Graduate Texts in Mathematics
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Graduate Texts in Mathematics

245

Graduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA

Advisory Board: Colin Adams, Williams College, Williamstown, MA, USA Alejandro Adem, University of British Columbia, Vancouver, BC, Canada Ruth Charney, Brandeis University, Waltham, MA, USA Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA Roger E. Howe, Yale University, New Haven, CT, USA David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USA Jill Pipher, Brown University, Providence, RI, USA Fadil Santosa, University of Minnesota, Minneapolis, MN, USA Amie Wilkinson, University of Chicago, Chicago, IL, USA

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.

For further volumes: http://www.springer.com/series/136

Rub´ı E. Rodr´ıguez • Irwin Kra • Jane P. Gilman

Complex Analysis In the Spirit of Lipman Bers Second Edition

123

Rub´ı E. Rodr´ıguez Facultad de Matem´aticas Pontificia Universidad Cat´olica de Chile Santiago, Chile

Irwin Kra Department of Mathematics State University of New York at Stony Brook Stony Brook, NY, USA

Jane P. Gilman Department of Mathematics and Computer Science Rutgers University Newark, NJ, USA

ISSN 0072-5285 ISBN 978-1-4419-7322-1 ISBN 978-1-4419-7323-8 (eBook) DOI 10.1007/978-1-4419-7323-8 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012950351 © Springer Science+Business Media New York 2007, 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

For Victor, Eleanor and Bob and to the memory of Mary and Lipman Bers

Preface to Second Edition

The second edition contains significant new material in several new sections. We have also expanded sections from the first edition (improving, we expect, the exposition throughout) and included new figures and exercises for added clarity, as well as, of course, corrected errors and typos in the previous version. Among the most important changes are: • We have expanded and clarified several sections from the first edition. • We have significantly enlarged the exercise sections. Some of the problems are routine, others challenging, and some require knowledge of other subjects usually covered in various first-year graduate courses. The problems are listed in more or less random order as far as their difficulty. • In both editions of this text, we use the approach to integration based on differential forms. In an alternative approach differential forms are a by-product of work on integration of functions motivated by ideas from standard treatments of integral calculus. That is the approach that Bers took in courses that he taught; it is also the approach used by Ahlfors. Of course, either of the two approaches are equally valid and lead to the same major results. In this second edition, we provide an appendix that outlines this alternative path to the main results. • New sections on Perron’s method for solving the Dirichlet problem, Green’s function, an alternative proof of the Riemann mapping theorem, and a description of the divisor of a bounded analytic function on the disc via infinite Blaschke products are included. • We prove the Bers theorem on isomorphisms between rings of holomorphic functions on plane domains. • A section on historical references prepared by Ranjan Roy has been added. In addition, the following items related to our work might be of interest to those reading this volume. • An answer manual for the exercises prepared by Vamsi Pritham Pingali is available, to instructors using the book for a course, from the publisher. • An electronic version of the book is available from the publisher. vii

viii

Preface to Second Edition

• One of the authors (IK) has created and maintains a section about the book on his web site. It contains, among other things, updated information and errata. http://www.math.sunysb.edu/irwin/bookcxinfo.html The other authors may also have information about the book on their web sites. It is our pleasure to thank Ranjan Roy for producing and allowing us to include in this volume his historical note. We are grateful to our colleagues and students who pointed out places for improvement in the first edition and in drafts of the second one. Among them: Bill Abikoff, Robert Burckel, Eduardo Friedman, Bryna Kra, Peter Landweber, Howard Masur, Sudeb Mitra, Lee Mosher, Robert Sczech, and Jacob Sturm. It is still true, of course, that errors and shortcomings may remain in the final version of this edition and these are entirely our responsibility. Spring 2012 New York, NY, USA Santiago, Chile Newark, NJ, USA

Irwin Kra Rub´ı E. Rodr´ıguez Jane P. Gilman

Preface to First Edition

This book presents fundamental material that should be part of the education of every practicing mathematician. This material will also be of interest to computer scientists, physicists and engineers. Because complex analysis has been used by generations of practicing mathematicians working in a number of different fields, the basic results have been developed and redeveloped from a number of different perspectives. We are not wedded to any one viewpoint. Rather we will try to exploit the richness of the development and explain and interpret standard definitions and results using the most convenient tools from analysis, geometry and algebra. Complex analysis has connections and applications to many other subjects in mathematics, both classical and modern, and to other sciences. It is an area where the classical and the modern techniques meet and benefit from each other. We will try to illustrate this in the applications we give. Complex analysis is the study of complex valued functions of a complex variable and its initial task is to extend the concept of differentiability from real valued functions of a real variable to these functions. A complex valued function of a complex variable that is differentiable is termed analytic, and the first part of this book is a study of the many equivalent ways of understanding the concept of analyticity. The equivalent ways of formulating the concept of an analytic function are summarized in what we term the fundamental theorem for functions of a complex variable. In dedicating the first part of this book to the very precise goal of stating and proving the fundamental theorem we follow a path in the tradition of Lipman Bers from whom we learned the subject. In the second part of the text we then proceed to the leisurely exploration of interesting consequences and applications of the fundamental theorem. We are grateful to Lipman Bers for introducing us to the beauty of the subject. The book is an outgrowth of notes from Bers’s original lectures. Versions of these notes have been used by us at our respective home institutions, some for more than 20 years, as well as by others at various universities. We are grateful to many colleagues and students who read and commented on these notes. Our interaction

ix

x

Preface to First Edition

with them helped shape this book. We tried to follow all useful advice and correct, of course, any mistakes or shortcomings they identified. Those that remain are entirely our responsibility. Newark, NJ, USA New York, NY, USA Santiago, Chile

Jane P. Gilman Irwin Kra Rub´ı E. Rodr´ıguez

Acknowledgement

The first author was supported in part by Fondecyt Grant # 1100767. The third author was supported in part by grants from the National Security Agency, from the Rutgers University Research Foundation, and from Yale University while a visiting Fellow.

xi

Contents

1

The Fundamental Theorem in Complex Function Theory . . . . . . . . . . . . 1.1 Some Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Where Do Series Converge? . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 A Problem on Partitions . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.3 Evaluation of Definite Real Integrals . .. . . . . . . . . . . . . . . . . . . . 1.2 The Fundamental Theorem of Complex Function Theory . . . . . . . . . 1.3 The Plan for the Proof.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Outline of Text.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Appendix: Historical Notes by Ranjan Roy . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 3 5 6 6

2

Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction and Preliminaries.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 Properties of Conjugation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Properties of Absolute Value .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.3 Linear Representation of C . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.4 Additional Properties of Absolute Value . . . . . . . . . . . . . . . . . . 2.1.5 Lines, Circles, and Half Planes . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.6 Polar Coordinates .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.7 Coordinates on C . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 More Preliminaries that Rely on Topology, Metrics, and Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Differentiability and Holomorphic Mappings . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Convention .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 The Cauchy-Riemann (CR) Equations . . . . . . . . . . . . . . . . . . . .

15 15 17 18 18 18 19 21 22

Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Complex Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Properties of Limits Superior and Inferior . . . . . . . . . . . . . . . . 3.1.2 The Radius of Convergence .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 More on Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

39 40 45 46 47

3

23 28 30 30

xiii

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Contents

3.3

The Exponential Function, the Logarithm Function, and Some Complex Trigonometric Functions . .. . . . . . . . . . . . . . . . . . . . 3.3.1 The Exponential Function .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 The Complex Trigonometric Functions . . . . . . . . . . . . . . . . . . . 3.3.3 The Definition of and the Logarithm Function . . . . . . . . . An Identity Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Zeros and Poles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

53 53 56 57 62 67

4

The Cauchy Theory: A Fundamental Theorem . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Line Integrals and Differential Forms . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Reparameterization . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 Subdivision of Interval . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.3 The Line Integral . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 The Precise Difference Between Closed and Exact Forms .. . . . . . . . 4.2.1 Caution .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Existence and Uniqueness.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Integration of Closed Forms and the Winding Number .. . . . . . . . . . . 4.4 Homotopy and Simple Connectivity . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 More on the Winding Number .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Cauchy Theory: Initial Version . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Appendix I: The Exterior Differential Calculus . . . . . . . . . . . . . . . . . . . . 4.8 Appendix II: An Alternative Approach to the Cauchy Theory .. . . . 4.8.1 Integration of Functions .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8.2 The Key Theorem . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

81 82 84 84 85 88 94 94 95 97 100 103 106 107 108 109

5

The Cauchy Theory: Key Consequences . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Consequences of the Cauchy Theory .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Cycles and Homology . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Jordan Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 The Mean Value Property .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Appendix: Cauchy’s Integral Formula for Smooth Functions.. . . . .

119 119 126 129 131 134

6

Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Functions Holomorphic on An Annulus . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Isolated Singularities .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Zeros and Poles of Meromorphic Functions . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Local Properties of Holomorphic Maps . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Evaluation of Definite Integrals . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 Appendix: Cauchy Principal Value .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

139 139 143 147 149 153 156 162

Sequences and Series of Holomorphic Functions . . .. . . . . . . . . . . . . . . . . . . . 7.1 Consequences of Uniform Convergence . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 A Metric on C.D/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.1 Properties of d . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.2 Properties of . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

171 171 174 175 176

3.4 3.5

7

Contents

xv

7.3 7.4 7.5

The Cotangent Function . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Compact Sets in H.D/ . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Runge’s Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5.1 Preliminaries for the Proof of Runge’s Theorem.. . . . . . . . . 7.5.2 Proof of Runge’s Theorem . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5.3 Two Major Lemmas . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1 7.5.4 Approximating . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . zc Conformal Equivalence and Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . 8.1 Fractional Linear (M¨obius) Transformations . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 Fixed Points of M¨obius Transformations .. . . . . . . . . . . . . . . . . 8.1.2 Cross Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Aut.D/ for D D b C, C, D, and H2 . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 The Riemann Mapping Theorem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.1 The Poincar´e Metric .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.2 Upper Half-plane Model . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.3 Unit Disc Model . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.4 Contractions and the Schwarz’s Lemma.. . . . . . . . . . . . . . . . . . 8.5 Finite Blaschke Products .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

179 183 187 189 190 191

Harmonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Harmonic Functions and the Laplacian . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Integral Representation of Harmonic Functions . . . . . . . . . . . . . . . . . . . . 9.3 The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3.1 Geometric Interpretation of the Poisson Formula .. . . . . . . . 9.3.2 Fourier Series Interpretation of the Poisson Formula .. . . . 9.3.3 Classical Reformulation of the Poisson Formula . . . . . . . . . 9.4 The Mean Value Property: A Characterization of Harmonicity .. . . 9.5 The Reflection Principle . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Subharmonic Functions .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.7 Perron Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8 The Dirichlet Problem (Revisited) . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.9 Green’s Function and RMT Revisited . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

229 230 232 235 237 239 240 243 244 245 249 251 256

10 Zeros of Holomorphic Functions . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Infinite Products .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Holomorphic Functions with Prescribed Zeros .. . . . . . . . . . . . . . . . . . . . 10.3 The Ring H.D/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Euler’s -Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.1 Basic Properties.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.2 Estimates for .z/ . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.3 The Formulae for the Function.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.5 Divisors and the Field of Meromorphic Functions . . . . . . . . . . . . . . . . . 10.6 Infinite Blaschke Products . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

267 268 272 276 280 280 284 287 289 290

8

9

193 199 200 202 202 205 207 211 212 214 218 219 221

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Contents

Bibliographical Notes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 297 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 299 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 301

Standard Notation and Commonly Used Symbols

Standard Terminology

Term

Meaning

LHS RHS Deleted neighborhood of z CR d D MMP MVP CkR or C .D or Œa; b/

Left-hand side Right-hand side Neighborhood with z removed Cauchy–Riemann equations Proper subset Subset, may not be proper Euclidean distance on C Hyperbolic distance on D Maximum modulus property Mean value property k-times differentiable real- (or complex-) valued functions on the domain D (or the interval Œa; b/ If and only if The complement of B in A fa 2 AI a … Bg Piecewise differentiable path Integration over the path .t / D z0 C Re2{t , 0 t 1 The empty set

iff AB pdp R jz0 jDR

;

xvii

xviii

Standard Notation and Commonly Used Symbols

A List of Symbols

Term Z Q R C b C { {R 0 into finitely many (more than one) infinite arithmetic progressions with distinct differences? The answer is no. It is obviously possible to construct such partitions if some differences are allowed to be equal. So assume that the differences are different and to the contrary that Z>0 D S1 [ S2 [ [ Sn ; where n 2 Z>1 , and for 1 i n, Si is an arithmetic progression with initial term ai and difference di , for 1 i < j n, Si \ Sj D ;, and 1 < d1 < d2 < < dn . Then 1 X X X X zi D zi C zi C C zi ; i D1

i 2S1

i 2S2

i 2Sn

and each series converges for jzj < 1. Summing the above geometric series we see that za1 z za2 zan D C C C for all z with jzj < 1: 1z 1 zd1 1 zd2 1 zdn

(1.1)

Choose a sequence of complex numbers2 fzk g of absolute value less than 1 with 2{ lim zk D e dn . Then

k!1

2{

e dn zk lim D 2{ k!1 1 zk 1 e dn

1 In the subsequent parts of this text (except the historical remarks) we usually use z, w, and c, among others, but not x or y (which usually denote the real and imaginary part of z) to denote a complex variable. 2 Notation for the polar form of a complex number is established in Chap. 2.

1.2 The Fundamental Theorem of Complex Function Theory

3

and

lim

k!1

zaki 1 zdki

D

e

2{ai dn

1e

for i D 1; 2; : : : ; n 1;

2{di dn

(all these quantities are finite) while lim

k!1

contradiction to (1.1).

zakn 1 zdkn

does not exist. This is an obvious

1.1.3 Evaluation of Definite Real Integrals Some integrals are difficult, perhaps impossible, to evaluate using methods usually studied in undergraduate calculus courses. Examples are Z

1 1

dx and 1 C x4

Z

2 0

d : 2 C sin

Using the residue theorem (see Sect. 6.6), it is quite easy to evaluate them.

1.2 The Fundamental Theorem of Complex Function Theory Theorem 1.1. Let D C denote a domain (an open connected set), and let f D u C { v W D ! C be a complex-valued function defined on D. The following conditions are equivalent (here u and v are real-valued functions of the complex variable z D x C {y, with x and y real): (1) The complex derivative f 0 .z/ exists for all z 2 D

.Riemann/I

that is, the function f is holomorphic on D. (2) The functions u and v are continuously differentiable and satisfy @v @u @v @u D and D : @x @y @y @x

(Cauchy–Riemann: CR)

Alternatively, the function f is continuously differentiable and satisfies @f D 0: @z

(CR–complex form)

4

1 The Fundamental Theorem in Complex Function Theory

e of D there exists a holomorphic (3) For each simply connected subdomain D e ! C such that F 0 .z/ D f .z/ for all z 2 D. e function F W D (4) The function f is continuous on D, and if is a (piecewise smooth) closed curve in a simply connected subdomain of D, then Z f .z/ dz D 0:

((1) H) (4): Cauchy’s theorem; (4) H) (1): Morera’s theorem) An equivalent formulation of this condition is: The function f is continuous on D and the differential form f .z/ dz is closed on D. (5) If fz 2 C W jz z0 j rg D with r > 0, then f .z/ D

1 2{

Z j z0 jDr

f ./ d z

.Cauchy0s integral formula/

for each z such that jz z0 j < r. (6) The nth complex derivative f .n/ .z/ exists for all z 2 D and for all integers n 0: (7) If fz W jz z0 j rg D with r > 0, then there exists a unique sequence of complex numbers fan g1 nD0 such that f .z/ D

1 X

an .zz0 /n

.Weierstrass/

nD0

for each z such that jz z0 j < r. Furthermore, the series converges uniformly and absolutely on every compact subset of fz W jz z0 j < rg. The coefficients an may be computed as follows. Z f ./ 1 d .Cauchy/ an D 2{ j z0 jDr . z0 /nC1 and f .n/ .z0 / : .Taylor/ nŠ [ (8) Choose a point zi 2 Ki , where Ki is the connected component decompoan D

i 2I

sition of the complement of D in C [ f1g, and let S D fzi I i 2 I g. Then the function f is the limit (uniform on compact subsets of D) of a sequence of rational functions with singularities only in S . .Runge/

1.3 The Plan for the Proof

5

1.3 The Plan for the Proof We prove the fundamental theorem by showing the following implications. .1/ , .2/ ) .3/ ) .4/ ) .5/ ) .6/ ) .1/I .5/ ) .7/ ) .1/ , .8/: It is of course possible to follow other paths through the various claims to obtain our main result. For the convenience of the reader, we describe where the various implications are to be found. At times the reader will need to slightly enhance an argument to obtain the required implication. .1/ ( .2/: Corollary 2.41. .1/ ) .2/: Theorem 2.33 and Corollary 5.8. .1/ ) .3/: Theorem 4.61 and Corollary 4.52. .3/ ) .4/: This is a trivial implication. See Lemma 4.14 and the definitions preceding it. .4/ ) .5/: Theorems 5.12 and 5.2. .5/ ) .6/: The proof of Theorem 5.5. .6/ ) .1/: This is a trivial implication. .5/ ) .7/: The proof of Theorem 5.5. .7/ ) .1/: Theorem 3.19. .1/ ) .8/: Theorem 7.37. .8/ ) .1/: Theorem 7.2. In standard texts, typically each of these implications is stated as a single theorem. The tag words in parentheses in the fundamental theorem are the names or terms that identify the corresponding results. The forward implication .1/ ) .n/ would be the theorem: “If f is a holomorphic function, then condition .n/ holds,” where n 2 f2; 3; 4; 5; 6; 7; 8g. For example, .1/ ) .2/ would be stated as “If f is holomorphic, then the Cauchy–Riemann equations hold”. The organization of all these conditions (potentially 56 theorems—some trivial) into a single unifying theorem is the hallmark of Bers’s mathematical style: clarity and elegance. Here it provides a conceptual framework for results that are highly technical and often computational. The framework comes from insight that, once articulated, will drive the subsequent mathematics and lead to new results. While the organization of the results into one unifying theorem is a distinctive characteristic of Bers’s mathematics, the treatments we give for some of the topics are not necessarily the ones that Bers used when he taught the course. In particular, his approach to integration theory did not normally start with differentials, and his emphasis on homotopy and homology were minimal. Currently, most graduate students will often either have studied some topology and some differential geometry before they take a complex analysis course or will be taking courses that cover these topics concurrently. Such students may skip over these parts of the text or skim them as a review. Additionally, our choice of topics beyond the fundamental theorem and

6

1 The Fundamental Theorem in Complex Function Theory

hyperbolic geometry has been guided by the tastes of the various authors. Some important theorems we have omitted or not treated in detail include the Picard Theorem 6.9, whose proof Bers sometimes included in his courses.

1.4 Outline of Text Chapter 2 contains the basic definitions. It is followed by a study of power series in Chap. 3. Chapter 4 contains the central material, the Cauchy theory, of the subject. We prove that the class of analytic functions is precisely the same as the class of functions having power series expansions, and we establish other parts of the fundamental theorem. Many consequences of the Cauchy theory are established in the next two chapters. Some readers may skip or skim parts of chaps. 2 and 3. In the second part of the text we proceed to the leisurely exploration of interesting ramifications and applications of the fundamental theorem. It starts with an exploration of sequences and series of holomorphic functions in Chap. 7, that also contains Runge’s theorem on approximations of holomorphic functions by rational functions. The Riemann mapping theorem (RMT) and the connection between function theory and hyperbolic geometry are the highlights of Chap. 8. The next chapter deals with harmonic functions, including a discussion of the Dirichlet problem and an alternative proof of the RMT. Zeros of holomorphic functions are discussed in the last chapter, which contains a study of the ring of holomorphic functions on a fixed domain, infinite Blaschke products, and an introduction to special functions. The latter is the beginning of the deep connections to classical and modern number theory.

1.5 Appendix: Historical Notes by Ranjan Roy The statements of the propositions comprising the fundamental theorem of complex function theory, given above, are accompanied by the names of the mathematicians to whom those results are attributed. The history of the discoveries of these and other mathematicians can not only provide interesting footnotes in a complex analysis course but can also complement our understanding of the substance of the topic. The study of the development of complex analysis is complicated but also enhanced by the fact that several results are named for mathematicians whose work was anticipated by earlier mathematicians. For example, Theorem 10.21, credited to the Danish mathematician J. Jensen (1859–1925) who discovered it in 1899, was proved by C. Jacobi (1804–1851) for polynomials with real coefficients in his 1827 paper, Ueber den Ausdruck der verschiedenen Wurzeln einer Gleichung durch bestimmte Integrale, and, as noted by E. Landau, the proof carries over to the general case. We also note that the mathematicians of past centuries often used a notation different from ours. For instance, for Gauss, Cauchy, and Weierstrass,

1.5 Appendix: Historical Notes by Ranjan Roy

7

x or y denoted complex variables; p again, Gauss and Weierstrass employed the symbol i , whereas Cauchy used 1 for “the” square root of 1. In presenting the specific details of their researches, it is often helpful to work within their notational perspective, as in these notes. C.F. Gauss (1777–1855) appears to be the first mathematician to state Cauchy’s (integral) theorem–indeed, to have a clear conception of integration in the complex domain. In his letter of RDecember 18, 1811 to his friend F.W. Bessel, Gauss discussed the meaning of x:dx for x D a C bi . He explained that, just as the realm of real numbers may be conceived of as a line, the set of complex numbers may be viewed as a two-dimensional plane with abscissa a and ordinate b. He then defined the integral over any (rectifiable) curve as the sum of infinitesimals .x/:dx, where R dx was an infinitesimal increment along the curve. Gauss wrote that the value of x:dx remained the same along the two paths as long as x ¤ 1 for all points x in the region between the two paths. To contrast this result R with one for which x D 1 inside the region, he defined log x by the integral x1 dx starting at 1 and ending at x ¤ 0. If the curve from 1 to x circumscribed the origin, then each circuit would add ˙2 i . He noted that this helped explain why log x was a multivalued function. Though Gauss promised to publish these remarkable theorems and their proofs at an appropriate occasion, he never did so; such instances of delayed publication provide further obstacles in our study of the mathematical past. Thus, it remained for A.L. Cauchy (1789–1857) to gradually work out the theory of complex integration. In 1814, he set out to rigorously establish some earlier results on real definite integrals of Euler, Laplace, and Legendre. They had obtained such results by formally replacing real with complex parameters within the integrals. But Cauchy proved a formula, equivalent to Cauchy’s theorem for a rectangular contour, and established a 1781 result of Leonhard Euler (1707–1783) on the Gamma function. Note that Euler’s result went unpublished until 1794. As early as 1729, Euler had discovered the Gamma function as an infinite product: .m C 1/ D

1m 2mC1 2m 3mC1 3m 4mC1 ; mC1 mC2 mC3

m 0I

(1.2)

then, in 1730, he showed that the infinite product (1.2) equaled the Gamma integral: R1 m t 0 . log x/ dx . Soon afterwards, Euler defined e and the exponential function e , and, in fact, when he returned to the study of the Gamma integral late in life, he took x D et and m D n 1 to get Z

1

.n/ D

t n1 et dt:

(1.3)

0

Observe that (1.3) is equivalent to formula (10.17), except that Euler took n > 0 to be real. We note that the notation .n/ was introduced in 1811 by Legendre and the modern notation for the limits of integration was first given by Fourier in 1818.

