Table Of ContentRaghavan Narasimhan
Complex Analysis in
One Variable
Springer Science+Business Media, LLC
Professor Dr. Raghavan Narasimhan
The University of Chicago
Department of Mathematics
5734 University Avenue
Chicago, IL 60637 (USA)
Library of Congress Cataloging in Publication Data
Narasimhan, Raghavan
Complex analysis in one variable.
Includes bibliographies and index.
1. Functions of complex variables. 2. Mathematical
analysis. 1. Title. II. Title: Complex analysis in 1
variable.
QA331.N27 1985 515.9 84-11151
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Narasimhan, Raghavan:
Complex analysis in one variable !
by Raghavan Narasimhan. -
Springer Science+ Business Media, LLC
ISBN 978-0-8176-3237-3 ISBN 978-1-4757-1106-6 (eBook)
DOI 10.1007/978-1-4757-1106-6
All rights reserved.
No part of this publication may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise,
without prior permission of the copyright owner.
© 1985 Springer Science+Business Media New York
Originally published by Birkhăuser Boston, Inc in 1985.
Softcover reprint ofthe hardcover Ist edition 1985
Dedicated to
K. CHANDRASEKHARAN
vii
Contents
Preface.. .. .. .. .. .. .. . .. .. .. . .. ... . ... . .. .. .. .... .. .. .. .. .. . .. .. xi
Notation and Terminology ........................................ XIV
Chapter 1. Elementary Theory of Holomorphic Functions. . . . . . . . . . . . . . 1
1. Some basic properties of <C-differentiable and holomorphic functions 2
2. Integration along curves ....................................... 9
3. Fundamental properties of holomorphic functions. . . . . . . . . . . . . . . .. 21
4. The theorems of Weierstrass and Montel. . . . . . . . . . . . . . . . . . . . . . . .. 32
5. Meromorphic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
6. The Looman-Menchoff theorem ................................ 43
Notes on Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50
References: Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51
Chapter 2. Covering Spaces and the Monodromy Theorem ............ 53
1. Covering spaces and the lifting of curves. . . . . . . . . . . . . . . . . . . . . . . .. 53
2. The sheaf of germs of holomorphic functions. . . . . . . . . . . . . . . . . . . .. 55
3. Covering spaces and integration along curves. . . . . . . . . . . . . . . . . . . .. 58
4. The monodromy theorem and the homotopy form of
Cauchy's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60
5. Applications of the monodromy theorem. . . . . . . . . . . . . . . . . . . . . . . .. 64
Notes on Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
References: Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
viii Contents
Chapter 3. The Winding Number and the Residue Theorem. . . . . . . . . . .. 70
1. The winding number .......................................... 70
2. The residue theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
3. Applications of the residue theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81
Notes on Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87
References: Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87
Chapter 4. Picard's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89
Notes on Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97
References: Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98
Chapter 5. The Inhomogeneous Cauchy-Riemann Equation and
Runge's Theorem ................................................ 100
1. Partitions of unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100
2. The equation ~~ = ¢ .......................................... 102
3. Runge's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 106
4. The homology form of Cauchy's theorem. . . . . . . . . . . . . . . . . . . . . . .. 115
Notes on Chapter 5 .............................................. 117
References: Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 118
Chapter 6. Applications of Runge's Theorem ........................ 119
1. The Mittag-Leffler theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119
2. The cohomology form of Cauchy's theorem ...................... 123
3. The theorem of Weierstrass .................................... 126
4. Ideals in Yf (Q) ............................................... 131
Notes on Chapter 6 .............................................. 141
References: Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 142
Chapter 7. The Riemann Mapping Theorem and Simple
Connectedness in the Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 144
1. Analytic automorphisms of the disc and of the annulus. . . . . . . . . . .. 144
2. The Riemann mapping theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 148
3. Simply connected plane domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 150
Notes on Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. 153
References: Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 154
Contents IX
Chapter 8. Functions of Several Complex Variables 156
Notes on Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 164
References: Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165
Chapter 9. Compact Riemann Surfaces ............................. 166
1. Definitions and basic theorems ................................. 166
2. Meromorphic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 171
3. The cohomology group Hl(U,CD) ................................ 173
4. A theorem from functional analysis ............................. 176
5. The finiteness theorem ......................................... 183
6. Meromorphic functions on a compact Riemann surface. . . . . . . . . . .. 185
Notes on Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 190
References: Chapter 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191
Chapter 10. The Corona Theorem ................................... 193
1. The Poisson Integral and the theorem of F. and M. Riesz .......... 194
2. The corona theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 204
Notes on Chapter 10 ............................................. 214
References: Chapter 10 ........................................... 215
Chapter 11. Subharmonic Functions and the Dirichlet Problem ......... 216
1. Semi-continuous functions ...................................... 216
2. Harmonic functions and Harnack's principle ..................... 219
3. Convex functions .............................................. 223
4. Subharmonic functions: Definition and basic properties. . . . . . . . . . .. 227
5. Subharmonic functions: Further properties and application to
convexity theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 236
6. Harmonic and subharmonic functions on Riemann surfaces . . . . . . .. 246
7. The Dirichlet problem ......................................... 246
8. The Rad6-Cartan theorem ..................................... 255
Notes on Chapter 11 ............................................. 260
References: Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 261
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 263
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 264
xi
Preface
This book is based on a first-year graduate course I gave three times at the
University of Chicago. As it was addressed to graduate students who intended
to specialize in mathematics, I tried to put the classical theory of functions of
a complex variable in context, presenting proofs and points of view which
relate the subject to other branches of mathematics. Complex analysis in one
variable is ideally suited to this attempt. Of course, the branches of mathema
tics one chooses, and the connections one makes, must depend on personal
taste and knowledge. My own leaning towards several complex variables will
be apparent, especially in the notes at the end of the different chapters.
