographies 321
Preface to the Series
Youngmenshouldprovetheorems,oldmenshouldwritebooks.
—FreemanDyson,quotingG.H.Hardy1
Reed–Simon2 starts with “Mathematics has its roots in numerology, ge-
ometry, and physics.” This puts into context the division of mathematics
into algebra, geometry/topology, and analysis. There are, of course, other
areas of mathematics, and a division between parts of mathematics can be
artificial. But almost universally, we require our graduate students to take
courses in these three areas.
This five-volumeseries began and, to some extent, remains a set of texts
forabasicgraduateanalysiscourse. InpartitreflectsCaltech’sthree-terms-
per-yearscheduleandtheactualcoursesI’vetaughtinthepast. Muchofthe
contentsofParts1and2(Part2isintwovolumes,Part2AandPart2B)are
common to virtually all such courses: point set topology, measure spaces,
Hilbert and Banach spaces, distribution theory, and the Fourier transform,
complex analysis including the Riemann mapping and Hadamard product
theorems. Parts 3 and 4 are made up of material that you’ll find in some,
but not all, courses—on the one hand, Part 3 on maximal functions and
Hp-spaces; on the other hand, Part 4 on the spectral theorem for bounded
self-adjointoperatorsonaHilbertspaceanddetandtrace, againforHilbert
space operators. Parts 3 and 4 reflect the two halves of the third term of
Caltech’s course.
1InterviewwithD.J.Albers,TheCollegeMathematicsJournal,25,no.1,January1994.
2M.ReedandB.Simon,Methodsof ModernMathematicalPhysics, I: FunctionalAnalysis,
AcademicPress,NewYork,1972.
ix