8

1 The Fundamental Theorem in Complex Function Theory

In his 1781 paper, Euler used (1.3) to get (in modern notation) Z

1

x n1 ekx dx D

0

.n/ ; with k > 0: kn

p He then boldly took k D p C q 1 with p > 0 and obtained Z

1

x n1 epx cos qx dx D

.n/ cos n fn

(1.4)

x n1 epx sin qx dx D

.n/ sin n ; fn

(1.5)

0

Z

1 0 1

where f D .p 2 C q 2 / 2 and tan D pq : He then deduced a number of remarkable and novel integrals as particular and limiting cases. For example: Z Z

1 0 1 0

cos x p dx D x epx

r

; 2

sin qx dx D ; x

Z

r

1

sin x p dx D x

1

sin x dx D : x 2

0

Z

0

; 2

Thus, by assuming k complex, Euler came upon results he regarded as amazing and he expounded on them in some detail, sensing them to be potentially very useful. Cauchy also found Euler’s results on the Gamma function significant, and the first part of his 1814 paper, M´emoire sur les int´egrales d´efinies, contained a proof of a particular case of his integral theorem, therebyRproviding a rigorous foundation p for Euler’s results. Cauchy started with the integral f dy, where y D M C N 1 D p M.x; z/ C N.x; z/ 1 ; x, and z real. He noted that @ @x

Z f .y/ dy D f .y/

@y ; dx

@ @z

Z f .y/ dy D f .y/

@y : @z

(1.6)

p Setting f .y/ D P C Q 1, he obtained p @M @N @N @M p @y D P Q C P CQ 1 S C T 1 ; @x @x @x @x @x (1.7) p @M @N @N @M p @y D P Q C P CQ 1 U C V 1 : f .y/ @z @z @z @z @z (1.8)

f .y/

From (1.6)–(1.8), and taking

@ @ @z @x

R

f .y/ dy D

@ @ @x @z

R

f .y/ dy; he had

1.5 Appendix: Historical Notes by Ranjan Roy

9

@T p @V p @U @S C C 1 D 1 : @z @z @x @x Equating the real and imaginary parts, Cauchy derived @S @U D ; @z @x

@T @V D : @z @x

(1.9)

Note that, by taking y D x C {z, Equations (1.9) become the Cauchy–Riemann equations. We remark that d’Alembert had found these equations in the course of his researches in fluid mechanics; similar equations had appeared in the work of Lagrange and Euler. But Cauchy was the first to understand the significance of these equations in distinguishing analytic from nonanalytic functions, though he came to this insight late in his life. To arrive at the particular case of his integral theorem, Cauchy next integrated the equations in (1.9) with respect to x and z within finite limits to obtain two equations, expressible in modern form as a single formula: Z f .y/ dy D 0;

(1.10)

C

where C represents a rectangular curve (or even a class of curvilinear quadrilaterals), though we note that Cauchy did not employ geometric language in his 1814 paper. Later in this paper, he extended this result to an infinite interval, R 1 and then, using the substitution y D .p C {q/x, he transformed the integral 0 f .y/ dy to an integral along a ray through p C {q. Euler’s results (1.4) and (1.5) followed as a corollary by taking f .y/ D y n1 ey . Cauchy’s 1814 Rbasic approach was Rto conceive of andR express the real and imaginary parts of f .z/ dz as separate: P dx Qdy and Qdx C P dy. But his 1825 paper, M´emoire sur les int´egrales d´efinies prises entre des limites imaginaires, took a much more geometric point of view. He considered points in the complex plane and R discussed contours within the complex plane. He gradually began to regard f .z/ dz as an entity in itself. In his 1825 paper, Cauchy presented a proof of his integral theorem (1.10) for a more general curve C and for functions he called finite and continuous. To his mind, such functions were continuously differentiable. He showed that the value of the integral from a point A to a point B did not change for two neighboring curves. Then by a homotopy type of argument, he extended the result to two non-neighboring curves. This argument is not rigorous by modern standards. Indeed, Cauchy himself may have been unsure of his reasoning; the paper contains his attempt to reformulate it in terms of the calculus of variations. And later, in his 1846 Sur les int´egrales qui s’´etendent a` tous les points d’une courbe ferm´ee, he expressed the integral as Z

Z

Z

f dz D C

.P dx Qdy/ C { C

Qdx C P dy; C

(1.11)

10

1 The Fundamental Theorem in Complex Function Theory

where C was a closed curve within which there was no singularity of f .z/. He applied Green’s theorem, Theorem 4.20, “

Z P dx C Qdy D ˙ C

D

@Q @P @x @y

dx dy

(where the plus sign was chosen if C had a positive orientation and D represented the region within C ) to each of the two real integrals. The Cauchy–Riemann equations in the form @P D @Q and @P D @Q then showed that the two integrals @x @y @y @x were equal to 0. In fact, it appears that Green’s theorem was first stated in Cauchy’s 1846 paper. It is possible that this work was inspired by the 1828 paper by George Green (1793– 1841) on the application of mathematics to electric and magnetic phenomena. Cauchy did not prove Green’s theorem; Riemann presented the first proof in his famous dissertation of 1851. Also in 1851, Cauchy noted that the Cauchy– Riemann equations implied analyticity; he assumed that the partial derivatives were continuous. We also note that Weierstrass was aware of the proof of Cauchy’s integral theorem by means of Green’s theorem as early as 1842. All these proofs of Cauchy’s integral theorem assumed the continuity of f 0 .z/. In 1883, E. Goursat (1858–1936) retained this assumption in his discovery of a new proof of the theorem for the case of a rectangular contour. But he soon realized that he did not require continuity, so that when he republished the proof in 1900, in the first issue of the Transactions of the American Mathematical Society, he removed this condition. To deal with functions with singularities inside the closed curve, Cauchy’s 1814 paper had a result amounting to a particular case of the residue theorem, Theorem 6.17. But it was in his 1826 paper, Sur un nouveau genre de calcul analogue au calcul infinit´esimal, that he formally defined a residue for a function with a pole. Note that the idea of an essential singularity was not then known. He further showed that residues could be applied to the evaluation of definite integrals. In 1830, leaving France for political reasons, Cauchy became professor of theoretical physics at the University of Turin. There he published an important paper on the power series expansions of analytic functions, significant portions of which were reprinted with some changes in his Exercices d’analyse of 1841. He first proved Cauchy’s integral formula f .x/ D

1 2

Z

x f .x/ dp; xx

(1.12)

p

where x D X ep 1 and jxj < X ; note that x did not represent the complex conjugate. He began the proof by showing that Z

f .x/ dp D 2f .0/:

(1.13)

1.5 Appendix: Historical Notes by Ranjan Roy

11

For this purpose, Cauchy observed that DX f .x/ D X p11 Dp f .x/; he then integrated with respect to X from 0 to X and with respect to p from to . The left-hand side simplified to Z

f .x/ f .0/ dp D

Z

f .x/ dp 2 f .0/;

and the right-handRside became zero, yielding (1.13). This in turn implied that when .x/ , taking f .0/ D 0 he had f .x/ dp D 0. He then replaced f .x/ by x f .x/f xx jxj < X , to arrive at Z

xf .x/ dp D xx

Z

xf .x/ dp x x Z x2 x D f .x/ 1 C C 2 C dp D 2f .x/: x x

This proved the integral formula (1.12). To obtain the Maclaurin series (a particular x case of the Taylor series), he expanded .xx/ in (1.12) as a geometric series 1 f .x/ D 2

Z

1 f .x/ x dp D 1 x 2

Z

x2 x f .x/ 1 C C 2 C dp: x x

The general term in the series was given by 1 2

Z

f .x/ xn n x

xn dp D 2

Z

f .x/ dp: xn

We mention that, although he could easily have done so, Cauchy did not establish the existence of the derivatives of f 0 .x/. Nevertheless, Cauchy applied repeated integration by parts to write 1 2

Z

f .x/ 1 dp D xn 2 n

Z

f 0 .x/ 1 dp D x n1 2.1 2 n/ .n/

Z

f .n/ .x/ dp:

Note that, by (1.13), the last integral could be set equal to f nŠ.0/ ; Cauchy therefore argued that the function f .x/ could be expanded as a Maclaurin series for those values of x whose modulus was less than the least value for which f .x/ was not continuously differentiable. More generally, he could have obtained the coefficient .n/ as f nŠ.a/ by expanding the series in powers of x a. This was and is known as the Taylor expansion of f .x/; Brook Taylor (1685–1731) proved this result for real functions in 1712 and published it 3 years later in his Methodus Incrementorum. We should perhaps not be surprised, however, that the first explicit statement of the Taylor series and the particular case of the Maclaurin series were given by

12

1 The Fundamental Theorem in Complex Function Theory

the father of calculus, Isaac Newton (1642–1727). He presented these results in unpublished portions of his De quadratura curvarum of 1691–92, in connection with series solutions of algebraic differential equations. We must also note that in a 1671 letter to John Collins, James Gregory (1638–1675) wrote down power series expansions of functions such as tan x and sec x by computing the derivatives of these functions. This implies that he was aware of the Taylor expansion 40 years before Taylor. The Laurent series given in Theorem 6.1, a generalization of the Taylor series, was derived by P. A. Laurent (1813–1854), a French military engineer. In 1843, he published a short note on the series expansion of a function analytic in an annulus. He generalized Cauchy’s integral formula to the annulus and deduced that the function had a series expansion, provided that infinitely many terms with negative powers of x a were included. Cauchy reported to the French Academy on this work, calling Laurent’s result a new theorem but mentioning that it could be deduced from his own 1840 work on the mean values of a function. Surprisingly, Karl Weierstrass (1815–1897) had already proved Laurent’s theorem in his 1841 paper, Darstellung einer analytischen Function einer complexen Ver¨anderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt. In this paper, Weierstrass expanded a function F .x/, for x inside an annulus around the origin, into a series of the form A0 C A1 x C C A x C C A1 x 1 C C A x C D

1 X

A x :

D1

In addition to the conditions of continuity and finiteness, the function was required to satisfy the condition that the difference of the two quotients F .x C hk/ F .x/ hk

and

F .x C k/ F .x/ ; k

where x was in the annulus and h was any complex number, became infinitely small when k became infinitely small. This condition would be satisfied by the class of continuously differentiable functions. Thus, a function analytic in the annulus had a Laurent series expansion. Weierstrass expressed the coefficient A as an integral around a circle in the annulus with center at the origin. From this he deduced that if jF .x/j M on jxj D a, then jA j M a . This implied that if F remained bounded in a neighborhood of the origin, then jA j M a . He then let a ! 0 to derive A D 0 so that F .x/ D A0 C A1 x C A2 x 2 C : Weierstrass’s paper containing this theorem was not published until 50 years later when it appeared in his collected works. It is clear, however, that he did his work independently, since he became aware of Cauchy’s results much later. It

1.5 Appendix: Historical Notes by Ranjan Roy

13

seems that the Laurent expansion was never mentioned in Weierstrass’s lectures. Weierstrass may have been dissatisfied with his proof of the Laurent expansion because it employed integration, whereas he regarded power series as conceptually more fundamental than differentiation or integration. Weierstrass’s 1876 proof of the Casorati–Weierstrass theorem (Theorem 6.8) did not employ Cauchy’s integral formula; the proof was not too elegant but very interesting. Weierstrass’s valuable insight was to use power series to define the analyticity of a function; this perspective became especially valuable when analysts began to take p-adic or other noncomplex numbers as the underlying field of constants. We note that the Casorati–Weierstrass theorem was first published in 1868 by F. Casorati (1835– 1890), who used integration in an annulus. However, Schwarz reported to Casorati that he had been present when Weierstrass discussed this theorem in his 1863 Berlin lectures. Then again, in 1868, the Russian complex analyst Y. Sokhotskii (1842–1927) employed Cauchy’s integral formula to independently prove the Casorati–Weierstrass theorem in his master’s thesis at St. Petersburg University; thus, the theorem is also named the Casorati–Sokhotskii–Weierstrass theorem. Bernhard Riemann (1826–1866) took the Cauchy–Riemann equations as the basis of his theory of analytic functions. He was primarily interested in the geometric properties of these functions, that is, in their orientation-preserving and conformal (or angle-preserving) properties. In his 1851 dissertation, Riemann defined simple connectivity of a domain and outlined a proof of the proposition that every simply connected proper subdomain of the complex plane could be biholomorphically (conformally) mapped onto the unit disc. This is the RMT, Theorem 8.20. The proof was not quite correct and also depended upon the unproven Dirichlet principle. It required the efforts of such outstanding mathematicians as H. A. Schwarz, H. Poincar´e, D. Hilbert, P. Koebe, and C. Carath´eodory to put Riemann’s ideas and intuitions on a solid basis. Poincar´e famously remarked that Riemann’s method was a method of discovery, whereas that of Weierstrass was a method of proof. Schwarz gave a proof of the RMT in his 1869 paper, Zur Theorie der Abbildung. The paper begins with a proof of Schwarz’s lemma for a biholomorphic function f , mapping the unit disc onto a region U and satisfying f .0/ D 0. This lemma was then forgotten until it was rediscovered in 1905 by Carath´eodory, who applied it to give a short proof of Landau’s extension of Picard’s Theorem 6.9. Landau’s extension states that if f .z/ D a0 C a1 z C az z2 C ; a1 ¤ 0 is holomorphic and does not take the values 0 and 1 in jzj < R, then R is bounded by a number that depends on the first two coefficients, a0 and a1 . The result as given in Theorem 5.34 is due to Carath´eodory. He reported that his initial unpublished proof of this theorem depended on the Poisson integral formula (9.3). But he showed this work to Schmidt, famous for his fundamental contributions to functional analysis, who provided a superior argument. This proof, utilizing the

14

1 The Fundamental Theorem in Complex Function Theory

maximum principle (Corollary 5.31), is presented in this and other modern texts. Carath´eodory discovered other applications of this lemma and in 1912 he named it after Schwarz. Morera’s theorem, contained in the fundamental theorem, was first proved in a paper of 1886, Un teorema fondamentale nella teo rica delle funzioni di una variabile complessa. The Italian mathematician, G. Morera (1856–1909), supplied another way to define analyticity; he proved that any locally integrable complex function was analytic. This is also known as the converse of Cauchy’s theorem. In another paper of 1886, Sulla rappresentazione delle funzioni di una variabile com plessa per mezzo di espressioni analitiche infinite, Morera used his theorem to reprove Weierstrass’s convergence theorem for compactly convergent series, established in his Zur Functionenlehre of 1880.

Chapter 2

Foundations

The first section of this chapter introduces the complex plane, fixes notation, and discusses some useful concepts from real analysis. Some readers may initially choose to skim this section. The second section contains the definition and elementary properties of the class of holomorphic functions—the basic object of our study.

2.1 Introduction and Preliminaries This section is a summary of basic notation, a description of some of the basic algebraic and geometric properties of the complex number system, and a disjoint collection of needed facts from real analysis (advanced calculus). We remind the reader of some of the formalities behind the standard notation which we usually approach quite informally. Not all concepts used as prerequisites are defined (among these are neighborhood, connected, path-connected, arc-wise connected, and compact sets); we assume that the reader has been exposed to them.1 We start with some standard notation: Z>0 Z Q R C b C: Here Z represents the integers, Z>0 the positive integers,2 Q the rationals (the integer n is included in the rationals as the equivalence class of the quotient n1 ), and R the reals. Whether one views the reals as the completion of the rationals or identifies

1 The reader may want to consult J. R. Munkres Topology (Second Edition), Dover, 2000, or J. L. Kelley, General Topology, Springer-Verlag, 1975 as well as definitions in Chap. 4. 2 In general Xcondition and fx 2 XI conditiong will describe the set of all x in X that satisfy the indicated condition.

R.E. Rodr´ıguez et al., Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics 245, DOI 10.1007/978-1-4419-7323-8 2, © Springer Science+Business Media New York 2013

15

16

2 Foundations

them with Dedekind cuts (we will not use these concepts explicitly), their most important property from the perspective of complex variables is the least upper bound property; that is, that every nonempty set of real numbers that has an upper bound has a least upper bound. The inclusion of R into the complex numbers C needs a bit more explanation. It is specified as follows: for z in C, we write z D x C { y with x and y in R, where the symbol { represents a square root of 1; that is, { 2 D 1. With these conventions we can define addition and multiplication of complex numbers using the usual rules for these operations on the reals3 : for all x; y; ; 2 R, .x C {y/ C . C {/ D .x C / C {.y C / and .x C {y/. C {/ D .x y/ C {.x C y/: The real numbers, R, are identified with the subset of C consisting of those numbers z D x C { y with y D 0; the imaginary numbers, {R, are those with x D 0. For z D x C { y in C with x and y in R we write x D C; with B in C¤0 and C real.

2.1 Introduction and Preliminaries

a

21

zw

b w ½

zw w ½ 0

z '

r

µ +'−2¼.

'

µ+' r½

r½

0 µ r

µ

z

Fig. 2.3 Vector multiplication. (a) Sum of arguments smaller than 2. (b) Sum of arguments larger than 2

2.1.6 Polar Coordinates A nonzero vector in C can be described by polar coordinates .r; / as well as by the rectangular coordinates .x; y/ we have been using. If z 2 C and z ¤ 0, then we can write z D x C { y D r .cos C { sin / ; y x where r D jzj and D arg z (an argument of z) D arcsin D arccos . r r Note that the last two identities are needed to define the argument and that arg z is defined up to addition of an integral multiple of 2. This is why we labeled an argument of z as opposed to the argument.4 If w D Œcos ' C { sin ' is another nonzero complex number, then, using the usual addition formulas for the sine and cosine functions, we have zw D .r/Œcos. C '/ C { sin. C '/: This polar form of the multiplication formula shows that the complex multiplication of two (nonzero) complex numbers is equivalent to the real multiplication of their moduli and the addition of their arguments, giving a geometric interpretation of how the operation of multiplication acts on vectors represented in polar coordinates; see Fig. 2.3. It also shows (again) that jz wj D jzj jwj. Polar coordinates also provide another way to view Fig. 2.2. In particular, it follows that if n 2 Z and z D r .cos C { sin / is a nonzero complex number, then zn D r n Œcos n C { sin nI

4

The number will be defined rigorously in Definition 3.34. Trigonometric functions will be introduced in the next chapter where some of their properties, including addition formulae, will be developed. For the moment, polar coordinates should not be used in proofs.

22

2 Foundations

it also proves the famous de Moivre’s formula: .cos C { sin /n D cos.n / C { sin.n / for n 2 Z and 2 R. Therefore, for each nonzero complex number z D r .cos C { sin / and each n in Z>0 , there exist precisely n complex numbers w such that wn D z; they are the n n-th roots of z, and are given by C 2k C 2k 1 C { sin ; r n cos n n with k D 0; 1; : : : ; n 1. Note that these n complex numbers are the vertices of a regular n-gon in the plane.

2.1.7 Coordinates on C We have already seen that we can use three sets of coordinates on C, as follows. 1. Rectangular .x; y/: Each equation x D constant (respectively y D constant) yields a line parallel to the imaginary axis (respectively real axis), while the equation x D y yields the line through the origin with slope equal to 1. 2. Complex .z; z/: Only one of these coordinates is needed to describe a point by the equation z D constant (or z D constant), while the equation z D z yields the real axis. 3. Polar .r; /: The equation r D a with a > 0 is a circle of radius a centered at the origin, whereas the equation D constant is a ray emanating from (but not including) the origin. The equation r D denotes a type of spiral ending at (but not passing through) the origin. The choice of the appropriate one among the various possible coordinates on C may simplify a problem. As an example we solve the following one. Let n be a positive integer, and suppose we want to find the set of points z in C that satisfy the equation zn D zn : (2.8) Using rectangular coordinates would lead us to solve .x C { y/n D .x { y/n ; which is doable but far from pleasant. Instead, we first note that certainly z D 0 satisfies (2.8). For z ¤ 0, we may use the polar coordinates: the equation we are trying to solve is then equivalent to

2.2 Topological Preliminaries

23

r n Œcos n C { sin n D r n Œcos n { sin n; which implies that n D k for some integer k. Thus we immediately see that the complete solution to (2.8) is the set of 2n rays D o from the origin (including the origin) with .2n 1/ 2 ;:::; : o 2 0 ; ; n n n

2.2 More Preliminaries that Rely on Topology, Metrics, and Sequences We collect some facts on sets of complex numbers and functions defined on them, that mostly follow from translating to the complex system the analogous results from real analysis. The formula d.z; w/ D jz wj, for z and w 2 C, defines a metric on C. Thus .C; d / is a metric space, with a metric that agrees with the Euclidean metric on R2 (under the linear representation of the complex plane described earlier). Definition 2.1. We say that a sequence (indexed by n 2 Z>0 ) fzn g of complex numbers converges to ˛ 2 C if given > 0, there exists an N 2 Z>0 such that jzn ˛j < for all n > N ; in this case we write lim zn D ˛:

n!1

A sequence fzn g of complex numbers is called Cauchy if given > 0, there exists an N 2 Z>0 such that jzn zm j < for all n; m > N . Theorem 2.2. If fzn g and fwn g are Cauchy sequences of complex numbers, then (a) fzn C ˛ wn g is Cauchy for all ˛ 2 C. (b) fzn g is Cauchy. (c) f jzn j g R0 is Cauchy. Proof. (a) It suffices to assume that ˛ ¤ 0. Given > 0, choose N1 such that for jzn zm j < 2 for all n; m > N1 and choose N2 such that jwn wm j < 2j˛j all n; m > N2 . Choose N D max fN1 ; N2 g. Then, for all n and m > N , we have j.zn C ˛ wn / .zm C ˛ wm /j jzn zm j C j˛j jwn wm j < : (b) It follows directly from jzn zm j D jzn zm j D jzn zm j. (c) We know that for all z and w in C we have

24

2 Foundations

jjzj jwjj jz wj : Applying this inequality to zn and zm in the sequence, we obtain jjzn j jzm jj jzn zm j ; and the result follows. t u Remark 2.3. The above arguments mimic arguments in real analysis needed to establish the corresponding results for real sequences. We will, in the sequel, leave such routine arguments as exercises for the reader. Corollary 2.4. fzn g is a Cauchy sequence of complex numbers if and only if f 0. The -ball about c, or the open disc with center c and radius , is the set Uc ./ D U.c; / D fz 2 CI jz cj < g; that is, the interior of the circle with center c and radius . Proposition 2.8. A subset A of C is bounded if and only if there exist a complex number c and a positive number R such that A U.c; R/: Remark 2.9. A proof is omitted for one of three reasons (in addition to the reason described in Remark 2.3): either it is trivial or it follows directly from results in real analysis or it appears as an exercise at the end of the corresponding chapter.5 The third possibility is always labeled as such; when standard results in real analysis are needed, there is some indication of what they are or where to find them. For example, the next two theorems are translations to C of standard metric results for R2 . It

5

Exercises can be found at the end of each chapter and are numbered by chapter, so that Exercise 2.7 is to be found at the end of Chap. 2.