The first three chapters deal largely with classical material which is avai
lable in the many books on the subject. I have tried to present this material as
efficiently as I could, and, even here, to show the relationship with other
branches of mathematics.
Chapter 4 contains a proof of Picard's theorem; the method of proof I have
chosen has far-reaching generalizations in several complex variables and in
differential geometry.
The next two chapters deal with the Runge approximation theorem and its
many applications. The presentation here has been strongly influenced by
work on several complex variables.
Although Chapter 8 is entitled "Functions of several complex variables,"
the book as a whole is about a single variable. This chapter is meant to
contrast the behavior in higher dimension with that in the plane.
xii Preface
Chapter 9, on Riemann surfaces, is meant to serve as an introduction to
tools which are of great importance, not only in the modern study of
Riemann surfaces, but also in algebraic geometry, in several complex va
riables and elsewhere.
Chapter 10 presents Tom Wolff's proof of the corona theorem. It is meant
to demonstrate the use of real variable methods in complex analysis, and
could be used as an introduction to the study of HP spaces, a subject that has
been much in evidence in recent work in Fourier analysis.
The last chapter is a return to classical material. Subharmonic functions
and their generalizations to several variables are of great importance.
There are notes at the end of each chapter which are partly historical and
partly an attempt to point out some directions in which the material of the
chapter has developed.
Chapters are divided into sections (§), and, in each section, definitions,
theorems, propositions and lemmas are each numbered consecutively. Refe
rences such as Theorem 1 are to the same section; § 2, Lemma 1, for example,
refers to the second section of the same chapter. A reference such as
Chapter 3, § 1, Definition 1 is self-explanatory. References to books and papers
in any chapter (which are few in the text, but more numerous in the notes) are
to the list given at the end of the chapter in which they occur.
As for prerequisites, it is assumed that the reader is well acquainted with
calculus in several variables and with point set topology (properties of locally
compact spaces, of connected components and the like). He will need some
basic definitions and results from linear algebra and the theory of rings and
ideals, as he will elementary properties of Lebesgue measure and the standard
convergence theorems for Lebesgue integrals. Finally, from functional analy
sis, he will need the Hahn-Banach theorem and the closed graph theorem (for
Banach spaces) and a few other elementary and easily accessible facts (such as
the finite dimensionality of a locally compact Banach space).
I prepared a handwritten version of the course I gave at Chicago. A few
people who saw this version (W. Beckner, K. Chandrasekharan, I. Kaplansky
among them) found it useful and suggested that it might be of some general
use. I am very grateful to them for their encouragement and for the many
suggestions they made.
I am also grateful to Klas Diederich who saw my handwritten notes and
suggested to Klaus Peters that Birkhauser might be interested in publishing
this book.
To K. Chandrasekharan and Irving Kaplansky, lowe special thanks. Wi
thout the suggestions and encouragement of the former, this book would
never have been completed. As for the latter, anyone glancing through the
Preface xiii
notes on Chapters 6 and 9 will find acknowledgment for specific results and
proofs he showed me. But he also read my notes very carefully, compared the
proofs given there with others in the literature and was always most helpful
with suggestions, references, and answers to questions.
