Basic Complex Analysis
A Comprehensive Course in Analysis, Part 2A
Barry Simon
,Basic Complex Analysis
A Comprehensive Course in Analysis, Part 2A
,
, http://dx.doi.org/10.1090/simon/002.1
Basic Complex Analysis
A Comprehensive Course in Analysis, Part 2A
Barry Simon
Providence, Rhode Island
, 2010 Mathematics Subject Classification. Primary 30-01, 33-01, 40-01;
Secondary 34-01, 41-01, 44-01.
For additional information and updates on this book, visit
www.ams.org/bookpages/simon
Library of Congress Cataloging-in-Publication Data
Simon, Barry, 1946–
Basic complex analysis / Barry Simon.
pages cm. — (A comprehensive course in analysis ; part 2A)
Includes bibliographical references and indexes.
ISBN 978-1-4704-1100-8 (alk. paper)
1. Mathematical analysis—Textbooks. I. Title.
QA300.S527 2015
515—dc23
2015009337
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy select pages for
use in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Permissions to reuse
portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink
service. For more information, please visit: http://www.ams.org/rightslink.
Send requests for translation rights and licensed reprints to .
Excluded from these provisions is material for which the author holds copyright. In such cases,
requests for permission to reuse or reprint material should be addressed directly to the author(s).
Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the
first page of each article within proceedings volumes.
c 2015 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
∞ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org/
10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15
, 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 xi
Preface to Part 2 xvii
Chapter 1. Preliminaries 1
§1.1. Notation and Terminology 1
§1.2. Complex Numbers 3
§1.3. Some Algebra, Mainly Linear 5
§1.4. Calculus on R and Rn 8
§1.5. Differentiable Manifolds 12
§1.6. Riemann Metrics 18
§1.7. Homotopy and Covering Spaces 21
§1.8. Homology 24
§1.9. Some Results from Real Analysis 26
Chapter 2. The Cauchy Integral Theorem: Basics 29
§2.1. Holomorphic Functions 30
§2.2. Contour Integrals 40
§2.3. Analytic Functions 49
§2.4. The Goursat Argument 66
§2.5. The CIT for Star-Shaped Regions 69
§2.6. Holomorphically Simply Connected Regions, Logs, and
Fractional Powers 71
§2.7. The Cauchy Integral Formula for Disks and Annuli 76
vii
,viii Contents
Chapter 3. Consequences of the Cauchy Integral Formula 79
§3.1. Analyticity and Cauchy Estimates 80
§3.2. An Improved Cauchy Estimate 93
§3.3. The Argument Principle and Winding Numbers 95
§3.4. Local Behavior at Noncritical Points 104
§3.5. Local Behavior at Critical Points 108
§3.6. The Open Mapping and Maximum Principle 114
§3.7. Laurent Series 120
§3.8. The Classification of Isolated Singularities;
Casorati–Weierstrass Theorem 124
§3.9. Meromorphic Functions 128
§3.10. Periodic Analytic Functions 132
Chapter 4. Chains and the Ultimate Cauchy Integral Theorem 137
§4.1. Homologous Chains 139
§4.2. Dixon’s Proof of the Ultimate CIT 142
§4.3. The Ultimate Argument Principle 143
§4.4. Mesh-Defined Chains 145
§4.5. Simply Connected and Multiply Connected Regions 150
§4.6. The Ultra Cauchy Integral Theorem and Formula 151
§4.7. Runge’s Theorems 153
§4.8. The Jordan Curve Theorem for Smooth Jordan Curves 161