2.2 Topological Preliminaries

25

should be clear from the context when the first possibility occurs. It is recommended that the reader ensures that he/she is able to supply an appropriate proof when none is given. Theorem 2.10 (Bolzano–Weierstrass). Every bounded infinite set S in C has at least one limit point; that is, there exists at least one c 2 C such that, for each > 0, the ball U.c; / contains a point z 2 S with z ¤ c. Theorem 2.11. A set K C is compact if and only if it is closed and bounded. We will certainly be using a number of consequences of compactness not discussed in this chapter (e.g., in a compact metric space, every sequence has a convergent subsequence) and also of connectedness, which we will not define here. Definition 2.12. Let f be a function defined on a set S in C. We assume that f is complex-valued, unless otherwise stated. Thus f may be viewed as either a map from S into R2 or into C and also as two real-valued functions defined on the set S . Let c be a limit point of S and let ˛ be a complex number. We say that the limit of f at c is ˛, and we write lim f .z/ D ˛ z!c

if for each > 0 there exists a ı > 0 such that jf .z/ ˛j < whenever z 2 S and 0 < jz cj < ı: Remark 2.13. The condition that c is a limit point of S ensures that there are points z in S arbitrarily close to (but different from) c so that f .z/ is defined there. Note that it is not required that f .c/ be defined. The above definition is again a translation of language from R2 to C. Thus we will be able to adopt many results (the next three theorems, in particular) from real analysis. In addition to the usual algebraic operations on pairs of functions f W S ! f C and g W S ! C familiar from real analysis, such as f Ccg with c 2 C, fg, and g (provided g does not vanish on S ; that is, if g.z/ ¤ 0 for any z 2 S or, equivalently, if no z 2 S is a zero of g), we will consider other functions constructed from a single function f , that are usually not emphasized in real analysis. Among them are the following: . 0 there exists an N 2 Z>0 such that jfn .z/ fm .z/j < for all z 2 S and all n and m > N: Note that in this case the limit function f is uniquely determined; it is the pointwise limit f .z/ D lim fn .z/, for all z 2 S . n!1

Theorem 2.23. Let ffn g be a sequence of functions defined on S C. If: .1/ ffn g converges uniformly on S . .2/ Each fn is continuous on S . Then the function f defined by f .z/ D lim fn .z/; z 2 S n!1

is continuous on S . Proof. Start with two points z and c in S . Then for each natural number n we have jf .z/ f .c/j jf .z/ fn .z/j C jfn .z/ fn .c/j C jfn .c/ f .c/j : Now fix > 0. By (1), the first and third term on the right-hand side are less than for n large. If we now fix c and n, it follows from (2) that the second term is less 3 than as soon as z is close enough to c. Thus f is continuous at c. t u 3 Definition 2.24. A domain or region in C is a subset of C which is open and connected. Remark 2.25. Note that a domain in C could also be defined as an open arcwise connected subset of C. (See also Exercise 2.20.) Also note that each point in a domain D is a limit point of D, and therefore it makes sense to ask, at each point in D, about the limit of any function defined on D.

28

2 Foundations

2.3 Differentiability and Holomorphic Mappings Up to now, the complex numbers were used mainly to supply us with a convenient alternative notation. This is about to change. The definition of the derivative of a complex-valued function of a complex variable mimics that for the derivative of a real-valued function of a real variable. However, we shall see shortly that the properties of the two classes of functions are quite different. Definition 2.26. Let f be a function defined in some disc about c 2 C. We say that f is (complex) differentiable at c provided f .c C h/ f .c/ h exists. In this case the limit is denoted by lim

(2.9)

h!0

f 0 .c/;

ˇ df ˇˇ df .c/; ; or .Df /.c/; dz dz ˇzDc

and is called the derivative of f at c. Remark 2.27. (1) It is important that h be an arbitrary complex number (of small nonzero modulus) in the above definition. (2) Note that f .c C h/ f .c/ f .z/ f .c/ D lim : z!c h zc (3) If f is differentiable at c, then f is continuous at c: The converse is not true in general; see Example 2.32.4. (4) We consider two identities for a function f defined in a neighborhood of c 2C: lim

h!0

f .c C h/ D a0 C .h/ with lim .h/ D 0; h!0

and f .c C h/ D a0 C a1 h C h.h/ with lim .h/ D 0: h!0

As in real analysis, the first of these says that f is continuous at c if and only if f .c/ D a0 ; the second says that f is differentiable at c if and only if f .c/ D a0 and f 0 .c/ D a1 . Whereas in the real case the second statement is sharp with regard to smoothness, we shall see that in the complex case, under appropriate conditions, it can be improved significantly. Notation 2.28. If the function f is differentiable on a domain D (i.e., at each point of D), then it defines a function f 0 W D ! C. Thus for every n 2 Z0 we can define inductively f .n/ , the n-th derivative of f , as follows:

2.3 Differentiability and Holomorphic Mappings

29

0 f .0/ D f , and if f .n/ is defined for n 0, then we set f .nC1/ D f .n/ whenever the appropriate limits exist. It is customary to abbreviate f .2/ and f .3/ by f 00 and f 000 , respectively. Of course, f .1/ D f 0 . Definition 2.29. Let f be a function defined in a neighborhood of c 2 C. Then f is holomorphic or analytic at c if it is differentiable in a neighborhood (perhaps smaller) of c. A function defined on an open set U is holomorphic or analytic on U if it is holomorphic (equivalently, differentiable) at each point of U . It should be emphasized that holomorphicity is always defined on open sets. A function f is called anti-holomorphic if fN is holomorphic. The usual rules of differentiation hold. Let f and g be functions defined in a neighborhood of c 2 C, let F be a function defined in a neighborhood of f .c/, and let a 2 C. Then (recall Remark 2.15): .f C ag/0 .c/ D f 0 .c/ C ag 0 .c/ .fg/0 .c/ D f .c/g 0 .c/ C f 0 .c/g.c/ 0 0 0 .F (the chain rule) ıf0 / .c/ D F 0.f .c//f .c/ f .c/ 1 .c/ D provided f .c/ ¤ 0 (d) f f .c/2 n (e) if f .z/ D z with n 2 Z (and z 2 C¤0 if n 0), then f 0 .z/ D n zn1

(a) (b) (c)

Remark 2.30. About the chain rule (c): If f .z/ D w is a differentiable function of z and if F .w/ D is a differentiable function of w, then we often write the chain rule as d d dw D : dz dw dz A “proof” follows. Let z0 be arbitrary in the domain of f , and set w0 D f .z0 / and 0 D F .w0 /. Note that w D f .z/ ! w0 as z ! z0 . Now .F ı f /0 .z0 / D

0 d .z0 / D lim z!z dz 0 z z0

D lim

z!z0

D

. 0 /.w w0 / 0 w w0 D lim lim w!w0 w w0 z!z0 z z0 .w w0 /.z z0 /

d dw .w0 / .z0 / D F 0 .w0 / f 0 .z0 /: dw dz

This “proof” has an error in it, what is it? Definition 2.31. A function defined on the complex plane is called entire if it is holomorphic on C, that is, if its derivative exists at each point of C.

30

2 Foundations

Example 2.32. We illustrate some of the concepts introduced with more or less familiar examples. 1. Every polynomial (in one complex variable) is entire. These (apparently) simple objects have fairly complicated behavior, that is studied, for example, as part of complex dynamics. P 2. A rational function is a function of the form R D , where P and Q are Q polynomials (in one complex variable), with Q not the zero polynomial. Note that the polynomial Q has only finitely many zeros (the number of zeros, properly counted, equals the degree of Q; see Exercise 3.19). The rational function R is holomorphic on C fzeros of Qg. az C b with a, b, c, and d fixed 3. A special case of Example 2.32.2 is R.z/ D cz C d complex numbers satisfying ad bc ¤ 0. These rational functions are called fractional linear transformations or M¨obius transformations and will be studied in detail in Sect. 8.1. They are the building blocks for much that will follow in this book—automorphisms of domains in the Riemann sphere, and Blaschke products, and as important ingredients for much current research in areas of complex analysis: Riemann surfaces, Fuchsian, and (the more general case of) Kleinian groups. 4. In real analysis it takes work to construct a continuous function on R that is nowhere differentiable. The situation with respect to complex differentiability is much simpler. The functions z 7! z and z 7! jzj are both continuous on C, but they are nowhere (complex) differentiable, since the corresponding limits (2.9) do not exist at any c in C.

2.3.1 Convention Whenever we write z D x C { y for variables and f D u C { v for functions, then we automatically mean that x D 0 such that N D U.c; "/ fcg, and, for any M > 0, there exists a z 2 N such that jg.z/j > M ; that is, such that jf .z/ wj < M1 . Thus w is a limit point of f .N / and the last statement in the theorem is proved. Now it suffices to prove that for all w 2 C and all " > 0, the function g is unbounded in U.c; "/ fcg. If g were bounded in such a neighborhood, it would have a removable singularity at z D c and thus would extend to a holomorphic function on U.c; "/; therefore f would be meromorphic there. t u A much stronger result can be established. We state it without proof.2 Theorem 6.9 (Picard). If f is holomorphic in 0 < jz cj < R2 and has an essential singularity at z D c, then there exists a w0 2 C such that for all w 2 C fw0 g; f .z/ D w has infinitely many solutions in 0 < jz cj < R2 . 1 shows the above theorem is sharp, with c D Example 6.10. The function exp z 0 and w0 D 0.

2

For a proof, see Conway’s book listed in the bibliography.

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6 Cauchy Theory: Local Behavior and Singularities

We now have a complete description of the behavior of a holomorphic function near an isolated singularity. Theorem 6.11. Assume that f is a holomorphic function in a punctured disc U 0 D U.c; R/ fcg around the isolated singularity c 2 C. Then (1) c is a removable singularity if and only if f is bounded in U 0 if and only if lim f .z/ exists and is finite. z!c

(2) c is a pole of f if and only if lim f .z/ D 1, in the sense of Definition 3.52. z!c

(3) c is an essential singularity of f if and only if f .U 0 / is dense in C. Proof. (1) follows from Theorem 6.7 and the definition of removable singularity. The Casorati–Weierstrass Theorem 6.8 shows that if c is an essential singularity then f .U 0 / is dense in C. We complete the proof by showing (2). If c is a pole of f of order N 1, then we know from the Laurent series expansion for f that there exists 0 < r < R and a holomorphic function g W U.c; r/ ! C such that g.z/ on U.c; r/ fcg and g.c/ ¤ 0. By continuity of g we may f .z/ D .z c/N M , and it assume that jg.z/j M > 0 for all z in U.c; r/. Then jf .z/j jz cjN follows that lim f .z/ D 1. Conversely, if lim f .z/ D 1, then c is not a removable z!c

z!c

singularity of f since for every M > 0 there exists ı > 0 such that jf .z/j M for 0 < jz cj < ı; and the Casorati–Weierstrass theorem implies that c is not an essential singularity of f . Thus it must be a pole. t u Example 6.12. For an entire function f .z/ D

1 X

an zn (we know that its radius of

nD0

convergence D C1), there are two possibilities: (a) Either there exists an N such that an D 0 for all n > N , in which case f is a polynomial of degree P N . If deg f D N 1, then f has a pole of order N at i 1, and f .z/ f .0/ D N i D1 ai z is the principal part of f at 1. If deg f D 0, then f is constant, of course. (b) Or f has an essential singularity at 1. We can now establish the following result. Theorem 6.13. Let f W C [ f1g ! C [ f1g. Then, (a) If f is holomorphic, it is constant, and (b) If f is meromorphic, it is a rational function. Proof. If f is holomorphic on C [ f1g, a compact set, it must be bounded. Since it is also an entire function, it must be constant, by Liouville’s theorem.

6.3 Residues

147

If f is meromorphic on C [ f1g, its set of poles must be finite, being isolated points in a compact set. Denote them by z1 ; : : : ; zk , which may include 1. If this set is empty, we are in the previous case. Otherwise, let Nj > 0 be the order of the pole at zj and let Pj .z/ be the principal part of f at zj . Then Pj .z/ D

8P < Nj

an;j nD1 .zzj /n ;

: PNj

nD1

It follows that f The result follows.

Pk

j D1 Pj

if zj 2 C;

an;j zn ; if zj D 1:

is holomorphic in C [ f1g and therefore constant. t u

6.3 Residues If f is a holomorphic function in a deleted neighborhood U of c in C [ f1g, then ! D f .z/ dz defines a holomorphic differential form on U with an isolated singularity at c. Definition 6.14. If the holomorphic function f has an isolated singularity at c 2 C, with Laurent series expansion (6.1) on jz cj < R2 for some R2 > 0; we define the residue of the differential form ! D f .z/ dz at c by the formula Res.f .z/ dz; c/ D Res.!; c/ D a1 : Remark 6.15. Note that if f has a simple pole at c (or more generally, if c .f / 1), then Res.f .z/ dz; c/ D lim .z c/f .z/: z!c

Our next result will give an alternate way to define the residue of the differential form f .z/ dz, and will show its invariance. It is an analogue of the Cauchy integral formula for the case of an isolated singularity. Theorem 6.16. Let A denote the annulus R1 < jz cj < R2 . If is a closed path C1 X in A and if f is holomorphic in A with Laurent series f .z/ D an .z c/n , nD1

then 1 2{

Z f .z/ dz D I.; c/ a1 :

148

6 Cauchy Theory: Local Behavior and Singularities

In the special case that R1 D 0 and I.; c/ D 1, we have 1 2{

Z f .z/ dz D Res.f .z/ dz; c/:

Proof. Write X

g.z/ D

an .z c/n and f .z/ D

n¤1

a1 C g.z/: zc

The function g has a primitive in the annulus A; namely, X n¤1

1 an .z c/n : nC1

Z g.z/ dz D 0, the result follows.

Since

t u

Theorem 6.17 (Residue Theorem). Let f be holomorphic in a domain D C except for isolated singularities at z1 ; : : : ; zn in D. If is a positively oriented Jordan curve homotopic to a point in D and all zj are in the interior of , then Z f .z/ dz D 2{

n X

Res.f .z/ dz; zj /:

j D1

Proof. Put a small positively oriented circle around each zj and use the extended version of Cauchy’s Theorems 5.26and 6.16. t u Let c and b 2 C. As above assume that c is an isolated singularity of a holomorphic function f defined in a deleted neighborhood U of c. Let h be a holomorphic function defined in a deleted neighborhood of b with values in U , and let be a positively oriented circle centered at b with sufficiently small radius. Then Res..f ı h/.z/ dh.z/; b/ D D

1 2{ 1 2{

Z

f .h.z//h0 .z/ dz

Z

f .w/ dw h. /

D I.h. /; c/ Res.f .w/ dw; c/:

(6.2)

The above invariance property of residues allows us to extend its definition to holomorphic differentials with an isolated singularity at infinity. If 1 is an isolated singularity of the holomorphic function f with Laurent series expansion f .z/ D

6.4 Zeros and Poles of Meromorphic Functions C1 X

149

an zn for jzj > R, then we define the residue of the differential form ! D

nD1

f .z/ dz at 1 by the formula Res.!; 1/ D a1 :

To justify the last definition (its consistency with previously defined concepts), we show that it satisfies the invariance property (6.2). This is in agreement with our earlier convention: to obtain invariants at z D 1 we change the variable from z to 1 w D , and then use the invariants defined for w D 0. z Thus, in this case, we have 1 1 Res.f .z/ dz; 1/ D Res f d ;0 : w w But 1 1 f d D w w D

C1 X nD1 C1 X

n

an w

!

1 dw w2

.ak2 /wk dw:

kD1

6.4 Zeros and Poles of Meromorphic Functions In this section we study several consequences of the residue theorem. Let D be a domain in C and f W D ! b C be a meromorphic function. This means that f is holomorphic except for isolated singularities in D, which are removable or poles (see Sect. 3.5). We have denoted the field of meromorphic functions on D by M.D/. Note that if is a positively oriented Jordan curve in D which is homotopic to a point in D, then the interior i. / of is contained in D. Furthermore, the number of zeros and the number of poles in the interior of of any nonconstant meromorphic function in M.D/ are both finite, since range [ i. / is a compact set. The next result counts the difference between these two numbers. We recall from Definition 3.54 that the meromorphic function f has order or multiplicity c .f / at c 2 D, if there exists a holomorphic function g defined near c such that g.c/ ¤ 0 and f .z/ D .z c/c .f / g.z/ near c.

150

6 Cauchy Theory: Local Behavior and Singularities

Theorem 6.18 (The Argument Principle). Let D be a domain in C and let f 2 M.D/. Suppose is a positively oriented Jordan curve in D which is homotopic to a point in D. Let c 2 C and assume that f .z/ ¤ c and f .z/ ¤ 1 for all z in range . Then Z X 1 f 0 .z/ z .f c/ D Z P; dz D 2{ f .z/ c z2i. /

where Z is the number of zeros of the function f c inside (counting multiplicities) and P is the number of poles of the function f inside (counting multiplicities). f 0 .z/ for z 2 D, then F 2 M.D/, and we claim that F has f .z/ c only simple poles, at the zeros and poles of f c, and that

Proof. If F .z/ D

Res.F .z/ dz; d / D d .f c/ for all d 2 D:

(6.3)

The theorem then follows immediately from (6.3) and the residue theorem. To verify our claim, it suffices to assume that c D 0 D d . If we set n D 0 .f /, then f .z/ D zn g.z/ with g holomorphic near 0 and g.0/ 6D 0. It follows that f 0 .z/ D nzn1 g.z/ C zn g 0 .z/ and hence

n g 0 .z/ f 0 .z/ f 0 .z/ D C near 0. Thus dz has residue n at zero. f .z/ z g.z/ f .z/

t u

Remark 6.19. The name “argument principle” attached to the previous result may be explained in the following way: let D be a domain in C and let f 2 M.D/ be a nonconstant function. Assume is a positively oriented Jordan curve in D which is homotopic to a point in D. Let c 2 C and suppose that f .z/ ¤ c and f .z/ ¤ 1 for all z in range . If Z denotes the number of zeros of the function f c inside (counting multiplicities) and P denotes the number of poles of f inside (counting multiplicities), then the argument of f c increases by 2.Z P / upon traversing f0 . Indeed, note that D .log.f c//0 and recall that f c log.f c/ D log jf cj C { arg.f c/: Therefore, Z

f 0 .z/ dz D f .z/ c

Z

Z d log jf .z/ cj C {

d arg.f .z/ c/:

The first integral on the rightmost side of the equation equals zero because z 7! log jf .z/ cj is a well-defined (single-valued) function on the range of . The second integral on the rightmost side equals the change in the argument of f c as one traverses .

6.4 Zeros and Poles of Meromorphic Functions

151

Corollary 6.20. Let f be a nonconstant holomorphic function on a neighborhood of c 2 C, ˛ D f .c/, and m D c .f ˛/. Then there exist r > 0 and " > 0 such that for all ˇ 2 C with 0 < jˇ ˛j < ", f ˇ has exactly m simple zeros in 0 < jz cj < r. Proof. Observe that m 1. Choose a positively oriented circle around c such that f ˛ vanishes only at c in cl i. / and f 0 .z/ ¤ 0 for all z 2 cl i. / fcg. If we consider the curve 1 D f ı , it follows from Theorem 6.18 that 1 I.1 ; ˛/ D 2{

Z

f 0 .z/ dz D m: f .z/ ˛

Let w D f .z/ and conclude that for every ˇ not in the range of 1 , we have 1 2{

Z

1 f 0 .z/ dz D f .z/ ˇ 2 {

Z 1

1 dw D I.1 ; ˇ/: wˇ

Now, there exists a ı > 0 such that jf .z/ ˛j ı for all z in range . Hence jˇ ˛j < ı implies that, for all z in range , jf .z/ ˇj D j.f .z/ ˛/ .˛ ˇ/j jf .z/ ˛j j˛ ˇj > 0: Thus f ˇ does not vanish on range for such ˇ. Since I.1 ; ˇ/ is constant on each connected component of the complement of the range of 1 in C, there is an " > 0 such that jˇ ˛j < " < ı ) I.1 ; ˇ/ D m: If ˇ ¤ ˛ and jˇ ˛j < ", then all the zeros of f ˇ in i. / are simple (since f 0 is not zero near, but not necessarily at, c), and therefore f ˇ has m simple zeros in i. / fcg D fz 2 CI 0 < jz cj < rg for some positive value of r. t u Corollary 6.21. A nonconstant holomorphic function is an open mapping. Proof. If f W D ! C is holomorphic on a domain D and is not a constant, we obtain from Corollary 6.20 that for any ˛ in f .D/ there exists > 0 such that U.˛; / f .D/, and the result follows. t u Corollary 6.22. An injective holomorphic function is a diffeomorphism from its domain onto its image. Remark 6.23. Corollary 6.20 characterizes when a holomorphic function f is locally injective: f is injective in a neighborhood of c if and only if m D c .f f .c// D 1. Together with Corollaries 6.21 and 6.22, we see that this condition is equivalent to being a local diffeomorphism. In particular, in the case that m D 1, f defines a bi-holomorphic map (a bijective holomorphic function; the local inverse of f is also holomorphic) between neighborhoods of c and f .c/. See also the discussion in the next section.

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6 Cauchy Theory: Local Behavior and Singularities

Theorem 6.24 (Rouch´e’s Theorem). Let f and g be holomorphic functions on a domain D. Let be a positively oriented Jordan curve with cl i. / contained in D, and assume that jf j > jgj on range . Then Zf Cg D Zf , where Zf denotes the number of zeros of f in i. /. Proof. It follows from Theorem 6.18 that Zf D I.f ı ; 0/. Now apply Theorem 4.60 with 1 D g ı and 2 D f ı . t u Example 6.25. Rouch´e’s theorem is very useful in locating zeros of a holomorphic function, as this example shows. Let h.z/ D z5 C z4 C 6z C 1. Then ˇ4 ˇ ˇ ˇ ˇz C 6z C 1ˇ 29 < 32 D ˇz5 ˇ for all jzj D 2, and

ˇ ˇ5 ˇz C z4 C 1ˇ 3 < 6 D j6zj

for all jzj D 1. Therefore h has its five zeros contained in jzj < 2, and four of them are contained in fz 2 CI 1 < jzj < 2g. Theorem 6.26 (Integral Formula for the Inverse Function). Let R > 0. Suppose f is holomorphic on jzj < R, f .0/ D 0, f 0 .z/ ¤ 0 for jzj < R, and f .z/ ¤ 0 for 0 < jzj < R. For 0 < r < R, let r be the positively oriented circle of radius r about 0, and let m D min jf j on r . Then Z 1 tf 0 .t/ dt g.w/ D 2{ r f .t/ w defines a holomorphic function in jwj < m with f .g.w// D w on jwj < m and g.f .z// D z for z 2 i.r / \ f 1 .jwj < m/: Proof. Observe that m > 0, and fix w0 with jw0 j < m. Then on the circle r we have jf .z/j m > jw0 j : Thus f and .f w0 / have the same number of zeros in i.r /, by Rouch´e’s theorem, and hence f .z/ w0 D 0 has a unique solution z0 in i.r /. Therefore it suffices to show the following: (1) g.w0 / D z0 if f .z0 / D w0 . (2) g is a holomorphic function on the disc jwj < m. To verify (1), note that it follows from the residue theorem that g.w0 / D Res.F .s/ ds; z0 /; where F .s/ D

s f 0 .s/ for jsj < R: f .s/ w0

6.5 Local Properties of Holomorphic Maps

153

Thus g.w0 / D lim .s z0 / s!z0

D lim

s!z0

s f 0 .s/ f .s/ f .z0 /

.2s z0 /f 0 .s/ C .s 2 sz0 /f 00 .s/ f 0 .s/

D z0 ; where the second equality follows from l’Hopital’s rule (see Exercise 3.25). Alternatively, to avoid the use of l’Hopital’s rule, we change the previous series of equalities to g.w0 / D lim .s z0 / s!z0

D lim .s z0 / s!z0

s f 0 .s/ f .s/ f .z0 / sf 0 .s/ 00 .s z0 / f 0 .z0 / C f 2.z0 / .s z0 / C

D z0 : To show (2) we note that jf .t/j > jwj on r and hence 1 D f .t/ w

Thus

ˇZ ˇ Since ˇˇ

1 w n 1 X

D : w f .t/ nD0 f .t/ f .t/ 1 f .t/ 1

Z 1 1 X n t f 0 .t/ g.w/ D w dt: nC1 2{ nD0 r f .t/

ˇ ˇ t f 0 .t/ ˇ M for some constant M that is independent of n, the dt ˇ mnC1 nC1 r f .t/ last power series has radius of convergence m. t u

6.5 Local Properties of Holomorphic Maps In this section we describe the behavior of an analytic function f near any point z0 in its domain of definition D, using results from the previous section. We use the following standard notation: z D x C { y; w D s C { t D f .z/ D u.x; y/ C { v.x; y/ for z 2 D:

154

6 Cauchy Theory: Local Behavior and Singularities

Proposition 6.27. Let D be a domain in C, z0 a point in D, and f a function holomorphic on D. Then the following properties hold: (1) If f 0 .z0 / ¤ 0, then f defines a homeomorphism of some neighborhood of z0 onto some neighborhood of f .z0 /. (2) If there exists n 2 Z1 such that 0 D f 0 .z0 / D D f .n/ .z0 / and f .nC1/ .z0 / ¤ 0; then f is n C 1 to 1 near z0 . (3) If f 0 .z0 / ¤ 0, then angles between tangent vectors to curves at z0 are preserved, and infinitesimal lengths at z0 are multiplied by jf 0 .z0 /j.3 More generally, if 0 D f 0 .z0 / D D f n .z0 / and f .nC1/.z0 / ¤ 0 for some n in Z1 , then angles between tangent vectors to curves at z0 are multiplied by n C 1. (4) Conversely, if g 2 C1 .D/ preserves angles, then g 2 H.D/. (5) The change in infinitesimal areas is given by multiplication by jf 0 .z0 /j2 . Proof. Let w0 D f .z0 /. We proceed to establish the various parts of the theorem. (1) The condition implies that z0 .f .z/ w0 / D 1, and it follows from Corollary 6.20 that there exist r > 0 and " > 0 such that for all w 2 C with 0 < jw w0 j < ", f w0 has exactly one simple zero in 0 < jz z0 j < r. In other words, f is injective near z0 . Now use Corollary 6.22 to conclude. (2) Let g.z/ D f .z/ w0 . It is enough to prove that g is n C 1 to 1 near z0 . But g.z0 / D 0 D g 0 .z0 / D D g .n/ .z0 /; and g .nC1/ .z0 / ¤ 0; and therefore, for jz z0 j small, we may write g.z/ D

X

am .z z0 /m .where anC1 ¤ 0/

mnC1 1 X

D .z z0 /nC1

akCnC1 .z z0 /k

kD0

D .z z0 /

nC1

.h.z//nC1

D .g1 .z//nC1 ; where h and g1 are holomorphic functions near z0 , g1 .z0 / D 0, and .g1 /0 .z0 / D h.z0 / ¤ 0, since .h.z0 //nC1 D anC1 . The existence of h is a consequence of

3

Both claims are stated in standard shorthand form. The first statement is reformulated in the proof that follows; the second can be reformulated as jf .z/ f .z0 /j D jf 0 .z0 /j jz z0 j C .z/, where limz!z0 .z/ D 0.

6.5 Local Properties of Holomorphic Maps

155

Exercise 5.1. By (1), g1 is a homeomorphism from a neighborhood of z0 to a neighborhood of 0. Since p.z/ D znC1 is clearly nC1 to 1 from a neighborhood of 0 to a neighborhood of 0, and since g D p ı g1 , it follows that g is n C 1 to 1 from a neighborhood of z0 to a neighborhood of g.z0 / D 0, as claimed. (3) Let us write f .z/ f .z0 / w : D lim f 0 .z0 / D lim z!z0 z!z0 z z z0 Assume first that f 0 .z0 / ¤ 0. Then f 0 .z0 / D e{ ; for some > 0; 2 R: If z W Œ0; 1 ! D is a C1 -curve with z.0/ D z0 and z0 .0/ ¤ 0, then w D f ı z is a C1 -curve with w.0/ D w0 , w0 .0/ ¤ 0. Furthermore, if we denote z D z z0 w (for z close to but different from z0 ) and w D f .z/ w0 , then w D z z implies that w arg w D arg z C arg ; z which together with lim arg

z!z0

imply that

w D z

arg w0 .0/ D arg z0 .0/ C arg f 0 .z0 /:

All uses of the multi-valued arg function need to be interpreted appropriately; we leave it to the reader to do so. The assertion about lengths means that the ratio of the length of w to the length of z tends to jf 0 .z0 /j as z tends to z0 . This follows immediately from ˇ ˇ ˇ w ˇ ˇ ˇ D : lim z!z0 ˇ z ˇ The argument for the case with vanishing derivative is almost identical to the one used to establish (2) and is hence left to the reader. (4) Let z W Œ0; 1 ! D be a C1 -curve with z0 .t/ ¤ 0 for all t. Then w D g ı z W Œ0; 1 ! g.D/ is also a C1 -curve and w0 .t/ D gz z0 .t/ C gz z0 .t/: Since g preserves angles, arg

w0 .t/ must be independent of arg z0 .t/. But z0 .t/

156

6 Cauchy Theory: Local Behavior and Singularities

w0 .t/ z0 .t/ D gz C gz 0 ; 0 z .t/ z .t/ and therefore gz 0. (5) We compute the Jacobian of the holomorphic map f at z0 : ˇ ˇ ˇ ux .z0 / vy .z0 / ˇ ˇ ˇ D ux .z0 /vy .z0 / uy .z0 /vx .z0 / J.f /.z0 / D ˇ vx .z0 / vy .z0 / ˇ ˇ ˇ2 D u2x .z0 / C v2x .z0 / D ˇf 0 .z0 /ˇ : t u Remark 6.28. The above property (2) of holomorphic mappings is also a consequence of Corollary 6.20. Much of the above discussion for (1) and (2), as well as the next corollary, are slight amplifications of the material in the previous section. We have collected them in Proposition 6.27 together with (3), (4), and (5) so as to have the complete local behavior description in one place. Corollary 6.29. A holomorphic function f is injective near a point z0 in its domain if and only if f 0 .z0 / ¤ 0 if and only if f is a homeomorphism near z0 .

6.6 Evaluation of Definite Integrals The residue theorem is a powerful tool for the evaluation of many definite integrals. We illustrate this with a few examples. For the third example, the reader might want to look at the appendix to the chapter. 1. The first integral to be evaluated is Z

1 1

x4

1 dx: C1 Z

This method will work for the evaluation of integrals of the form

C1

F .x/

1 satisfying 1 .F /

dx, where F is a rational function with no singularities on R 2. 1 dz. To apply the residue We will obviously want to integrate F .z/ dz D 4 z C1 theorem we must carefully choose the path of integration. Let R > 1; choose R to be the portion on R from R to CR followed by the upper half of the circle jzj D R, as in Fig. 6.1. Since {

z4 C 1 D .z e 4 /.z e

3{ 4

{

/.z C e 4 /.z C e

3{ 4

/;

6.6 Evaluation of Definite Integrals

157 γR

−R

R

Fig. 6.1 The path of integration for first example

the function F has (possibly) nonzero residues only at these four roots of unity, and we conclude from the residue theorem that Z 1 { 3{ 4 4 C Res F; e : dz D 2{ Res F; e 4 R z C 1 The residues are easy to compute: ˇ ˇ ˇ Res.F; e / D { ˇ 2 .z C {/.z C e 4 / 1

{ 4

and Res.F; e

3{ 4

{ zDe 4

D

1 p p 2. 2 { 2/

ˇ ˇ 1 ˇ /D D p p : 3{ ˇ 2. 2 C { 2/ .z2 {/.z C e 4 / ˇzDe 3{ 4 1

Next we estimate the absolute value of the integral over the semicircle fzI =z 0; jzj D Rg: ˇZ ˇ ˇ ˇ

0

ˇ ˇ R R { e{ d ˇˇ 4 ! 0 as R ! C1: 4 4 { R e C1 R 1

p 1 2 dx D . 4C1 x 2 1 2. A second class of integrals that can be evaluated by the residue theorem consists of those of the form Z 2 I D Q.cos ; sin / d; Z

1

We conclude that

0

where Q is a rational function of two variables with no singularities on the unit circle S 1 D fzI jzj D 1g. To apply the residue theorem, we express I as an integral of a holomorphic function over the unit circle. We use the change of variables z D ei ; from where dz D e{ { d D { z d

158

6 Cauchy Theory: Local Behavior and Singularities

and cos D

z C z1 e{ e{ z z1 e{ C e{ D ; sin D D : 2 2 2{ 2{

Example 6.30. Let 0 < b < a and evaluate Z 2 Z 1 I D d D a C b cos 0 jzjD1 Z

1 dz 1 1 .{ z/ a C b z C z 2

2 { dz 2 C 2az C b bz jzjD1 X 2 { D 2 { ; z : Res bz2 C 2az C b

D

jzj0 . It is this discontinuity that is the key to the method used here. • By the residue theorem 1 2{

Z F .z/dz D Res .F .z/dz; {/ C Res .F .z/dz; {/

e{ 2 ˛ 1 e{ 2 ˛ 3 C D { e{ 2 ˛ e{ 2 ˛ : 2{ 2{ 2

D

3

162

6 Cauchy Theory: Local Behavior and Singularities

• The above integral is a sum: Z

Z F .z/dz D

R

r ˛ e{0 ˛ d.re{0 / C 1 C r 2 e2{0

Z

20 0

R˛ e{˛ d.Re{ / 1 C R2 e2{

Z 0 r e ˛ e{˛ {0 d.re / C d.e{ / C 2 2{0 2 2{ 20 1 C e R 1Cr e Z R Z 20 r˛ e{.˛C1/ 1C˛ De{0 .˛C1/ dr C {R d 2 2{0 1 C R2 e2{ 0 1Cr e Z R r˛ dr e{.20 /.1C˛/ 2 2{0 1Cr e Z 20 {.1C˛/ e 1C˛ d: { 1 C 2 e2{ 0 Z

˛ {.20 /˛

• We need some estimates: ˇ ˇ Z 20 ˇ ˇ e{.˛C1/ 1 ˇ ˇ 1C˛ 2 ! 0 as R ! 1; d ˇ R1C˛ 2 ˇR 2 e2{ ˇ ˇ 1 C R R 1 0 ˇ ˇ Z 20 {.˛C1/ ˇ ˇ e 1 ˇ 1C˛ ˇ d 2 ! 0 as ! 0: ˇ ˇ 1C˛ 2 2{ ˇ ˇ 1C e 1 2 0 R • The value of F .z/dz is independent of the choices of , 0 , and R. We can hence let the first two of these constants approach zero and the last approach infinity to obtain Z 1 Z 1 r ˛ r˛ 1 {˛ { 3 ˛ 2{.1C˛/ 2 2 D e dr e dr: 2{ { e 2 1 C r2 1 C r2 0 0 • The last identity yields at once that Z

1 0

e{ 2 ˛ x˛ dx D ; 1 C x2 1 C e{ ˛

a real number, as expected.

6.7 Appendix: Cauchy Principal Value We have studied integration of holomorphic forms ! along paths . For some applications it is useful to allow ! to have some singularities, simple poles in our case, on . We describe a path toward this goal of evaluating integrals of differential forms with some singularities.

6.7 Appendix: Cauchy Principal Value

163

Fig. 6.4 A modification of a Jordan path

Definition 6.31. We consider an oriented pdp C, and a differential form ! D f .z/ dz that is holomorphic in a neighborhood D of Rexcept for isolated singularities on the range of that are simple poles. Because ! is not defined, Rwe want to R introduce two paths ı and d;ı , the second is disconnected, for which ! and ı d;ı ! are defined and finite. Let w1 ; w2 ; : : : ; wm be the set of singularities of ! on . Let 2 R>0 be the minimum of the finite set consisting of one half the distances between the various wk and the radii of the largest discs about these points contained in D. Let 0 < ı < . For each k, let Ck be the positively oriented circle of radius ı and center wk . This circle intersects in a finite set of points. Let aı;k be the last point before wk in this intersection and bı;k the first point after wk . These two points define two arcs on Ck . Choose one of these arcs; call it ˛ı;k , the arc subtended at wk , and give it the orientation consistent with that of . Let 2 < ˛ı;k < 2 be the angle of this arc measured from aı;k . The ı-modification ı of the curve is obtained by replacing for each k the segment of between aı;k and bı;k by the arc ˛ı;k . The disconnected ı-modification d;ı of the curve is obtained by removing for each k the segment of between aı;k and bı;k . See Figs. 6.4 and 6.5. We define the principal value Z Z pr. v. ! D lim !;

e

e

ı!0 d;ı

provided the limit exists. Remark 6.32. Each of the sets faı;k g, fbı;k g, and f˛ı;k g is bounded. Hence we can construct a sequence fın g that converges to zero with the property that all three sequences faın ;k g, fbın ;k g, and f˛ın ;k g have limits denoted by ak , bk ˛k , respectively. Once these limits are known to exist, it is easy to see that the three nets4 faı;k g, fbı;k g, and f˛ı;k g also converge to the appropriate limits.

4

This topological concept and its properties are not discussed here.

164

6 Cauchy Theory: Local Behavior and Singularities

Fig. 6.5 The modification localized (after a translation and rotation)

Lemma 6.33. Under the hypothesis of the last definition, there exists a constant M > 0 independent of ı such that Z

Z !

! D {

ı

d;ı

m X

˛ı;k Res.!; wk / C ı.ı/;

kD1

where j.ı/j < M . Hence !

lim

ı!0

!

Z

Z ı

! d;ı

D {

m X

˛k Res.!; wk /:

(6.4)

kD1

Proof. Only the first identity needs verification. Fix k. By a translation and rotation, we may assume that wk D 0 and ak 2 R0 . How many zeros does the function g.z/ D Œf .z/ eŒf .z/ have? 6.10. Evaluate

Z jzjD3

f 0 .z/ dz; f .z/ 1

z3 where f .z/ D 2 2z C z2 C . 81 6.11. Suppose f is holomorphic for jzj < 1 and f What can be said about f 000 .0/?

7 1 D 3 for n D 2; 3; : : : n n

Exercises

167 23

6.12. Let f be an entire function such that jf .z/j jzj 3 for all jzj > 10. Compute f .8/ .10:001/. 6.13. Evaluate the following real integrals using residues: Z

1

1

Z

cos x dx ; 1 C x2

1 1

6.14. Find all Laurent series of the form

1 X

sin x dx: 1 C x2

an zn for the functions

1

• f .z/ D • f .z/ D

z2 ; .1 z/2 .1 C z/

1 ; .z 1/.z 2/.z 3/

and • f .z/ D

2 z2 : z.1 z/.2 z/

6.15. If f is entire and satisfies jf 00 .z/ 3j 0:001 for all z 2 C, f .0/ D 0, f .1/ D 2, f .1/ D 4, what is f .{/? 6.16. If f is an entire function such that 2 for all z 2 C and f .{/ D {C2, what is f .{/? 6.17. If f is holomorphic on 0 < jzj < 2 and satisfies f . n1 / D n2 and f . n1 / D n3 for all n 2 Z>0 , what kind of singularity does f have at 0? 6.18. Let D be a bounded domain in C with smooth boundary. Assume f is a nonconstant holomorphic function in a neighborhood of the closure of D such that jf j is constant on @D, say, jf j D c on @D. Show that f takes on each value d such that jd j < c at least once in D. 6.19. Suppose f is holomorphic in a neighborhood of the closure of the unit disc. Show that for jzj 1 f .z/.1 jzj2 / D

1 2{

Z j jD1

1 zN

f . / d ;

z

and conclude that the following inequality holds: jf .z/j .1 jzj2 /

1 2

Z 0

2

ˇ ˇ ˇf .exp{ /ˇ d:

168

6 Cauchy Theory: Local Behavior and Singularities

6.20. Let f be an entire function. Suppose there exist positive constants A and B such that jf .z/j A C B jzj10 for all z 2 C. Show that f is a polynomial. What is its degree? 6.21. Suppose f is meromorphic in a neighborhood of the closed unit disc and that jf .z/j D 1 for jzj D 1. Find the most general such function. 6.22. Let C denote the positively oriented unit circle. Consider the function 21 2 z26 1 3 1 2 f .z/ D C exp z z z : 81 2 3 Evaluate the following integrals: Z

Z

f 0 .z/ dz ;

f .z/ dz ; C

C

Z C

f 0 .z/ dz: f .z/

1 1 2 Dn ,f D 6.23. If f is holomorphic for 0 < jzj < 1 and satisfies f n n 2 2n for n D 2; 3; 4; : : :, what can you say about f ? 6.24. Suppose f is entire and f .z/ ¤ t 2 for all z 2 C and for all t 2 R. Show that f is constant. 1 6.25. If f is holomorphic for 0 < jzj < 1 and satisfies f D n2 and n 1 D 2n2 , n D 2; 3; 4; : : :, find lim inf jf .z/ 2j. f z!0 2 Z 1 sin2 2x dx using residues. 6.26. Find x2 0 6.27. Prove the following extension of the maximum modulus principle. Let f be holomorphic ˇ andˇ bounded on jzj < 1, and continuous on jzj 1 except maybe at z D 1. If ˇf .e{ /ˇ A for 0 < < 2 then jf .z/j A for all jzj < 1. 6.28. Let D denote the unit disc and let ffn g be a sequence of holomorphic functions in D such that lim fn D f uniformly on compact subsets of D. Suppose that each n!1

fn takes on the value 0 at most seven times on D (counted with multiplicity). Prove that either f 0 or f takes on the value 0 at most seven times on D (counted with multiplicity). 6.29. Show that the function f .z/ D z C 2z2 C 3z3 C 4z4 C is injective in the unit disc D D fz 2 CI jzj < 1g. Find f .D/. 6.30. Suppose f is a nonconstant function holomorphic on the unit disc fz 2 CI jzj < 1g and continuous on fz 2 CI jzj 1g such that for all 2 R, the

Exercises

169

value f .e{ / is on the boundary of the triangle with vertices 0, 1, and {. Is there a 1 z0 with jz0 j < 1 such that f .z0 / D 10 .1 C {/? Is there a z0 with jz0 j < 1 such that 1 f .z0 / D 2 .1 C {/? 6.31. Is there a function f holomorphic for jzj < 1 and continuous for jzj 1 that satisfies f .e{ / D cos C 2{ sin ; for all 2 R‹ R1 2 6.32. The integral I D 0 ex dx can be evaluated using both real and complex analytic methods. R1 R1 2 2 1. By working with 0 ex dx 0 ey dy and Fubini’s theorem, evaluate I . 2. Evaluate I using residues by considering a change of variables t D x 2 and R1 2 evaluating 0 ex cos 2mx dx by integrating along a proper closed curve. 6.33. Formulate and prove forms of the residue theorem and the argument principle for domains in the sphere C [ f1g. 6.34. Describe the type of singularity at 1 for the following functions f W C¤0 !C: 1 (a) f .z/ D exp z n (b) f .z/ D z , n 2 Z sin.z/ (c) f .z/ D z 6.35. Prove that the inverse function of an injective entire function f cannot be entire unless f is a polynomial of degree one.

Chapter 7

Sequences and Series of Holomorphic Functions

We now turn from the study of a single holomorphic function to the study of collections of holomorphic functions. In the first section we will see that under the appropriate notion of convergence of a sequence of holomorphic functions, the limit function inherits several properties that the approximating functions have, such as being holomorphic. In the second section we show that the space of holomorphic functions on a domain can be given the structure of a complete metric space. We then apply these ideas and results to obtain, as an illustrative example, a series expansion for the cotangent function. In the fourth section we characterize the compact subsets of the space of holomorphic functions on a domain. This powerful characterization is used in Sect. 7.5, to study approximations of holomorphic functions and, in particular, to prove Runge’s theorem, which describes conditions under which a holomorphic function can be approximated by rational functions with prescribed poles. The characterization will also be used in Chap. 8 to prove the Riemann mapping theorem.

7.1 Consequences of Uniform Convergence on Compact Sets We begin by recalling some notation and introducing some new symbols. Let D be a domain in C; we denote by C.D/ the vector space of continuous complex-valued functions on D, and recall that H.D/ C.D/ is the vector space of holomorphic functions on D (see Definition 3.57). We say that a compact disc cl Uz .r/ has rational center if z D x C { y with x and y in Q. Proposition 7.1. A necessary and sufficient condition for a sequence of functions ffn g C.D/ to converge uniformly on all compact subsets of D is for the sequence to converge uniformly on all compact discs with rational centers and rational radii contained in D.