Chapter 5. More Consequences of the CIT 167
§5.1. The Phragmén–Lindelöf Method 168
§5.2. The Three-Line Theorem and the Riesz–Thorin
Theorem 174
§5.3. Poisson Representations 177
§5.4. Harmonic Functions 183
§5.5. The Reflection Principle 189
§5.6. Reflection in Analytic Arcs; Continuity at Analytic
Corners 196
§5.7. Calculation of Definite Integrals 201
Chapter 6. Spaces of Analytic Functions 227
§6.1. Analytic Functions as a Fréchet Space 228
§6.2. Montel’s and Vitali’s Theorems 234
, Contents ix
§6.3. Restatement of Runge’s Theorems 244
§6.4. Hurwitz’s Theorem 245
§6.5. Bonus Section: Normal Convergence of Meromorphic
Functions and Marty’s Theorem 247
Chapter 7. Fractional Linear Transformations 255
§7.1. The Concept of a Riemann Surface 256
§7.2. The Riemann Sphere as a Complex Projective Space 267
§7.3. PSL(2, C) 273
§7.4. Self-Maps of the Disk 289
§7.5. Bonus Section: Introduction to Continued Fractions
and the Schur Algorithm 295
Chapter 8. Conformal Maps 309
§8.1. The Riemann Mapping Theorem 310
§8.2. Boundary Behavior of Riemann Maps 319
§8.3. First Construction of the Elliptic Modular Function 325
§8.4. Some Explicit Conformal Maps 336
§8.5. Bonus Section: Covering Map for General Regions 353
§8.6. Doubly Connected Regions 357
§8.7. Bonus Section: The Uniformization Theorem 362
§8.8. Ahlfors’ Function, Analytic Capacity and the Painlevé
Problem 371
Chapter 9. Zeros of Analytic Functions and Product Formulae 381
§9.1. Infinite Products 383
§9.2. A Warmup: The Euler Product Formula 387
§9.3. The Mittag-Leffler Theorem 399
§9.4. The Weierstrass Product Theorem 401
§9.5. General Regions 406
§9.6. The Gamma Function: Basics 410
§9.7. The Euler–Maclaurin Series and Stirling’s
Approximation 430
§9.8. Jensen’s Formula 448
§9.9. Blaschke Products 451
§9.10. Entire Functions of Finite Order and the Hadamard
Product Formula 459
A Comprehensive Course in Analysis, Part 2A
Barry Simon
,Basic Complex Analysis
A Comprehensive Course in Analysis, Part 2A
,
, http://dx.doi.org/10.1090/simon/002.1
Basic Complex Analysis
A Comprehensive Course in Analysis, Part 2A
Barry Simon
Providence, Rhode Island
, 2010 Mathematics Subject Classification. Primary 30-01, 33-01, 40-01;
Secondary 34-01, 41-01, 44-01.
For additional information and updates on this book, visit
www.ams.org/bookpages/simon
Library of Congress Cataloging-in-Publication Data
Simon, Barry, 1946–
Basic complex analysis / Barry Simon.
pages cm. — (A comprehensive course in analysis ; part 2A)
Includes bibliographical references and indexes.
ISBN 978-1-4704-1100-8 (alk. paper)
1. Mathematical analysis—Textbooks. I. Title.
QA300.S527 2015
515—dc23
2015009337
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy select pages for
use in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Permissions to reuse
portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink
service. For more information, please visit: http://www.ams.org/rightslink.
Send requests for translation rights and licensed reprints to .
Excluded from these provisions is material for which the author holds copyright. In such cases,
requests for permission to reuse or reprint material should be addressed directly to the author(s).
Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the
first page of each article within proceedings volumes.