R.E. Rodr´ıguez et al., Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics 245, DOI 10.1007/978-1-4419-7323-8 7, © Springer Science+Business Media New York 2013

171

172

7 Sequences and Series of Holomorphic Functions

Proof. Every compact set contained in D can be covered by finitely many such discs. t u It is clear that if a sequence of functions ffn g C.D/ converges uniformly to a function f on all compact subsets of D, then for all z in D we have pointwise convergence: lim fn .z/ D f .z/. The converse is not true: uniform convergence on n!1 all compact subsets of D is stronger than pointwise convergence. To see this observe that we know from Theorem 2.23 that if a sequence of functions ffn g C.D/ converges uniformly to a function f on all compact subsets of D, then f 2 C.D/. On the other hand, it is easy to construct an example of a sequence of continuous functions converging at every point of the domain to a discontinuous function (see Exercise 7.1). We proceed to describe some consequences of uniform convergence on all compact subsets of D, also called locally uniform convergence, for H.D/. The first of these is that H.D/ is closed under locally uniform convergence. Theorem 7.2. If ffn g H.D/ and ffn g converges uniformly on all compact subsets of D, then the limit function f is holomorphic on D. Proof. We already know that f 2 C.D/. Let be any closed curve homotopic to a point in D. Then, by Cauchy’s theorem, Z fn .z/ dz D 0:

By uniform convergence it follows that Z

Z f .z/ dz D lim

N !1

fn .z/ dz D 0;

and then, by Morera’s theorem, f is holomorphic on D. Corollary 7.3. If ffn g H.D/ and

1 X

t u

fn converges uniformly on all compact

nD1

subsets of D, then the limit function (also denoted by

1 X

fn / is holomorphic on D.

nD1

Theorem 7.2 has no analog in real variables: it is easy to see (at least pictorially) that the absolute value function on R, which has no derivative at 0, can be uniformly approximated by differentiable functions (see Exercise 7.1). A more extreme example was constructed by Weierstrass, that of a continuous function defined on Œ0; 1 which is nowhere differentiable and uniformly approximated by polynomials. We will shortly see that uniform convergence of a sequence of holomorphic functions on all compact subsets of their common domain of definition implies uniform convergence of the derivatives on the same sets. This is another feature

7.1 Consequences of Uniform Convergence

173

of holomorphic functions not shared by real differentiable functions: it is easy to construct a sequence of differentiable functions converging uniformly on a closed interval with the property that the sequence of derivatives does not converge uniformly there. We leave this construction to the reader as Exercise 7.2. Theorem 7.4. If ffn g H.D/ and fn ! f uniformly on all compact subsets of D, then fn0 ! f 0 uniformly on all compact subsets of D. Proof. Since f 2 H.D/, it is enough to check uniform convergence of the derivatives on all compact subdiscs R D, with @R D positively oriented. For z 2 i. /, we have 1 f .z/ D 2{ 0

Z

f .w/ 1 dw D lim n!1 2{ .w z/2

Z

fn .w/ dw D lim fn0 .z/I n!1 .w z/2

this convergence is uniform in any smaller compact subdisc, such as e D fz 2 i. /I inffjz wj I w 2 @Rg ı > 0g; R t u

with ı sufficiently small.

Theorem 7.5. Let ffn g be a sequence of holomorphic functions on D such that fn ! f uniformly on all compact subsets of D. If fn .z/ ¤ 0 for all z 2 D and all n 2 Z>0 , then either (a) f is identically zero, or (b) f .z/ ¤ 0 for all z 2 D. Proof. Assume that there is c 2 D with f .c/ D 0 and that f is not identically zero. Then there exists a circle with center c such that cl i. / D and f .z/ ¤ 0 for all z 2 cl i. / fcg. By the argument principle, the number N of zeros of f in i. / is given by Z 1 f 0 .z/ N D dz 1: 2{ f .z/ But Z

f 0 .z/ dz D lim n!1 f .z/

Z

fn0 .z/ dz D 0: fn .z/

t u

Remark 7.6. An equivalent formulation for this theorem is the following, sometimes referred to as Hurwitz’s theorem. Theorem 7.7. Let ffn g be a sequence of holomorphic functions on D such that fn ! f uniformly on all compact subsets of D, and assume f is not identically zero on D. For every disc U such that cl U D with the property that f ¤ 0 on @U , there exists N 2 Z>0 such that f and fn have the same number of zeros in U for all n N .

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7 Sequences and Series of Holomorphic Functions

Definition 7.8. Let f 2 H.D/. We call f simple, univalent, or schlicht if it is one-to-one (injective) on D; thus a homeomorphism onto f .D/. Theorem 7.9 (Hurwitz). Assume D is a domain in C. If ffn g is a sequence in H.D/ with fn ! f uniformly on all compact subsets of D and fn is schlicht for each n, then either f is constant or schlicht. Proof. Assume that f is neither constant nor schlicht; thus, in particular, there exist z1 and z2 in D with z1 ¤ z2 and f .z1 / D f .z2 /. For each n 2 Z>0 , set gn .z/ D fn .z/ fn .z2 / on the domain D 0 D D fz2 g. Then gn 2 H.D 0 /, gn never vanishes in D 0 and gn ! g D f f .z2 / uniformly on all compact subsets of D 0 . But g is not identically zero and vanishes at z1 ; we have thus obtained a contradiction to Theorem 7.5. t u

7.2 A Metric on C.D/ We introduce, for use in the proof of the compactness theorem of this chapter and in the proof of the Riemann mapping theorem in Chap. 8, a metric on C.D/ for any domain D in C. The metric on C.D/ will have the property that convergence in the -metric is equivalent to uniform convergence on all compact subsets of D. For K compact in D and f 2 C.D/, set jjf jjK D maxfjf .z/j I z 2 Kg: Consider the set of compact (closed) discs contained in D with rational radii and rational centers. There are countably many such discs and they cover D. Call this collection of discs fDi gi 2Z>0 . For n 2 Z>0 , let (1) Kn D

[

Di ;

i n

then fKn g is an exhaustion of D; that is, (2) Each Kn is compact; (3) Kn KnC1 for all n 2 Z>0 ; and (4)

[

int Kn D D;

n2Z>0

where int K denotes the interior of the set K.

7.2 A Metric on C.D/

175

From now on, we shall use only properties (2), (3), and (4) of our exhaustion and not how these sets were constructed. Remark 7.10. A crucial consequence of these properties that we will use often is that given an exhaustion fKn g of D, each compact subset K of D is contained in Kn for some n. For f 2 C.D/ and i 2 Z>0 , we set Mi .f / D jjf jjKi ; and note that Mi C1 Mi : We define d.f / D

1 X

2i min.1; Mi .f //

i D1

1 X

2i D 1:

(7.1)

i D1

7.2.1 Properties of d For all f and g 2 C.D/: (1) (2) (3) (4)

d.f / 0, and d.f / D 0 if and only if f 0 d.f C g/ d.f / C d.g/ For each i , 2i min.1; Mi .f // d.f / For each i , d.f / Mi .f / C 2i

Proof. Properties (1) and (3) are immediate from the definition of d . To prove (2), observe that d.f C g/ D

1 X

2i min.1; Mi .f C g//

i D1

1 X

2i min.1; Mi .f / C Mi .g//

i D1

1 X

2i Œmin.1; Mi .f // C min.1; Mi .g//

i D1

D d.f / C d.g/: For property (4) d.f / D

X j i

2j min.1; Mj .f // C

X j >i

2j min.1; Mj .f //

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7 Sequences and Series of Holomorphic Functions

X

2j Mj .f / C

j i

0 @

X

1

X

2j

j >i

2j A Mi .f / C 2i

j i

Mi .f / C 2i :

t u

Finally, we define the metric on C.D/ we have been seeking: .f; g/ D d.f g/:

(7.2)

We list some immediate properties of ; only one of them requires proof.

7.2.2 Properties of For all f , g, and h in C.D/, the following hold: (1) (2) (3) (4) (5)

.f; g/ 0 and .f; g/ D 0 if and only if f D g .f; g/ D .g; f / .f; g/ .f; h/ C .h; g/ .f C h; g C h/ D .f; g/; that is, is translation invariant .f; g/ 1; that is, is a bounded metric

Proof of property (3) .f; g/ D d.f g/ D d.f h C h g/ d.f h/ C d.h g/: t u Note that properties (1)–(3) say that is a metric on C.D/. Theorem 7.11. Convergence in the -metric in C.D/ is equivalent to uniform convergence on all compact subsets of D. Proof. Let ffn g C.D/ and assume that ffn g is -convergent. Since for every compact set K D there is an i in Z>0 such that K Ki , it suffices to show uniform convergence on Ki for each i . Given 0 < < 1, we can choose N 2 Z>0 large so that .fm ; fn / D d.fm fn / d.fm f / C d.f fn / < 2i for all m; n N .

7.2 A Metric on C.D/

177

Now 2i min.1; Mi .fm fn // d.fm fn / < 2i ; and thus Mi .fm fn / < < 1I that is, jjfm fn jjKi < : The above inequality implies that the sequence ffn g converges uniformly on Ki . If 1, then use 0 D 34 and proceed as above. We have actually shown more than claimed: if ffn g is a -Cauchy sequence in C.D/, then there exists an f 2 C.D/ such that fn ! f uniformly on all compact subsets of D. Conversely, assume that fn ! f uniformly on Ki for all i . Thus lim Mi .f fn / D 0 for all i:

n!1

Given > 0, first choose i such that 2i < and next choose N such that Mi .f 2 fn / < for all n N . Then 2 d.f fn / Mi .f fn / C 2i

0 of D. Corollary 7.13. is a complete metric on C.D/. Because of Theorem 7.11, we can reformulate the results of the previous section in terms of the metric . In particular, Theorems 7.2 and 7.4 can now be phrased as in the following corollary. We already remarked that H.D/ C.D/. We let jH.D/ denote the restriction of the metric to H.D/. Corollary 7.14. H.D/ is a closed subspace of .C.D/; /. As such, .H.D/; jH.D/ / is a complete metric space. Furthermore, f 7! f 0 is a continuous linear operator from H.D/ to itself. There is an alternate description of the topology induced by , that is, the topology of the metric space .C.D/; /. Namely, Definition 7.15. Given f 2 C.D/, K compact D and > 0, we define Nf ./ D fg 2 C.D/I .g; f / < g

178

7 Sequences and Series of Holomorphic Functions

and Vf .K; / D fg 2 C.D/I jjg f jjK < g: Remark 7.16. For any f 2 C.D/, a basis for the neighborhood system at f (with respect to the topology induced by on C.D/) is given by the sets Nf ./, with > 0. That is, the open sets U C.D/ that contain f are precisely those for which there exists > 0 such that Nf ./ U . We will now show that the same is true for the collection fVf .K; /I K compact D and > 0g: Theorem 7.17. For any f 2 C.D/, a basis for the neighborhood system at f (with respect to the topology induced by on C.D/) is given by the sets Vf .K; /. Proof. It is enough to show that (1) Given Vf .K; /, there exists an Nf .ı/ Vf .K; /, and (2) Given Nf .ı/, there exists a Vf .K; / Nf .ı/. To show (1), we assume without loss of generality that 0 < < 1. Choose i such that K Ki and set ı D 2i . If g 2 Nf .ı/, then d.g f / < 2i . Thus 2i min.1; Mi .g f // < 2i and then Mi .g f / D jjg f jjKi < : But jjg f jjK jjg f jjKi I that is, g 2 Vf .K; /. To show (2), choose i such that 2

i

ı < . For g 2 Vf 2

Mi .g f /

0, there exists N D N.K; / in Z>0 such that jjf fn jjK < for all n > N . We can apply these concepts to convergence of series of meromorphic functions.

7.3 The Cotangent Function

179

Definition 7.19.PLet ffn g be a sequence in M.D/, the meromorphic functions on D. We say that fn converges uniformly (absolutely) on a subset A of D if there 1 X fn exists an integer N such that fn is holomorphic on A for all n > N and N C1

converges uniformly (absolutely) on A.

P Theorem 7.20. Let ffn g M.D/. fn converges uniformly on all compact P If P subsets of D, then the series f D fn is a meromorphic function on D, and fn0 converges uniformly on all compact subsets to f 0 . t u

Proof. The proof is trivial.

7.3 The Cotangent Function As an application of the ideas developed in the last two sections, we establish a series expansion formula for the cotangent function. Theorem 7.21. For all z in C Z the following equalities hold: cot z D

1 cos z D C sin z z

X n2Z; n6D0

1 1 C zn n

1 X 1 1 D C 2z : z z2 n2 nD1

(7.3)

We first observe that the meromorphic function F .z/ D

cos z sin z

has its poles at the integers, and that each of these poles is simple with residue equal 1 X 1 , but this one does to 1. It would seem more natural to sum the series zn nD1 not converge (Exercise 7.5). X X 1 1 z C D converges absolutely and We claim that zn n n.z n/ n2Z; n6D0

n¤0

uniformly on all compact subsets of C. To verify this claim, assume that jzj R with R > 0. Then X jnj2R

X R jzj jnj jn zj jnj .jnj R/ jnj2R

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7 Sequences and Series of Holomorphic Functions

(N+ 12 ) (1+ı) z

(N+ 12 ) (−1−ı)

Fig. 7.1 The square CN

X jnj2R

X 1 R ˇ n ˇ 2R < C1: ˇ ˇ jnj 2 jnj2 n¤0

We can now verify the expansion (7.3) for cot z. Proof of Theorem 7.21. For >0 , let CN be the positively oriented boundary N 2Z of the square with vertices N C 12 .˙1 ˙ {/; see Fig. 7.1. Then Z X 1 cot t cot t dt D ;t : Res 2{ CN t z t z t 2 i.CN /

Here z 2 C is fixed: we take z 2 i.CN / and z … Z. The poles of the function H.t/ D cot t occur at t D z and at t D n 2 Z, and they are all simple. Furthermore, we t z see that cot t Res ; z D cot z t z and

1 cot t cos t 1 ; n D lim .t n/ D : Res t !n t z sin t t z .n z/

Thus we have 1 2{

Z CN

N 1 X 1 cot t dt D cot z C t z nDN n z " # N 1 X 1 1 1 D cot z C ; nDN n z n z n¤0

7.3 The Cotangent Function

181

where the last equality holds because N X 1 D 0: n nDN n¤0

Hence it suffices to prove Lemma 7.22. We have

Z lim

N !1 CN

cot t dt D 0: t z

Remark 7.23. Once the lemma is verified, we will also have obtained " # 1 1 1 X 1 1 1 X 2z cot z D ; D z nD1 n z n C z z nD1 n2 z2 where the last series converges uniformly and absolutely on all compact subsets of C Z. Proof of Lemma 7.22. We proceed in stages: (1) 1 2{ As usual, for G.t/ D 1 2{

Z CN

cot t dt D 0: t

cot t , we have t

Z CN

X cot t dt D Res.G.t/dt; t/ t t 2i.CN /

D Res.G.t/dt; 0/ C

N X nDN n¤0

1 : n

The last sum is clearly zero, and the residue of G.t/ dt at zero is 0 because G is an even function. # # " " Z Z Z 1 cot t 1 z (2) dt D dt D dt, cot t cot t t z t t.t z/ CN t z CN CN where the first equality holds by (1). (3) There exists an M > 0 (independent of N ) such that jcot tj M for all t 2 CN : For t D u C {v,

182

7 Sequences and Series of Holomorphic Functions

jcos tj2 D cos2 u C sinh2 v; jsin tj2 D sin2 u C sinh2 v; and thus jcot tj2 D

cos2 u C sinh2 v sin2 u C sinh2 v

On the vertical sides of CN we have u D ˙N ˙

1 2

:

and hence

1 cos2 u D cos2 ˙N ˙ D 0; 2 1 D 1; and sin2 u D sin2 ˙N ˙ 2 jcot tj2 D

sinh2 v 1 C sinh2 v

On the horizontal sides of CN , v D ˙N ˙

1: 1 2

and hence

1 C sinh2 ˙N ˙ 12 ! 1 as N ! 1: jcot tj sinh2 ˙N ˙ 12 2

Thus there exists an M > 0 such that jcot tj M for t on the horizontal sides of CN , and the claim is proved. (4) If we denote by L.CN / the length of CN , then ˇZ ˇ ˇ ˇ

CN

ˇ ˇZ ˇ ˇ cot t ˇˇ ˇˇ z dt ˇ D ˇ dt ˇˇ cot t t z t.t z/ CN Z M jzj jdtj CN jtj jt zj M jzj ˇ ˇ ˇ L.CN / ˇ ˇN C 1 ˇ ˇN C 1 ˇ jzj 2 2 M jzj ˇ ˇ ˇ 4 .2N C 1/ ! 0 as N ! 1: Dˇ ˇN C 1 ˇ ˇN C 1 ˇ jzj 2 2 t u

We have thus established the formula (7.3) for the cotangent function. Differentiating the series (7.3) term by term, we obtain the following expansion.

7.4 Compact Sets in H.D/

183

Corollary 7.24. For all z 2 C Z, 2 2

sin z

D

1 X

1 : .z n/2 nD1

In particular, setting z D 12 , 1 X 2 1 D : 4 .2n 1/2 nD1

7.4 Compact Sets in H.D/ We return to the study of C.D/ with the -metric. A metric space X is compact if and only if every sequence in X has a subsequence which converges to a point in X , and a subset X of Rn is compact if and only if it is closed and bounded, a result we generalize to H.D/. Definition 7.25. Let A C.D/. We say that A is bounded in the strong sense or strongly bounded if for all compact K D and all > 0 there exists a > 0 such that A V0 .K; / D fg 2 C.D/I g D f with jjf jjK < g: Remark 7.26. For a subset A in a metric space .X; /, one defines diam A D supf.f; g/I f and g 2 Ag: Usually one says that A is bounded if diam A < C1. For a bounded metric (as in our case), this concept is not very useful. Hence we have introduced the concept of “strongly bounded sets.” Lemma 7.27. A set A C.D/ is strongly bounded if and only if for each compact K D, there exists an M.K/ > 0 such that jjf jjK M.K/ for all f 2 A; that is, A is strongly bounded if and only if the functions in A are uniformly bounded on each compact subset of D. Proof. We leave the proof as Exercise 7.6.

t u

Theorem 7.28. A compact subset A C.D/ is closed and strongly bounded. Proof. Since A is a compact subset of the metric space C.D/, it is closed. If K D is compact, then the function f 7! jjf jjK is continuous on the compact set A. Thus it is a bounded function, and hence A is strongly bounded. t u

184

7 Sequences and Series of Holomorphic Functions

Lemma 7.29. Let c 2 C and D D U.c; R/ for some R > 0. Assume A H.D/ is strongly bounded, and let ffk gk2Z> 0 A. The sequence ffk g converges uniformly .n/ on all compact subsets of D if and only if lim fk .c/ exists (in C) for all integers k!1

n 0.

Proof. If fk ! f uniformly on all compact subsets of D, then for every .n/ nonnegative integer n, fk ! f .n/ uniformly on all compact subsets of D; in .n/ particular, fk .c/ ! f .n/ .c/, as a set consisting of one point is certainly compact. Conversely, it suffices to show that ffk g converges uniformly on cl U.c; r/ with 0 < r < R. Choose r0 such that r < r0 < R. Since A is strongly bounded, there exists an M D M.r0 / such that jfk .z/j M for jz cj r0 and all k > 0: Write fk .z/ D

X

an;k .z c/n for jz cj < RI

n0

then Cauchy’s inequalities (5.3) tell us that jan;k j

M for all n and k: r0n

Assume jz cj r. Then ˇ ˇ1 1 ˇ ˇX X ˇ n nˇ an;k .z c/ an;m .z c/ ˇ jfk .z/ fm .z/j ˇ ˇ ˇ nD0 nD0 ˇ ˇN n 1 ˇ ˇX X r ˇ nˇ ˇ .an;k an;m /.z c/ ˇ C 2M ˇ ˇ r0 nD0 nDN C1 for all N; k; m. Let > 0 and choose N0 2 Z>0 such that 2M

n 1 X r for N > N0 : < r0 2 nDN C1

Finally, choose N1 such that k; m N1 implies ˇN ˇ 0 ˇX ˇ ˇ ˇ ˇ .an;k an;m /.z c/n ˇ < : ˇ ˇ 2 nD0

7.4 Compact Sets in H.D/

185

This will be achieved by requiring, for example, that jan;k an;m j

0 whose closures are contained in D. For each i 2 Z>0 and each n 2 Z0 , consider the mapping ni W H.D/ ! C; ni .f / D f .n/ .zi /I the maps fni g are C-linear and continuous. Given a sequence ffk gk2Z>0 in A, we consider the set of numbers .n/

ni .fk / D fk .zi /: We show that there exists B Z>0 , jBj D 1, such that .n/

lim fk .zi / exists for all n and i:

k2B k!C1

(7.4)

Assertion (7.4) suffices to prove the theorem; for then, by Lemma 7.29, the sequence ffk gk2B converges uniformly on the closed disc cl U.zi ; ri / for each i , which implies that the same sequence converges uniformly on all compact subsets of D. Since A is closed, lim fk 2 A. Thus every sequence in A has a subsequence k2B k!C1

converging to a point of A, and A is hence compact. To establish (7.4), we use the “Cantor diagonalization” method.1

1

This method is often used in analysis.

186

7 Sequences and Series of Holomorphic Functions

Since A is strongly bounded, for each i , there exists an M.i / 2 R>0 such that jf .z/j M.i / for all z in cl U.zi ; ri / and all f in A. Thus ˇ M.i / ˇ ˇ ˇ .n/ nŠ: ˇfk .zi /ˇ rin Now (as i and n vary) f 7! ni .f / form a countable set of mappings. Renumber these mappings as f1 ; 2 ; : : : ; m ; : : :g: For m D 1, f1 .fk /gk2Z>0 is a bounded sequence of complex numbers, and therefore there exists a subsequence B1 of Z>0 such that lim 1 .fk / exists: k2B1 k!C1

For m D 2, f2 .fk /gk2B1 f2 .fk /gk2Z>0 is a bounded sequence of complex numbers. Therefore there exists a subsequence B2 of B1 such that lim 2 .fk / exists: k2B2 k!C1

Continue to obtain a nested sequence of sets B1 B2 Bm with the property that lim m .fk / exists:

k2Bm k!C1

At last, diagonalize (justifying the name of the procedure); that is, let B D fn1 ; n2 ; : : : ; nm ; : : : g; where nm is the m-th term of Bm . Then lim m .fk / exists for all m 2 Z>0

k2B k!C1

7.5 Runge’s Theorem

187

because fnm ; nmC1 ; : : :g Bm :

t u

Corollary 7.31. A set A H.D/ is compact if and only if it is closed and bounded in the strong sense. Definition 7.32. A set A H.D/ is relatively compact if cl A is compact. This definition clearly makes sense in much more general settings. Corollary 7.33 (Montel’s Theorem). Every strongly bounded subset of H.D/ is relatively compact. Note that the converse to Montel’s theorem also holds. Definition 7.34. Let A be a strongly bounded set in H.D/ and let ffk gk2Z>0 be a sequence in A. We say that f 2 H.D/ is adherent to ffk gk2Z>0 if it is a limit point of this sequence; that is, if for every " > 0 there exists k 2 Z>0 such that 0 < .f; fk / < ". Remark 7.35. If fxk gk2Z>0 is a sequence in a compact metric space X , then there exists a convergent subsequence of fxk gk2Z>0 ; furthermore, if every subsequence of fxk gk2Z>0 that converges has the same limit, then the sequence fxk gk2Z>0 converges. Theorem 7.36 (Vitali’s Theorem). Let D be a domain in C, and assume that the elements in a sequence ffk gk2Z>0 H.D/ are uniformly bounded on compact subsets of D. Let S D and assume that S has a limit point in D. If lim fk .z/ k!1

exists (pointwise) for all z 2 S , then the sequence ffk g converges uniformly on compact subsets of D. Proof. The assumptions imply that the set A D ffk gk2Z>0 H.D/ is strongly bounded. It follows from Montel’s theorem that its closure is compact, and therefore every subsequence of the sequence ffk g has a converging sub-subsequence. Thus there exists an f in H.D/ adherent to A. Say that f and g are both adherent to ffk g; then f .z/ D lim fk .z/ D g.z/ for z 2 S k!1

and thus f D g on D. By our previous remark, ffk g converges to f in the -metric. t u

7.5 Runge’s Theorem We consider the problem of approximating holomorphic functions by rational functions. We regard a nonconstant polynomial as a rational function whose only pole is at infinity. The ability to uniformly approximate a holomorphic function

188

7 Sequences and Series of Holomorphic Functions

depends on the region where the function is being approximated, as well as upon the function itself. The strongest statement about uniform approximation of holomorphic functions that we prove is Runge’s approximation theorem. A number of proofs appear in the literature; ours is a variant of these.2 We have already proved a form of Runge’s theorem for an open disc : a holomorphic function on has a power series expansion at the center of the disc; for every positive integer n, we obtain a polynomial of degree n by discarding all the higher order terms in the series. These polynomials converge to the function uniformly on any compact subset of the disc. On the other hand, we also know that uniform polynomial approximation does not hold in general. For instance, consider a punctured disc D fz 2 CI 0 < jz cj < Rg, with R > 0 and c arbitrary, and the analytic function on defined by 1 f .z/ D (we take advantage of the fact that it is a rational function whose only zc pole is at c — not in , of course); if f were uniformly approximated by a sequence of polynomials fpn g in the closed annulus K D f0 < r jz cj < Rg, r C exp.2{t/ for 0 t 2 we would obtain the then by taking .t/ D 2 contradiction that Z Z 0 D lim pn .z/ dz D f .z/ dz D 2{: n!1

However, truncation of the Laurent series expansion for f on shows that f is indeed uniformly approximated on K by rational functions whose poles lie outside . This fact is generalized to arbitrary open sets by the next. Theorem 7.37 (Runge). Let K be a compact subset of C and let S be a subset of b C K that intersects nontrivially each connected component of b C K. If f is a holomorphic function on an open set containing K, then it can be uniformly approximated on K by rational functions with simple poles lying on S ; that is, for every > 0 there exists a rational function R with possibly simple poles only in S such that jf .z/ R.z/j < for all z 2 K: Runge’s theorem is the implication .1/ H) .8/ of the fundamental theorem. The converse follows from Theorem 7.2. We can always choose for S a smallest set consisting of one point from each connected component of b C K. For the important special case where b C K is connected and S is chosen as S D f1g, Runge’s theorem asserts that each function that is analytic in an open neighborhood of K can be uniformly approximated in K by a sequence of polynomials. 2

We follow a course outlined by S. Grabiner, A short proof of Runge’s theorem, Am. Math. Monthly 83 (1976), 807–808, and rely on arguments appearing in Conway’s book listed in our bibliography.

7.5 Runge’s Theorem

189

An outline for the proof of Runge’s theorem is given in Sect. 7.5.2, after some needed preliminaries from real analysis and topology given in the next subsection. The proof of the theorem depends on three major lemmas that are stated and proved in the subsequent subsections; the first two are given in Sect. 7.5.3 and the third in Sect. 7.5.4.

7.5.1 Preliminaries for the Proof of Runge’s Theorem We rely on some new terminology and notation. We also call the reader’s attention to some elementary topological concepts that we will need. If F and K are subsets of C with F closed and K compact, then the distance between these two sets is the nonnegative real number d.F; K/ D inffjz wj I z 2 F and w 2 Kg; that is easily seen to satisfy d.F; K/ D 0 if and only if F \ K ¤ ;: In particular, if F consists of only one complex number c, we set d.c; K/ D d.fcg; K/ D inffjc wj I w 2 Kg: If A and B are connected subsets of C that are not disjoint, then A [ B is connected. If C is a connected component of the set A, then C is an open subset of A. Let K and S be the sets described in the hypothesis of Runge’s theorem, and define B.K; S / to be the set of continuous complex-valued functions on K that are uniform limits of sequences of rational functions with poles only in S . Sums and products of elements of B.K; S / are obviously also elements of B.K; S /, as are products of constants by elements of B.K; S /. We summarize the (additional) properties of this algebra in the following lemma. The proof is left as an exercise. Lemma 7.38. The algebra B.K; S / contains the rational functions with poles in S , and is closed under uniform limits in K. Runge’s theorem asserts that every holomorphic function on a neighborhood of K belongs to B.K; S /. To establish this, we will also need the following topological result. Lemma 7.39. Let U and V be open subsets of C with V U and @V \ U D ;. If H is a connected component of U and H \ V ¤ ;, then H V .

190

7 Sequences and Series of Holomorphic Functions

Proof. Let c be a point in H \ V and let G be the connected component of V containing c. It is enough to show that G D H . Now H [G is connected, contained in U and contains c. Since H is the component of U , containing c, G H . Furthermore, @G @V and so @G \ H D ;. This implies that H G D H cl G and, therefore, that H G is open in H . Since G is also open in H , the conclusion that G D H follows. t u

7.5.2 Proof of Runge’s Theorem We outline the steps in the proof of Runge’s theorem. The details needed to fill in the outline will be completed in the next two subsections. Let , K, S , and f be as in the statement of Runge’s theorem. The proof of the theorem (that f 2 B.K; S /) consists of four steps. We establish: (1) (Lemma 7.40 of the next section) There exists a finite collection of oriented line segments 1 , 2 , . . . , n in K such that f .z/ D

n Z 1 X f .w/ dw for all z 2 K: 2{ j D1 j w z

Z

f .w/ dw can be w z j 1 approximated uniformly on K by finite sums of rational functions z 7! P zc , where P is a polynomial and c is a point in S . Wehence drop the subscript j 1 from the notation. It will be convenient to regard P z1 as a polynomial in z. The observation that every uniformly convergent series in h on K (in particular P .h/) belongs to B.K; S / whenever h 2 B.K; S / will be used repeatedly in our arguments that follow. Z f .w/ (3) (Lemma 7.41) The line integral dw can be uniformly approximated on wz K by Riemann sums of the form (2) It suffices to prove that each of the above integrals

X k

ak ; with ak 2 C and bk 2 range : bk z

ak can be approximated uniformly on K by bk z appropriate finite sums, and thus belongs to B.K; S /.

(4) (Lemma 7.42) Each summand

7.5 Runge’s Theorem

191

7.5.3 Two Major Lemmas We establish two results: the first gives an extension of Cauchy’s integral formula, and the second provides an approximation by a rational function to a function defined by an integral. Lemma 7.40. Let K be a compact subset of C and let be an open set containing K. Then there exists a finite collection of oriented line segments 1 , 2 , . . . , n in K such that for every holomorphic function f on , n Z 1 X f .w/ f .z/ D dw 2{ i D1 i w z

(7.5)

for all z 2 K. Proof. After enlarging K if necessary, we may assume that K D cl.int.K//: For example, we enlarge K if it consists of a single point. To simplify statements we adopt the standard convention that a curve W Œa; b ! b C and its range f.t/I t 2 Œa; bg are both called . For any positive real number ı we consider a rectangular grid of horizontal and vertical lines in the plane C so that consecutive parallel lines are at distance ı apart. We let R1 ; R2 ; : : : ; Rm be the rectangles in the grid that have nonempty intersection with K. Since K is compact, there are only a finite number of such rectangles. We can (and from now on do) choose ı such that Rj for all j ; if D C any ı > 0 suffices, and otherwise it is enough to consider any 0 < ı < 12 d.K; C /, since z 2 Rj implies that p d.z; K/ < 2ı. As usual, the boundary of Rj is denoted by @Rj and is oriented in the counterclockwise direction. The integrals of a continuous form along the common boundaries of any pair of contiguous Ri and Rj cancel out (as in the proof of Goursat’s Theorem 4.61). This last observation implies S that we can choose a set S of curves whose ranges are a subset of the sides in m j D1 @Rj , and such that the set S D fi I 1 i ng satisfies (1) If i is in S, it lies on a side of only one Rj . (2) If i is in S, then it is disjoint fromSK. (3) For any continuous function g on m j D1 @Rj , we have m Z X j D1 @Rj

g.z/dz D

n Z X i D1

g.z/dz:

(7.6)

i

Each i is an oriented line segment in K. It remains to prove that equation (7.5) holds with these i . If z 2 K and z is not on the boundary of any of the rectangles, then the function

192

7 Sequences and Series of Holomorphic Functions

w 7! g.w/ D

m [ 1 f .w/ @Rj ; ; w2 2{ w z j D1

is continuous. Thus, we have, by (7.6), m Z n Z 1 X 1 X f .w/ f .w/ dw D dw: 2{ j D1 @Rj w z 2{ i D1 i w z

Assume that z belongs to the interior of exactly one of the Rj , call it Rt . If j ¤ t, then z … Rj and Z 1 f .w/ dw D 0I 2{ @Rj w z also, since z 2 Rt , by Cauchy’s integral formula, we have Z 1 f .w/ dw D f .z/: 2{ @Rt w z Thus (7.5) holds for all z 2 Rt . Since range i does not intersect K, both sides of this equation are continuous functions of z on K, and they agree on the set of points z in K that are not on the boundary of any rectangle Rj , a dense subset of K. Thus they agree for all z 2 K. t u Lemma 7.41. Let be a pdp and let K be a compact set disjoint from the range of . If f is a continuous function on and is any positive real number, then there exists a rational function R, with only simple poles, all lying on range , such that ˇZ ˇ ˇ f .w/ ˇ ˇ ˇ ˇ w z dw R.z/ˇ < for all z 2 K: Proof. We assume that is not a constant, and thus that its length is positive. Since K and the image of are disjoint, d.K; range / > 0, and we can choose a number r with 0 < r < d.K; range /: If is parameterized by Œ0; 1, then for all 0 s; t 1 and all z 2 K we have ˇ ˇ ˇ f ..t// f ..s// ˇˇ ˇ ˇ .t/ z .s/ z ˇ 1 jf ..t//.s/ f ..s//.t/ z .f ..t// f ..s///j r2 1 2 jf ..t//j j.s/ .t/j C j.t/j jf ..s// f ..t//j r C jzj jf ..s// f ..t//j :

7.5 Runge’s Theorem

193

Since and f are continuous functions and K is a compact set, there is a constant C > 0 such that jzj < C2 for all z 2 K, j.t/j C2 and jf ..t//j C for all t 2 Œ0; 1. Thus for all s and t in Œ0; 1 and all z 2 K we have ˇ ˇ ˇ f ..t// f ..s// ˇˇ C ˇ : .t/j C ..s// f ..t//j j.s/ jf ˇ .t/ z .s/ z ˇ r 2 Since both and f ı are uniformly continuous on Œ0; 1, there is a partition of Œ0; 1 with 0 D t0 < t1 < < tn D 1 such that ˇ ˇ ˇ f ..t// f ..tj // ˇˇ ˇ ˇ .t/ z .t / z ˇ < L. / j

(7.7)

for tj 1 t tj , 1 j n, and all z 2 K, where L. / denotes the length of (recall Definition 4.62). Define the function R as follows. For z ¤ .tj /, j D 1; : : : ; n, R.z/ D

n X j D1

f ..tj //

.tj / .tj 1 / : .tj / z

Then R is a rational function whose poles are simple and contained in the set f.t1 /; .t2 /; : : : ; .1/gI in particular, they are contained in range . Now inequality (7.7) gives ˇ ˇ ˇ ˇX ˇZ ˇ Z tj ˇ ˇ ˇ n ˇ f .w/ f ..t // f ..t// j 0 ˇ ˇ ˇ .t/ˇˇ dt ˇ w z dw R.z/ˇ D ˇ .t/ z .t / z j ˇj D1 tj 1 ˇ

n Z X tj ˇˇ 0 ˇˇ .t/ dt D L. / j D1 tj 1

for all z 2 K.

t u

7.5.4 Approximating

1 zc

Of central importance in the proof of Runge’s theorem is Lemma 7.42. For every c 2 C K, the rational function g W z 7! .z c/1 2 B.K; S /:

194

7 Sequences and Series of Holomorphic Functions

Proof. The proof consists of several claims that will need verification. • Let us choose R 2 R>0 so that K U.0; R/. Let z0 2 C be arbitrary with jz0 j > R and let h be a rational function with a pole only at z0 (thus if z0 2 C, h is finite at infinity). Then h can be approximated uniformly on K by polynomials; in particular, h 2 B.K; f1g/. This assertion is obvious if z0 D 1. In general,3 the 1 X an zn . This series converges uniformly Taylor series for h at 0 is of the form nD0

on every compact sub-disc of fjzj < jz0 jg centered at 0, hence certainly on K. Since its individual terms an zn are in B.K; f1g/ and this space is closed under uniform convergence on K, h 2 B.K; f1g/. • Assume that 1 2 S . We claim that B.K; .S f1g/ [ fz0 g/ B.K; S /: Indeed, if f 2 B.K; .S f1g/ [ fz0 g/ and fRj g is a sequence of rational functions with poles in .S f1g/ [ fz0 g uniformly approximating f on K, we can write f D lim Rj D lim .R1;j C R2;j /; j !1

j !1

where R1;j has all its poles (if any) in S f1g and R2;j has a unique pole (if any) at z0 . Here and below limits are understood in the sense of “uniformly on K”, and estimates on absolute values of functions are on the set K. We have shown in the previous assertion that R2;j 2 B.K; f1g/ and that there exist polynomials Pi;j (they belong to B.K; f1g/) such that lim Pi;j D R2;j :

i !1

For each j , choose i sufficiently large so that jR2;j Pi;j j

0. Choose j sufficiently large so that j >

3

ˇ ˇ 2 and ˇf .R1;j C R2;j /ˇ < : 2

The argument that follows also applies for z0 D 1.

7.5 Runge’s Theorem

195

Then ˇ ˇ ˇ ˇ ˇf .R1;j C Pi;j /ˇ D ˇf .R1;j C R2;j / C .R2;j Pi;j /ˇ ˇ ˇ ˇ ˇ ˇf .R1;j C R2;j /ˇ C ˇR2;j Pi;j ˇ

0g. Note that both D and H2 should be regarded as discs in e C, since they are bounded by circles in b C: the unit circle S 1 and the extended real line b R D R [ f1g, respectively. The next result shows that these two discs in b C are conformally equivalent. Corollary 8.15. If w.z/ D onto D.

z{ for z 2 H2 , then w is a conformal map of H2 zC{

Proof. All M¨obius transformations, in particular w, are conformal. A calculation shows that w maps b R D R [ f1g onto S 1 (the unit circle centered at 0) and w.{/ D 0. By connectivity considerations, it follows that w.H2 / D D. t u

8.2 Aut.D/ for D D b C, C, D, and H2

205

b C, D, and H2 8.2 Aut.D/ for D D C, Theorem 8.16. A function f W C ! C belongs to Aut.C/ if and only if there exist a and b in C, a ¤ 0, such that f .z/ D az C b for all z 2 C. Proof. The if part is trivial. For the only if part, note that f is an entire function, and we can use its Taylor series at zero to conclude that f .z/ D

1 X

an zn for all z 2 C:

nD0

If 1 were an essential singularity of f , then f .jzj > 1/ would be dense in C. But f .jzj > 1/ \ f .jzj < 1/ is empty since f is injective. Thus 1 is either a removable singularity or a pole of f ; in any case, there is a nonnegative integer N such that an D 0 for all n > N and aN ¤ 0; that is, f is a polynomial of degree N . If N were bigger than one or equal to zero, then f would not be injective. t u Theorem 8.17. Aut.b C/ Š PSL.2; C/. Thus the last arrow in the exact sequence (8.3) corresponds to a surjective map. Proof. We need only show that Aut.b C/ is contained in the M¨obius group. Let f be an element of Aut.b C/. If f .1/ D 1, then f is a M¨obius transformation by Theorem 8.16. If f .1/ D c ¤ 1, then consider the M¨obius transformation A.z/ D 1 and conclude that B D A ı f in Aut.b C/ and fixes 1; therefore B is a M¨obius zc 1 transformation. But then so is f D A ı B. t u We now provide a characterization of the elements of Aut.D/; it shows that they form a subgroup of Aut.b C/. Another useful characterization is given in Exercise 8.5. Theorem 8.18. A function B defined on D is in Aut.D/ if and only if there exist a and b in C such that jaj2 jbj2 D 1 and B.z/ D for all z 2 D.

az C b bz C a

206

8 Conformal Equivalence and Hyperbolic Geometry

Proof. The if part: Assume that B is of the above form and observe that a is different from zero. We show that B 2 Aut.D/. This follows from the following easy to prove facts: (1) Mappings B of the given form constitute a group under composition. In az b az C b with jaj2 jbj2 D 1, then B 1 .z/ D has particular, if B.z/ D bz C a bz C a the same form as B. (2) jzj D 1 if and only if jB.z/j D 1. ˇ ˇ ˇb ˇ (3) jB.0/j D ˇˇ ˇˇ < 1. a (4) B.D/ is connected. Thus, from (2), either B.D/ is contained in D or B.D/\D is empty. From (3) we see that B.D/ D. It follows from (1) that B 1 .D/ D. (5) Obviously D D B ı B 1 .D/, which implies that B.D/ D D. The only if part: Let f 2 Aut.D/ and w D f .z/. Then f 1 2 Aut.D/ and z D f 1 .w/. (6) If f .0/ D 0, then it follows by the Schwarz’ lemma applied first to f 1 and then to f that ˇ ˇ jzj D ˇf 1 .w/ˇ jwj D jf .z/j jzj for all z 2 D: The same lemma implies that there exists a 2 R such that f .z/ D e{ z for all z 2 D. So we can take a D e{ 2 and b D 0 to conclude that f has the required form. zc . (7) If f .0/ D c ¤ 0, then 0 < jcj < 1 and we set C.z/ D 1 cz It follows from Exercise 8.5 (see also Exercise 2.2) that the M¨obius transformation C belongs to Aut.D/. Since C ı f fixes the origin, it follows from (6) that C ıf is of the required form, and therefore so is f D C 1 ı.C ıf / by (1). t u Just as in Sect. 8.1 we defined PSL.2; C/ as the quotient of SL.2; C/ by ˙I and then proved that it is isomorphic to the group Aut.b C/, we can define the group PSL.2; R/ D SL.2; R/=f˙I g of appropriate matrices with real coefficients modulo plus or minus the identity matrix and obtain the following description: Theorem 8.19. Aut.H2 / Š PSL.2; R/. Proof. Consider the conformal map w W H2 ! D given in Corollary 8.15. Then Aut.H2 / D w1 Aut.D/ w:

8.3 The Riemann Mapping Theorem

207

By the preceding theorem, any element f of Aut.D/ may be written as f .z/ D

az C b bz C a

with jaj2 jbj2 D 1. Denote a D a1 C { a2 , b D b1 C { b2 . Then .w1 ı f ı w/.z/ D

.a1 C b1 /z C b2 a2 .a2 C b2 /z C b1 a1

with .a1 C b1 /.b1 a1 / C .a2 C b2 /.a2 b2 / D jaj2 jbj2 D 1; thus we have 1 2 associated to any element w ı f ı w of Aut.H / the image in PSL.2; R/ of the .a1 C b1 / b2 a2 in SL.2; R/. matrix a2 C b2 b1 a1 ab Conversely, every matrix in SL.2; R/ induces a M¨obius transformation cd az C b given by S.z/ D . Since S preserves the circle b R and cz C d =S.{/ D

ad bc 1 D 2 2 2 c Cd c C d2

is positive, we conclude that S belongs to Aut.H2 /.

t u

8.3 The Riemann Mapping Theorem We now combine the results about M¨obius transformations of the previous two sections with results from Chap. 7 about compact and bounded families of holomorphic functions to show that every simply connected domain D in C, other than C itself, is conformally equivalent to the unit disc; any conformal map from D onto the unit disc D will be called a Riemann map. Recall that a set A is a proper subset of a set B if A B (thus, in particular, A ¤ B). Theorem 8.20 (Riemann Mapping Theorem). Let D be a nonempty proper simply connected open subset of C, and let c 2 D. Then there exists a unique conformal map f W D ! D with f .c/ D 0, f 0 .c/ > 0, and f .D/ D D. Proof. We are looking for a map for the pair .D; c/. The argument has two parts, existence and uniqueness. Existence. We first reduce to a special case.