c 2015 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
∞ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org/
10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15
, 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 xi
Preface to Part 2 xvii
Chapter 1. Preliminaries 1
§1.1. Notation and Terminology 1
§1.2. Complex Numbers 3
§1.3. Some Algebra, Mainly Linear 5
§1.4. Calculus on R and Rn 8
§1.5. Differentiable Manifolds 12
§1.6. Riemann Metrics 18
§1.7. Homotopy and Covering Spaces 21
§1.8. Homology 24
§1.9. Some Results from Real Analysis 26
Chapter 2. The Cauchy Integral Theorem: Basics 29
§2.1. Holomorphic Functions 30
§2.2. Contour Integrals 40
§2.3. Analytic Functions 49
§2.4. The Goursat Argument 66
§2.5. The CIT for Star-Shaped Regions 69
§2.6. Holomorphically Simply Connected Regions, Logs, and
Fractional Powers 71
§2.7. The Cauchy Integral Formula for Disks and Annuli 76
vii
,viii Contents
Chapter 3. Consequences of the Cauchy Integral Formula 79
§3.1. Analyticity and Cauchy Estimates 80
§3.2. An Improved Cauchy Estimate 93
§3.3. The Argument Principle and Winding Numbers 95
§3.4. Local Behavior at Noncritical Points 104
§3.5. Local Behavior at Critical Points 108
§3.6. The Open Mapping and Maximum Principle 114
§3.7. Laurent Series 120
§3.8. The Classification of Isolated Singularities;
Casorati–Weierstrass Theorem 124
§3.9. Meromorphic Functions 128
§3.10. Periodic Analytic Functions 132
Chapter 4. Chains and the Ultimate Cauchy Integral Theorem 137
§4.1. Homologous Chains 139
§4.2. Dixon’s Proof of the Ultimate CIT 142
§4.3. The Ultimate Argument Principle 143
§4.4. Mesh-Defined Chains 145
§4.5. Simply Connected and Multiply Connected Regions 150
§4.6. The Ultra Cauchy Integral Theorem and Formula 151
§4.7. Runge’s Theorems 153
§4.8. The Jordan Curve Theorem for Smooth Jordan Curves 161
Chapter 5. More Consequences of the CIT 167
§5.1. The Phragmén–Lindelöf Method 168
§5.2. The Three-Line Theorem and the Riesz–Thorin
Theorem 174
§5.3. Poisson Representations 177
§5.4. Harmonic Functions 183
§5.5. The Reflection Principle 189
§5.6. Reflection in Analytic Arcs; Continuity at Analytic
Corners 196
§5.7. Calculation of Definite Integrals 201
Chapter 6. Spaces of Analytic Functions 227
§6.1. Analytic Functions as a Fréchet Space 228
§6.2. Montel’s and Vitali’s Theorems 234
, Contents ix
§6.3. Restatement of Runge’s Theorems 244
§6.4. Hurwitz’s Theorem 245
§6.5. Bonus Section: Normal Convergence of Meromorphic
Functions and Marty’s Theorem 247
Chapter 7. Fractional Linear Transformations 255
§7.1. The Concept of a Riemann Surface 256
§7.2. The Riemann Sphere as a Complex Projective Space 267
§7.3. PSL(2, C) 273
§7.4. Self-Maps of the Disk 289
§7.5. Bonus Section: Introduction to Continued Fractions
and the Schur Algorithm 295
Chapter 8. Conformal Maps 309
§8.1. The Riemann Mapping Theorem 310
§8.2. Boundary Behavior of Riemann Maps 319
§8.3. First Construction of the Elliptic Modular Function 325
§8.4. Some Explicit Conformal Maps 336
§8.5. Bonus Section: Covering Map for General Regions 353
§8.6. Doubly Connected Regions 357
§8.7. Bonus Section: The Uniformization Theorem 362
§8.8. Ahlfors’ Function, Analytic Capacity and the Painlevé
Problem 371
Chapter 9. Zeros of Analytic Functions and Product Formulae 381
§9.1. Infinite Products 383
§9.2. A Warmup: The Euler Product Formula 387
§9.3. The Mittag-Leffler Theorem 399
§9.4. The Weierstrass Product Theorem 401
§9.5. General Regions 406
§9.6. The Gamma Function: Basics 410
§9.7. The Euler–Maclaurin Series and Stirling’s
Approximation 430
§9.8. Jensen’s Formula 448
§9.9. Blaschke Products 451
§9.10. Entire Functions of Finite Order and the Hadamard
Product Formula 459