208

8 Conformal Equivalence and Hyperbolic Geometry

First Reduction: It suffices to assume that D is bounded. Proof. Since D ¤ C, we can choose b 2 C D; since D is simply connected there is a branch g.z/ of log.z b/ on D. Thus eg.z/ D z b for all z 2 D: The function g is injective: for if g.w/ D g.z/, then w b D z b. Furthermore, if d 2 D, then g.z/ g.d / ¤ 2{ for all z 2 D. Otherwise z b D eg.z/ D eg.d /C2{ D eg.d / D d b: Since g.D/ is an open set, we can choose d 2 D and ı > 0 such that jw g.d /j < ı ) w 2 g.D/: Thus jw g.d / 2{j < ı ) w … g.D/; since otherwise w D g.z0 / for some z0 in D, and jw g.d / 2{j D jŒg.z0 / 2{ g.d /j < ı implies that g.z0 / 2{ D g.z1 / for some z1 in D, a contradiction. Now 1 F .z/ D g.z/ g.d / 2{ is a conformal map from D onto F.D/, and the simply connected domain F .D/ is 1 . t u contained in the bounded set cl U 0; ı We are reduced to solving the mapping problem for .F .D/; F .c//, and may thus assume that D is bounded. Second Reduction: We may also assume that c D 0. Proof. The map G W z 7! z c takes the bounded domain D onto its (bounded) translate D 0 . If F is a solution of our problem for .D 0 ; 0/, then F ı G solves the problem for .D; 0/. t u Thus we now assume that D is bounded, simply connected and c D 0 2 D. Proof of the Theorem Under These Assumptions. We define F D ff 2 H.D/I f is either conformal or identically zero; f .0/ D 0; f 0 .0/ 2 R; f 0 .0/ 0 and jf .z/j < 1 for all z 2 Dg:

8.3 The Riemann Mapping Theorem

209

Our first observation is that F is nonempty. Of course, f 0 is in F . This is not good enough for much. Since D is bounded, there exists an M > 0 such that jzj M for all z 2 D:

(8.4)

z Hence, for every a 2 R such that a > M , the function f .z/ D for z in D belongs a to F . Next we show that F is compact using Corollary 7.31.Each element f of F satisfies jjf jjK 1 for all compact sets K D; hence F is bounded in the strong sense. To show that F is closed, let ffn g F be a sequence such that fn ! f uniformly on all compact subsets of D. Then f 2 H.D/, by Theorem 7.2, and since each fn vanishes at 0, so does f . It is now convenient to consider two cases: 1. fn 0 for infinitely many distinct n. In this case f 0 and hence certainly f 2 F. 2. fn 0 for only finitely many n. In this case we may assume that each fn is a conformal map; then fn0 .0/ > 0 for all n, and thus f 0 .0/ 0. Hurwitz’s Theorem 7.9 says that f is either constant (hence identically zero) or univalent (that is, one-to-one). Since jfn .z/j < 1 for all z 2 D, we conclude that jf .z/j 1 for all z 2 D. If jf .z0 /j D 1 for some z0 2 D, then jf j 1 by the maximum modulus principle; this is a contradiction to f .0/ D 0. Thus jf .z/j < 1 for all z 2 D, and we conclude that f 2 F ; thus F is closed, and therefore compact. We now complete the proof of the existence part. If S D ff 0 .0/I f 2 F g; then S R0 . We claim that S is bounded from above. Indeed, choose > 0 so that cl U.0; / D. If ./ D e{ for 0 2 is the circle centered at 0 with radius , then for any f 2 F we have Z 1 f .z/ 0 dz f .0/ D 2{ z2 and thus ˇ 0 ˇ ˇf .0/ˇ 1 2 D 1 2 2 1 2 S for all a > M , a as we saw above. Also, there exists a sequence ffn g F such that If D sup S , then

1 M

> 0, with M as in (8.4), because lim fn0 .0/ D :

n!1

Since F is compact, there exists a convergent subsequence ffnk g with lim fnk D f 2 F . Since f 0 .0/ D , f is a conformal map.

k!1

210

8 Conformal Equivalence and Hyperbolic Geometry

Since f .D/ D, to show that f .D/ D D, we assume for contradiction that f .D/ ¤ D and construct an h 2 F with h0 .0/ > , thus contradicting the fact that D sup S . Namely, if f .D/ ¤ D, then there exists w0 D re{ with 0 < r < 1 such that w0 2 @f .D/. We now construct h as follows: 1. Let g1 .z/ D e{ f .z/. The map g1 is the map f followed by a rotation through the angle ; g1 sends w0 to r. r g1 .z/ 2. Let p.z/ D . The map p is g1 followed by an automorphism of D that 1 rg1 .z/ sends r to 0 (see Exercise 8.5). Note that p.z/ ¤ 0 for all z 2 D, since p.z/ D 0 if and only if g1 .z/ D r if and only if f .z/ D w0 . 1 3. Let q.z/ D p.z/ 2 , where we choose the branch of the square root2 with q.0/ D 1 r 2 > 0. The map q is injective because q.z1 / D q.z2 / if and only if p.z1 / D p.z2 /. Furthermore, jq.z/j < 1 for all z 2 D. 1 r 2 q.z/ . The map g2 is q followed by an automorphism of D 4. Let g2 .z/ D 1 1 r 2 q.z/ 1 that sends r 2 to 0. 5. Let h.z/ D e{ g2 .z/. The map h is g2 followed by a rotation through the angle . Conclusion: h is a univalent mapping of D into D. We calculate h.0/ and h0 .0/. In order to use the chain rule we need to see what happens to zero under all the maps used to construct h. It is easily checked that 1

g1 .0/ D 0; p.0/ D r; q.0/ D r 2 ; g2 .0/ D 0; and h.0/ D 0: ˛z C ˇ Aside: Let ˛; ˇ; ; and ı be complex numbers. If A.z/ D for all z 2 C z C ı ı ˛ı ˇ , then A0 .z/ D . . z C ı/2 The calculation of the derivative of h at zero proceeds as follows: h0 .0/ D e{ g20 .0/ D e{

r 1 1 2

q 0 .0/ D e{

r 1 0 q .0/ .1 r/2

.1 r q.0//2 e{ 1 1 1 1 C r 2 e{ 12 0 D p.0/ p .0/ D r 2 g 0 .0/ r 1 2 r 1 2 .1 rg1 .0//2 1

D

r C1 1 { 1 1 r 2 1 { e e D : 1 2 r 1 r 12 1 2r 2

By Exercise 5.1 there certainly exists a holomorphic function q whose square is p. Hence q is also such a function. These are the two branches of the square root of p.

2

8.4 Hyperbolic Geometry

211

Finally r C1

> 1 if and only if 0 < r < 1; 1 2r 2 arriving at the contradiction h0 .0/ > that finishes the existence proof. Uniqueness. This is a straightforward argument using the Schwarz’s lemma.

t u

Corollary 8.21. If D is a nonempty simply connected domain in b C, then D is conformally equivalent to one and only one of the following domains: (i) b C, (ii) C, or (iii) D. The case (i), (ii), or (iii) occurs when the boundary of D consists of no points, one point, or more than one point, respectively. In the last case the boundary of D contains a continuum (a homeomorphic image of a closed interval containing more than one point). Proof. Existence: If D ¤ b C, we may first reduce to the case D C by observing 1 that if D contains 1, we can choose c 2 C D, and setting F .z/ D we zc have that F .D/ C is a nonempty simply connected domain not containing 1 and conformally equivalent to D. If the result holds for F .D/, then it also holds for D. If D is a proper subset of C, the result follows from the RMT. Uniqueness: We need to prove that no two of the simply connected domains b C, b C, and D are conformally equivalent. But C is compact, and hence cannot be conformally equivalent to either C or D. On the other hand, a conformal map from C onto D would be a nonconstant entire bounded function, a contradiction to Liouville’s theorem. t u

8.4 Hyperbolic Geometry Let D be a simply connected domain in the extended complex plane with two or more boundary points. In this section we establish that such a domain3 carries a conformally invariant metric, known as the Poincar´e or hyperbolic metric. These domains are called hyperbolic; they are all conformally equivalent to the unit disc, by the RMT. We show that conformal equivalences between these domains preserve the hyperbolic metric; that is, they are isometries (distance preserving maps) with respect to the hyperbolic metrics on the respective domains. Endowed with these equivalent metrics, the upper half-plane H2 and the unit disc D become models for non-Euclidean (also known as hyperbolic or Lobachevsky) geometry. As we

The metric may be defined on all domains in b C with two or more boundary points, not necessarily simply connected; to prove this takes us beyond the scope of this book.

3

212

8 Conformal Equivalence and Hyperbolic Geometry

have shown, the groups Aut.H2 / and Aut.D/ of conformal automorphisms of these domains consist of M¨obius transformations, a class of maps much easier to study than the group of conformal automorphisms of an arbitrary D. It is a remarkable fact that these M¨obius functions constitute the full group of orientation-preserving isometries of H2 and D with their respective hyperbolic metrics. We conclude this section using Schwarz’s lemma and the hyperbolic metric to establish a deep connection between complex analysis and geometry. Namely, holomorphic maps between hyperbolic domains are either isometries or contractions with respect to their hyperbolic metrics. We first define the Poincar´e metric in a general setting; that is, on an arbitrary simply connected domain D with two or more boundary points (Sect. 8.4.1). We subsequently study it in more detail on H2 and D, where specific computations are most easily carried out (Sects. 8.4.2 and 8.4.3). The results that follow from these computations transfer to the general setting because of the conformal equivalence established in the RMT. Finally in Sect. 8.4.4 we establish the result about contractions.

8.4.1 The Poincar´e Metric Definition 8.22. Let D be a simply connected domain in the extended complex plane with two or more boundary points. We define the (infinitesimal form of the) Poincar´e metric on D D .z/ jdzj as follows. First, in the unit disc, set D .z/ D

2 1 jzj2

; z 2 D:

(8.5)

Next, for arbitrary D, choose a Riemann map W D ! D and define D by ˇ ˇ D .w/ D D ..w// ˇ 0 .w/ˇ ; w 2 D:

(8.6)

Our first task is to show that D .w/ is well defined for all simply connected domains4 D and all w 2 D. Toward this end, let A be a conformal automorphism of D. Recall that there exist complex numbers a and b with jaj2 jbj2 D 1 such that A.z/ D

4

With two or more boundary points.

az C b bz C a

; z 2 D:

8.4 Hyperbolic Geometry

213

An easy calculation now shows that ˇ ˇ D .A.z// ˇA0 .z/ˇ D D .z/; z 2 D:

(8.7)

Let w0 2 D be arbitrary and suppose that and are two Riemann maps of D onto D, with .w0 / D z0 and .w0 / D t0 for two points z0 and t0 2 D. We need to show that ˇ ˇ ˇ ˇ D .z0 / ˇ 0 .w0 /ˇ D D .t0 / ˇ 0 .w0 /ˇ : But A D ı 1 is in Aut.D/ and A.z0 / D t0 ; it now follows from (8.7) that ˇ ˇ ˇ ˇˇ ˇ D .z0 / ˇ 0 .w0 /ˇ D D .A.z0 // ˇA0 .z0 /ˇ ˇ 0 .w0 /ˇ ˇ ˇˇ ˇ D D .A.z0 // ˇ 0 . 1 .z0 //. 1 /0 .z0 /ˇ ˇ 0 .w0 /ˇ ˇ ˇ D D .t0 / ˇ 0 .w0 /ˇ : Remark 8.23. (1) If w 2 D is arbitrary and we choose the Riemann map to satisfy .w/ D 0, then ˇ ˇ D .w/ D 2 ˇ 0 .w/ˇ : (2) It is easy to see, using (8.6) and Corollary 8.15, that H2 .z/ D

1 for all z 2 H2 : =z

(8.8)

The important invariance property of our metric is described in our next result. Proposition 8.24. For every conformal map f defined on D, ˇ ˇ f .D/ .f .z// ˇf 0 .z/ˇ D D .z/ for all z 2 D: Proof. If is a Riemann map (for D), so is ı f 1 (for f .D/).

t u

Any infinitesimal metric on D allows us to define lengths of paths in D, and hence a distance function on the domain. We work, of course, with the length element ds D D .z/ jdzj : Definition 8.25. We define the hyperbolic length of a piecewise differentiable curve in D by Z lD . / D D .z/ jdzj I

214

8 Conformal Equivalence and Hyperbolic Geometry

and if z1 and z2 are any two points in D, the hyperbolic (or Poincar´e) distance between them by D .z1 ; z2 / D infflD . /I is a pdp in D from z1 to z2 g:

(8.9)

We leave to the reader (Exercise 8.13) the verification that D defines a metric on D. An isometry from one metric space to another is a distance preserving map between them. It follows from Proposition 8.24 that for every conformal map f defined on D and every pdp in D, lf .D/ .f ı / D lD . / and f .D/ .f .z1 /; f .z2 // D D .z1 ; z2 / for all z1 and z2 2 DI that is, is conformally invariant and f is an isometry between D and f .D/ with respect to the appropriate hyperbolic metrics. In particular, every element of Aut.D/ is an isometry for the hyperbolic metric on D.

8.4.2 Upper Half-plane Model jdzj We know from Remark 8.23 that in H2 we have ds D =.z/ . The hyperbolic length 2 of an arbitrary curve in H and the hyperbolic distance between two points in H2 may be hard to calculate directly from their definitions; an indirect approach is technically less complicated. We show that given any two distinct points in H2 , they lie on either a unique Euclidean circle centered on the real axis or on a unique straight line perpendicular to the real axis. The corresponding portion of the circle or straight line lying in H2 is called a hyperbolic line or geodesic; the unique portion of the geodesic between the two points is called a geodesic path or geodesic segment. The name is justified by showing that the hyperbolic length of a geodesic segment realizes the hyperbolic distance between its two end points. A straight line in C is a circle in C [ f1g passing through infinity (see Exercise 3.21). It is not useful, in general, to assign centers to these circles. However, if such a line intersects R in one point and is perpendicular to R at that point, we consider that point to be the center of the circle. In the current context, we shall be interested only in lines perpendicular to R and use the related fact that a Euclidean circle with center on the real axis is perpendicular to the real axis.

Definition 8.26. For a circle C in C [ f1g centered on the real axis, the part of C lying in the upper half plane is called a hyperbolic line or a geodesic in H2 . The reason for the terminology will shortly become clear. The following lemma establishes the existence of a geodesic path between two points; the proof of its uniqueness follows.

8.4 Hyperbolic Geometry

215

O perpendicular to R through two pairs of points in the upper Fig. 8.2 Unique circles (in C) half plane

Lemma 8.27. For every pair z and w of distinct points in H2 , there exists a unique circle centered at the real axis passing through them, and a unique geodesic in H2 passing through them. e to be the Euclidean line through z and w. Otherwise, Proof. If 0, we can choose an r such that ˇB 1 .w/ˇ 1 for all w of absolute value r. Without loss of generality we can choose r jzj. Hence, by the maximum principle for jwj r, it follows that jF .w/j D

1 jf .w/j .B/1 : 1 r 0 .1 / jB .w/j

r 0 .B/1

Since is arbitrary and the inequality holds for all jzj r < 1, the claim is proved. By Schwarz’s lemma we obtain jF .z/j jzj, or, equivalently, ˇ ˇˇ ˇ jf .z/j jzj ˇz 0 .B/1 ˇ ˇB 1 .z/ˇ D jB.z/j for all z 2 D, proving the first inequality in (a). For the inequality about derivatives, note that by the first inequality, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ f .z/ f .ai / ˇ ˇ f .z/ ˇ ˇ B.z/ ˇ ˇ B.z/ B.ai / ˇ ˇ ˇDˇ ˇˇ ˇDˇ ˇ: ˇ ˇ ˇz a ˇ ˇz a ˇ ˇ ˇ z ai z ai i i Taking limits as z approaches ai yields the second inequality. f is analytic on D, BA and its modulus is at most 1 on D. If jf .a/j D jB.a/j for some a 2 D A, then the f modulus of at a is 1; therefore this function is constant and (b) follows. BA To prove (b), note that it follows from (a) that the function

224

8 Conformal Equivalence and Hyperbolic Geometry

To prove (c), we may assume that ai D 0. By l’Hopital’s rule, ˇ ˇ ˇˇ . / ˇˇ ˇf ˇ ˇ f .0/ ˇ ˇ ˇ ˇ B .0/ˇ D ˇˇ . / ˇˇ D 1; A BA .0/ t u

and (8.16) follows.

The particular case when A consists of one point leads to the following interesting result, a generalization of Schwarz’s lemma that the reader may have encountered before as an exercise. Corollary 8.48 (Schwarz-Pick Lemma). Let f be a holomorphic self-map of the open unit disc D. If a 2 D, then ˇ ˇ ˇ ˇ ˇ f .z/ f .a/ ˇ ˇ z a ˇ ˇ ˇˇ ˇ ˇ ˇ 1 az ˇ N ˇ 1 f .a/f .z/

ˇ ˇ 1 jf .a/j2 for all z 2 D and ˇf 0 .a/ˇ : 1 jaj2

Furthermore, equality for one z 6D a in the first inequality or equality in the second inequality implies that f is an automorphism of D. Proof. Apply the previous theorem to the function Bf .a/ ı f .

t u

This corollary may also be deduced by applying Schwarz’s lemma to the function Bf .a/ ıf ıBa1 (see Exercise 5.4) or, more interesting, is equivalent to Theorem 8.43 (Exercise 8.22).

Exercises

ab 8.1. A matrix A D with a; b; c; d 2 C, ad bc ¤ 0, acts on b C by cd azCb . Show that for each t 2 C6D0 , A and tA induce the same action. z 7! czCd C are permuted. What effect 8.2. Suppose the four distinct points z1 ; z2 ; z3 ; z4 in b will this have on the cross ratio .z1 ; z2 ; z3 ; z4 /? 8.3. Prove in detail that the angles at z1 and z2 in Fig. 8.1 are equal precisely when the four points lie on a circle in b C. 8.4. In the text we established, using a geometric argument, that the cross ratio of four distinct points fz1 ; z2 ; z3 ; z4 g on the Riemann sphere C [ f1g is real if and only if the four points lie on a circle. This exercise presents an alternate, purely analytic, way of establishing this result. 1. Show that every M¨obius transformation maps the extended real line R [ f1g onto a circle in C [ f1g.

Exercises

225

2. Use the last assertion to show that .z1 ; z2 ; z3 ; z4 / is real if and only if the four distinct points fz1 ; z2 ; z3 ; z4 g lie on a circle in C [ f1g. 8.5. Show that f 2 Aut.D/ if and only if there exist in R and a 2 D such that f .z/ D e{

za for all z 2 D: 1az

8.6. This exercise explores the transitivity properties of groups of M¨obius transformations and provides alternate formulations and proofs for many of the properties of M¨obius transformations developed in this chapter by other methods. 1. We begin with the study of PSL.2; C/ viewed as the group of conformal automorphisms of b C, and let a; b; c be three distinct points in b C. • Assume that all the points are finite. Show that the M¨obius transformation A.z/ D

c.a b/z C a.b c/ ; z2b C; .a b/z C .b c/

maps the ordered triple 0; 1; 1 in b C to the ordered triple a; b; c in C. • Show that by appropriate limiting processes applied to the formula for A, we can extend the above result to ordered triples of distinct points a; b; c in b C. What are the resulting formulae? • Show by direct computation that the identity is the only M¨obius transformation that fixes 0, 1, and 1. • Determine the analogues of the above formulae for mappings of an arbitrary triple in b C to another such triple. Thus establishing once again the unique triple transitivity of PSL.2; C/ which can be summarized as: given two ordered triples of distinct points .a0 ; b0 ; c0 / and .a; b; c/ in b C, there exists a unique A 2 PSL.2; C/ mapping the first ordered triple to the second one. • We now allow appropriate repetitions. Say our triples are .a0 ; a0 ; c0 / and .a; b; c/. There are certain limitations on b that should be described. For a0 and a finite, (in this case b 6D 0) the requirement for A should be that A.a0 / D a, A0 .a0 / D b, and A.c0 / D c, of course. Show that there exists a unique such A and determine its formula. Extend the discussion to allow a0 to be infinite. • Extend the previous analysis to triples .a0 ; a0 ; a0 / and .a; b; c/. 2. We can view PSL.2; R/ as a group of automorphisms of b R. Formulate and establish the analogues of the previous part to this setting. 3. More relevant for our work is to study PSL.2; R/ as a group of automorphisms R. Toward this aim, let a0 and a 2 H2 , and b0 and b 2 b R. of H2 [ b • Show that there is a unique A 2 PSL.2; R/ mapping the ordered pair .a0 ; b0 / to the ordered pair .a; b/. What are the appropriate formulae for A?

226

8 Conformal Equivalence and Hyperbolic Geometry

• Let .a0 ; b0 / and .a; b/ be pairs of distinct points in H2 . Show that there is a unique A 2 PSL.2; R/ mapping the ordered pair .a0 ; b0 / to the ordered pair .a; b/ iff H2 .a0 ; b0 / D H2 .a; b/: Determine the formula for A. • Extend the last claim to allow repetitions. 4. Translate the previous set of exercises on automorphisms of H2 to automorphisms of D. 8.7. Formulate and prove (as a consequence of Theorem 8.20) the RMT for appropriate simply connected domains D b C. Include the possibility that c D 1. 8.8. This exercise deals with the construction of specific Riemann maps. 1. Find a conformal map from the disc fz 2 CI jzj < 1g onto fz 2 CI jzj < 1; =z > 0g. 2. Let D be the domain in the extended complex plane b C exterior to the circles jz 1j D 1 and jz C 1j D 1. Find a Riemann map of D onto the strip S D fz 2 CI 0 < =z < 2g. 3. Find a conformal map from the domain in b C defined by fz 2 CI jz 1j > 1; jz C 1j > 1g [ f1g onto the upper half plane. 4. For each n D 1; 2; 3; : : :, find a conformal map from the infinite angular sector 0 < Arg z < onto the unit disc. n 5. Find { the Riemann0 map f from the strip 0 < =z < 1 onto the unit disc satisfying f 2 D 0 and f 2{ > 0. 6. Find a conformal map from the domain fz 2 CI jzj < 1 and z ¤ t for 0 t < 1g onto fw 2 CI jwj < 1g. 7. Find a conformal map from the upper half plane onto the unit disc minus the nonnegative real numbers. 8.9. If f .z/ D w is a Riemann map from the domain jArg zj < jwj < 1 and if f .1/ D 0 and f 0 .1/ > 0, find f .2/.

100

onto the domain

8.10. If w D g.z/ maps the quadrant fz D x C {y 2 CI x > 0; y > 0g conformally onto jwj < 1 with g.1/ D 1, g.{/ D 1 and g.0/ D {, find jg 0 .1 C {/j. 8.11. If f is holomorphic for jzj < 1 and satisfies jf .z/j < 1 for jzj < 1 and f .0/ D f 12 D 0, show that

Exercises

227

ˇ ˇ ˇ 2z 1 ˇ ˇ ˇ jf .z/j ˇz 2z ˇ

for all jzj < 1:

8.12. Suppose ffn g is a sequence of holomorphic functions in jzj < 1 that satisfy 0 and jfn .0/ {j

0 and all jzj < 1:

Show that ffn g contains a subsequence that converges uniformly on compact subsets of the unit disc. 8.13. Supply the details of the proof that H2 defines a metric in H2 that is invariant under the M¨obius group PSL.2; R/. 8.14. Let z and w be points in H2 with z and w the corresponding points as in Definition 8.31; set D.z; w/ D j log.z ; w ; w; z/j. Show that D defines a metric in the upper half plane. 8.15. Let z and w be points in the upper half plane. Prove that D.z; w/ D jz wj C jz wj defines a metric in the upper half plane. log jz wj jz wj 8.16. Prove that D .0; z/ D log

1 C jzj for all z 2 D: 1 jzj

(8.17)

8.17. Show that for all z1 D x1 C { y1 and z2 D x2 C { y2 2 H2 , H2 .z1 ; z2 / D arccosh

.x2 x1 /2 C .y2 y1 /2 : 1C 2y1 y2

8.18. This exercise establishes some of the basic facts about geodesics in H2 with respect to the hyperbolic metric in slightly different form than developed in the text. It should be established using the formula (8.8) for the infinitesimal form for the metric, but ignoring all facts about geodesics that we already established. 1. Show that {R>0 is a geodesic in H2 . 2. Let be a pdp in H2 . Prove that is a geodesic in H2 iff for all c and d 2 range , the length of the piece of between c and d equals H2 .c; d /. 3. Show that an arbitrary geodesic in H2 is a semi-circle (this includes infinite line segments) perpendicular to R. 4. Let c and d be two distinct points in H2 . Show that there exists a unique geodesic passing through them. 5. Formulate and prove the corresponding fact about a given single point and a given direction from it.

228

8 Conformal Equivalence and Hyperbolic Geometry

8.19. This exercise contrasts the transitivity of PSL.2; C/ acting on C [ f1g (established in the text) with that of PSL.2; R/ acting on H2 (to be established here). It should be compared with Exercise 8.6. ab 1. Show that A 2 PSL.2; R/ fixes { iff it is of the form ˙ , with a, b 2 R b a and a2 C b 2 D 1. 2. Let 2 R. Show that there exists a unique A 2 PSL.2; R/ with A.{/ D { and 2 arg A0 .{/. 3. Let z and w 2 H2 and 2 R. Show that there exists a unique A 2 PSL.2; R/ with A.z/ D w and 2 arg A0 .z/. 4. Let fa; bg and fz; wg be two pairs of distinct points in H2 . Show that there exists a unique A 2 PSL.2; R/ with A.a/ D z and A.b/ D w iff H2 .a; b/ D H2 .z; w/.

{ 8.20. Assume f is a bounded holomorphic function on the unit disc and f D 2

{ D 0. Show that there exists a bounded holomorphic function G on D f 2 such that z 2{ z C 2{ f .z/ D G.z/; for all z 2 D: 1 C 2{ z 1 2{ z 8.21. Let D b C be a disc. Show that Aut.D/ consists of M¨obius transformations. 8.22. Establish the equivalence of Theorem 8.43 and Corollary 8.48.

Chapter 9

Harmonic Functions

This chapter is devoted to the study of harmonic functions. They are closely connected to holomorphic maps, since these as well as their real and imaginary parts are harmonic. The study of harmonic functions is important in physics and engineering, and there are many results in the theory of harmonic functions that are not directly connected with complex analysis. In the early parts of this chapter we consider that part of the theory of harmonic functions that grows out of the Cauchy theory. Mathematically this is quite pleasing. One of the most important aspects of harmonic functions is that they solve a boundary value problem, known as the Dirichlet problem. An example is the problem of finding a function that is continuous on a closed disc, that assumes prescribed values on the boundary of the disc, and is harmonic in the interior of the disc. An important tool in the solution is the Poisson formula. In the first section we define harmonic functions and the Laplacian of a function. In the second we obtain integral representations for harmonic functions that are analogous to the Cauchy integral formula, including the Poisson formula; in the third we use these integral representations to solve the Dirichlet problem. The third section includes three interpretations of the Poisson formula: a geometric interpretation, a Fourier series interpretation, and a classical one that involves differential forms. In the fourth section we characterize harmonic functions by their mean value property (MVP). The fifth section deals with the reflection principle for holomorphic and real-valued harmonic functions, a simple but useful extension tool. In the second part of this chapter, starting with the sixth section, we study the class of subharmonic functions and use them to give a more complete solution to the Dirichlet problem, introduce the Green’s function, and provide an alternate proof of the Riemann mapping theorem (RMT). The first two sections in this part introduce the new class of functions and study their basic properties. Perron families of subharmonic functions are studied in the next section, followed by a section in which the Dirichlet problem is revisited. The final section is devoted to the main aim of the second part of this chapter—existence proofs for Green’s function and an alternate proof of the RMT. R.E. Rodr´ıguez et al., Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics 245, DOI 10.1007/978-1-4419-7323-8 9, © Springer Science+Business Media New York 2013

229

230

9 Harmonic Functions

9.1 Harmonic Functions and the Laplacian We begin with Definition 9.1. Let D be a domain in C and g 2 C2 .D/. We define g, the Laplacian of g, by g D

@2 g @2 g C 2I 2 @x @y

(Laplace)

is also called the Laplacian or the Laplace operator. If D is a domain in C and g 2 C2 .D/, we say that g is harmonic in D if it satisfies Laplace’s equation g D 0 in D. Remark 9.2. The analogous of harmonic functions in one real variable are linear and constant functions. The last two definitions have a number of immediate consequences: (1) It is obvious from the definition of the Laplacian as a linear operator on C2 complex-valued functions that it preserves real-valued functions. It is useful to have equivalent formulae for it (Exercise 9.1): D

@2 @2 @2 @2 D 4 ; C D 4 @x 2 @y 2 @z@z @z@z

(9.1)

and in polar coordinates .r; / D

1 r2

@ @2 @ r C 2 : r @r @r @

(2) Recall that for f 2 C1 .D/, f is holomorphic on D if and only if

(9.2) @f D 0 in @z

@g is holomorphic. @z In particular, holomorphic functions are harmonic and (2) gives an easy way to construct analytic functions from harmonic ones. f is harmonic if and only if f is. f is harmonic if and only if 0 such that ˇ ˇ jP Œu1 .z/j < for ˇz e{ 0 ˇ < ı. It follows that 2 jP Œu.z/j jP Œu1 .z/j C jP Œu2 .z/j < ˇ ˇ as long as jzj < 1 and ˇz e{ 0 ˇ < ı. This is the required continuity statement.

t u

9.3.1 Geometric Interpretation of the Poisson Formula This interpretation is due to Schwarz; the presentation follows Ahlfors.1

Recipe. To find P Œu.z/ we replace u.e{ /—the value of the function u at the point e{ —by its value u.e{ / at the point e{ on the unit circle opposite to the point to e{ with respect to z (i.e., on the intersection of the line through e{ and z with the unit circle), and average these values over the unit circle (see Fig. 9.2). This conclusion comes from reinterpreting the second formula (9.8) defining P Œu.z/ as follows: fix a point z inside the unit circle and a point e{ on the unit circle. Let e{ be the unique point on the unit circle which also lies on the straight line through z and e{ . High school geometry (similar triangles, see Fig. 9.2) or a calculation (i.e., using the law of cosines) yields ˇ ˇ ˇ ˇˇ ˇ 1 jzj2 D ˇe{ zˇ ˇe{ zˇ :

1

See [1].

238

9 Harmonic Functions

eıθ

*

Fig. 9.2 The similar triangles

Since the three points z, e{ , and e{ lie on a straight line and z is between the e{ z is a negative real number; it follows from this other two points, the ratio { e z observation that

1 jzj2 D .e{ z/ .e{ z/:

(9.9)

To verify this last equality note that the real number ˇ2 e{ z ˇˇ { e{ z ˇ .e{ z/ .e{ z/ z D ˇe ˇ { { e z e z

is negative and has the same absolute value as .e{ z/ .e{ z/. An alternate argument to obtain (9.9) follows. By definition there is a real number t such that e{ D t e{ C .1 t/ zI jzj2 1 it is easy to compute that t D ˇ ˇ . ˇe{ zˇ2 It now follows from e{ z D t .e{ z/ that ˇ ˇ2 .e{ z/ .e{ z/ D .e{ z/ t .e{ z/ D t ˇe{ zˇ D jzj2 1: We now regard as a function of , with z fixed, and differentiate equality (9.9) logarithmically to obtain

9.3 The Dirichlet Problem

239

e{ e{ d D d : e{ z e{ z Hence we see that (because is an increasing function of ) that ˇ ˇ ˇ e{ z ˇ d ˇ ˇ D ˇ { ˇD ˇ e z ˇ d

1 jzj2 ˇ ˇ : ˇe{ zˇ2

We have thus shown that 1 P Œu.z/ D 2

Z

2

1 u.e / d D 2

{

0

Z

2

u.e{ / d: 0

The recipe follows.

9.3.2 Fourier Series Interpretation of the Poisson Formula We again consider the case of the unit disc and proceed to compute the power series expansion of P Œu.z/ at the origin. Noting that P Œu.z/ D

1 2

Z

2 0

1 zz u.e{ / d for jzj < 1; .e{ z/ .e{ z/

we start with an expansion of the Poisson kernel

.e{

1 zz 1 zz D { { z/ .e z/ .1 e z/ .1 e{ z/ X D .1 zz/ e{ n zn e{ m zm n;m0

D .1 zz/

X

e{ .mn/ zn zm

n;m0

D1C

1 X

e{ n zn C

nD1

1 X

e{ m zm

mD1

and since the last two series converge uniformly and absolutely on all compact subsets of the unit disc, we conclude P Œu.z/ D a0 C

1 X nD1

an zn C

1 X mD1

bm zm ;

240

9 Harmonic Functions

where, for n 2 Z0 and m 2 Z>0 , 1 an D 2

Z

2

e 0

{ n

1 u.e / d; bm D 2

Z

2

e{ m u.e{ / d:

{

(9.10)

0

We thus have the following procedure for extending a given continuous function u on the unit circle to a continuous function on the closed unit disc that is harmonic on the interior of the disc. First compute the Fourier series of u: u.e{ / D

1 X

an e{n C

nD0

1 X

bm e{m ;

mD1

where the Fourier coefficients an and bm are given by (9.10). In this series replace e{n by zn , for each n 2 Z0 , and e{m by zm , for each m 2 Z>0 .

9.3.3 Classical Reformulation of the Poisson Formula For the next reformulation, we start with Definition 9.16. If ! D P dx C Q dy is a differential form, we define !, the conjugate differential of !, by

! D Q dx C P dy:

If D is a simply connected domain in C and u 2 C2 .D/ is real-valued, we know that u is harmonic on D if and only if u is the real part of an analytic function f on D. In this case, df D f 0 .z/ dz D .ux {uy /.dx C { dy/ D .ux dx C uy dy/ C { .uy dx C ux dy/ D du C { du is an exact differential on D and du D dv, where v is a harmonic conjugate of u on D. Thus du and du are exact differential forms on D whenever u is a real-valued harmonic function on a simply connected domain D. In what follows we work with cycles rather than curves. The definitions of cycles and cycles homologous to zero can be found in Sect. 5.2. In the general case, for a harmonic (including complex-valued) function u on an arbitrary (not necessarily simply connected) domain D, the form du D ux dx C uy dy is always exact on D, and its conjugate differential du D uy dx C ux dy is closed since .u/yy D uxx . We conclude that

9.3 The Dirichlet Problem

241

Z

du D 0

(9.11)

for all harmonic functions u on D and all cycles in D that are homologous to zero on that domain. We can now turn to the classical reformulation of the Poisson formula. Assume that is a regular curve with equation z D z.t/ (regular means that z0 .t/ 6D 0 for all t). The direction of the tangent line to the curve at z.t/ is determined by the angle ˛ D Arg z0 .t/ and dx D jdzj cos ˛; dy D jdzj sin ˛: The normal line at z.t/, which points to the right of the tangent line, has direction ˇ D ˛ 2 . The normal derivative of u is the directional derivative of u in the direction ˇ: @u D ux cos ˇ C uy sin ˇ D ux sin ˛ uy cos ˛: @n Thus we see that du D

@u jdzj, and (9.11) can be rewritten as @n Z @u jdzj D 0: @n

@u @u D . It is important to realize that if is the circle fjzj D rg, then @n @r We prove an important generalization of (9.11). Theorem 9.17. If u1 and u2 are harmonic functions on D, then u1 du2 u2 du1 is a closed form on D. Proof. To establish this assertion, it involves no loss of generality to assume that the functions are real-valued (see Exercise 9.4), and hence we may also assume (because the issue is local) that each function uj has a single-valued harmonic conjugate vj ; thus u1 du2 u2 du1 D u1 dv2 u2 dv1 D u1 dv2 C v1 du2 d.u2 v1 /: The last expression d.u2 v1 / is, of course, exact, and u1 dv2 C v1 du2 D = ..u1 C { v1 /.du2 C { dv2 // :

242

9 Harmonic Functions

Now u1 C { v1 is an analytic function and du2 C { dv2 is the total differential of an analytic function. By Cauchy’s theorem, their product is a closed form, and hence we have shown that Z u1 du2 u2 du1 D 0

for all cycles which are homologous to zero in D. In classical language the above formula reads Z @u2 @u1 u2 u1 jdzj D 0: @n @n t u Let us take for D the annulus fz 2 CI R1 < jzj < R2 g and apply the above formula to the functions z 7! u1 .z/ D log r (in polar coordinates) and an arbitrary harmonic function u2 D u on D. We take for the cycle C1 C2 where Cj is the circle fjzj D rj g oriented counter clockwise; here R1 < r1 < r2 < R2 . On any @u d. Hence we also see circle fjzj D rg, with R1 < r < R2 , we have du D r @r that Z Z Z Z @u @u log r1 d d r1 u d D log r2 r2 u d @r @r C1 C1 C2 C2 Z

or

r

log r jzjDr

@u d @r

Z u d D B jzjDr

is a constant (independent of r). Applying the same argument to the functions u1 D 1 (constant function) and u2 D u, we obtain that Z @u d D A r @r jzjDr is constant over the annulus D, and hence is equal to zero (let r ! 0) if u is harmonic in the disc fz 2 CI jzj < R2 g. Thus, for a function u harmonic in an annulus, the arithmetic mean over concentric circles jzj D r is a linear function of log r 1 2

Z u d D A log r C B I jzjDr

if u is harmonic in the disc jzj < R2 or bounded in the punctured disc 0 < jzj < R2 , then A D 0 and the arithmetic mean is constant. In the latter case, if u is harmonic in the disc, then B D u.0/ by continuity (the reader should know other proofs of this fact.) The fact that A may be 0 in the above discussion has consequences:

9.4 The Mean Value Property: A Characterization of Harmonicity

243

Theorem 9.18. If u is a bounded harmonic function on the punctured disc 0 < jz aj < R, then (a) u extends to be harmonic on the disc jz aj < R (b) For real-valued u, the extension has a harmonic conjugate. We leave the proof to the reader, see Exercise 9.17. Changing the origin, we see that if u is harmonic in U.z0 ; R/, then for 0 < r < R, u.z0 / D

1 2

Z

2

u.z0 C r e{ / dI

(9.12)

0

this is the MVP for harmonic functions, that was already established in Corollary 9.7 as a consequence of the fact that real-valued harmonic functions are locally real parts of analytic functions. From it one also obtains the area MVP u.z0 / D

1 2{r 2

“ u.z/ dz dz:

(9.13)

jzz0 jr

Remark 9.19. If u W S 1 ! S 1 is a homeomorphism, then P Œu is also a homeomorphism, from fzI jzj < 1g onto itself. This useful observation is not at all obvious, and not established here.

9.4 The Mean Value Property: A Characterization of Harmonicity Harmonic functions satisfy the MVP, as we have seen in Corollary 9.7. As a matter of fact this property characterizes harmonic functions. The proof below is based on the solution to the Dirichlet problem. Theorem 9.20. A continuous complex-valued function that satisfies the MVP is harmonic. Proof. Let f be a continuous function on a domain D, let c 2 D and let r0 > 0 be sufficiently small so that cl U.c; r0 / D and f satisfies (5.7) for all r r0 . It suffices to assume that f is real-valued. Let v be the continuous function on fjz cj r0 g that is harmonic on fjz cj < r0 g and agrees with f on fjz cj D r0 g. Then f v has the MVP in fjz cj < r0 g, and thus attains its maximum and minimum on fjz cj D r0 g. Since f D v on fjz cj D r0 g, we conclude that f D v on fjz cj r0 g and thus that f is harmonic there. t u

244

9 Harmonic Functions

9.5 The Reflection Principle We start with the simplest form of the general principle we are to establish. Let be a nonempty region in the complex plane which is symmetric about the real axis; that is, zN 2 if and only if z 2 (see Fig. 9.3). Such a region must intersect the real axis nontrivially, and it is a disjoint union of three sets: D C [ [ ; where C D fz 2 I =z > 0g; D \ R; and D fz 2 I =z < 0g: Remark 9.21. A function z 7! f .z/ on a symmetric region is harmonic (analytic) if and only if the function z 7! f .Nz/ is (see Exercise 9.2). We concentrate on the holomorphic case. Assume that f 2 H. / and f is real on at least one segment of ; then f .z/ D f .Nz/ for all z 2 . Proof. The function z 7! g.z/ D f .z/ f .Nz/ is analytic on and vanishes on a subset of with a limit point in (namely, on ); g is thus identically zero on . t u The same conclusion holds if we merely assume that f 2 C. C [ /, is analytic on C and real on , since in this case the extension of f to defined by f .z/ D f .Nz/ for z 2 satisfies the previous hypothesis. We now strengthen this statement considerably. Theorem 9.22. Let be a nonempty region in the complex plane that is symmetric about the real axis. If v is a real-valued and continuous function on C [ , and it is harmonic on C and zero on , then v has a harmonic extension to that satisfies the symmetry condition v.z/ D v.Nz/.

Ω+ σ Ω−

Fig. 9.3 A symmetric region

9.6 Subharmonic Functions

245

Moreover, if v is the imaginary part of an analytic function f in H. C /, then f has an analytic extension to that satisfies the symmetry condition f .z/ D f .Nz/. Proof. We use the symmetry to extend v to all of . We show that the resulting extension (also called v) is continuous on , harmonic on C [ , and vanishes on . Harmonicity of v in is a local property; therefore we only need to show that v is harmonic in a neighborhood of each point x 2 . For this, consider an open disc D with center at x whose closure is contained in . Let V be the unique function that is continuous on cl D, harmonic on its interior, and agrees with v on the boundary of D. Since v restricted to @D satisfies the symmetry condition v.z/ D v.Nz/, so does the function V (on cl D). Hence V vanishes on cl D C \ R. The function V v is continuous on cl D C , harmonic on D C , and vanishes on its boundary; hence it is identically zero on cl D C . Similarly, V v D 0 on cl D . We conclude that V D v on D, and we have shown that v is harmonic on . The function f has a symmetric extension to C [ (that satisfies f .z/ D f .Nz/). We only know that its imaginary part can be extended to all of (and that the extension vanishes on ). We must use information on the imaginary part of f to draw conclusions about its real part. Again, the problem is local, and we work with the disc D defined above. The real-valued harmonic function v on D has a harmonic conjugate u on this disc. The fact that harmonic conjugates are unique up to addition of real constants allows us to normalize so that u D 0 such that cl U.z0 ; r0 / D, u satisfies equality (9.12). Subharmonic functions are similarly characterized: If u is real-valued and continuous in D, then u is subharmonic in D if and only if for all z0 2 D and all r0 > 0 such that cl U.z0 ; r0 / D, u satisfies the mean value inequality u.z0 /

1 2

Z

2

u.z0 C r e{ / d for all 0 r < r0 :

(9.15)

0

Proof. First assume u is subharmonic in D, and consider z0 and r0 as above. Let ' denote the function (solution to the Dirichlet problem) that is continuous in cl U.z0 ; r0 /, harmonic in U.z0 ; r0 /, and coincides with u on @U.z0 ; r0 /. Then u.z/ '.z/ for all z with jz z0 j < r0 . In particular, u.z0 / '.z0 /, and '.z0 / is precisely the RHS of (9.15) by the MVP for harmonic functions. We have established the “only if ” part of the claim. To establish the “if ” part, assume u is real-valued and continuous in D, and for all z0 and r0 as above, u satisfies (9.15). Let G be a bounded subdomain of D such that @G D, and let ' denote a continuous real-valued function on G [ @G such that ' is harmonic in G, and u.z/ '.z/ for all z in @G. We need to show that u.z/ '.z/ for all z in G. If this were not so, then the function v D u ', being continuous in cl G, would attain a positive maximum M at some point in G, and the set S D fz 2 cl G W v.z/ D M g would be a nonempty closed set contained in G. Since both S and @G are compact, there exists a point c in S minimizing the distance from S to @G. Furthermore, on every circumference fjz cj D rg, for small positive values of r, there would exist an arc ˛r such that v < M on ˛r . Therefore

9.6 Subharmonic Functions

1 2

Z

247

2

u.c C r e{ / d '.c/ D 0

1 2

Z

2

v.c C r e{ / d 0

< M D u.c/ '.c/I from this it follows that 1 2

Z

2

u.c C r e{ / d < u.c/ 0

for all small positive values of r, a contradiction.

t u

3. It follows from the mean value inequality (9.15) that a subharmonic function satisfies the maximum principle. It also follows from 2 that subharmonicity is a local property. 4. Let u be a subharmonic function in D with G and ' as in Definition 9.23. Condition (ii) of the definition can now be strengthened to: (ii)’ If u.z/ '.z/ for all z in @G, then either u.z/ D '.z/ or u.z/ < '.z/ for all z in G: For suppose there exists c in G such that u.c/ D '.c/. Then '.c/ D u.c/

1 2

1 2

Z

2

u.c C r e{ / d Z

0 2

'.c C r e{ / d D '.c/ 0

for all small positive values of r, and hence u D ' near c. Thus fz 2 G W u.z/ D '.z/g is an open set, that we are assuming to be nonempty. On the other hand, u and ' are continuous functions on cl G, and hence fz 2 G W u.z/ < '.z/g is an open set, which must then be empty. 5. Let D be a domain in C. If fun gn2N is a sequence of subharmonic functions on D that converges to a function u on D uniformly on all compact subsets of D, then u is subharmonic in D. We leave the proof as an exercise for the reader. 6. If u1 and u2 are subharmonic functions in D, then the function u D maxfu1 ; u2 g is also subharmonic in D. Proof. It is clear that u is continuous in D. Let c 2 D. Without loss of generality we assume u.c/ D u1 .c/. Then

248

9 Harmonic Functions

u.c/ D u1 .c/

1 2

1 2

Z

2

u1 .c C r e{ / d Z

0 2

u.c C r e{ / d 0

for all small positive values of r, and the result follows.

t u

This result provides many examples of subharmonic functions (that are not harmonic): If h is a real-valued harmonic function, then so is h, and therefore hC D maxfh; 0g and jhj D maxfh; hg are subharmonic. In particular, u.z/ D jzj2 is an example of a subharmonic function in C (that is not harmonic). 7. Harmonic functions are of class2 C2 , and they are characterized by the vanishing of their Laplacian. We have seen above that subharmonic functions need not be differentiable, but we now prove that a real-valued function u of class C2 in a domain D is subharmonic if and only if u 0 in D. Proof. First assume that u 2 C2R .D/ and u.z/ > 0 for all z in D. Let G be a bounded subdomain of D such that cl G D, and let ' be a continuous realvalued function in cl G such that ' is harmonic in G, and u.z/ '.z/ for all z in @G. Assume u.z/ — '.z/ for all z in G. Then v D u ' is a continuous function in cl G, and it attains its maximum at a point c in G. But v is of class C2 in G, and hence v.c/ 0; but v D u ' D u and we have obtained a contradiction. Therefore u is subharmonic. In the general case u 2 C2R .D/ and u.z/ 0 for all z in D, set u .z/ D u.z/ C jzj2 for each positive and all z in D. Then u is of class C2 in D, with u > 0 in D, and hence subharmonic in D for each positive . It is also clear that u converges uniformly to u on compact subsets of D as goes to zero, and therefore u is subharmonic in D. For the converse assume u 2 C2R .D/ and u is subharmonic in D. Suppose that there exists c in D such that u.c/ < 0. Then u is superharmonic in a neighborhood of c, and hence harmonic in that neighborhood, from where it follows that u.c/ D 0, a contradiction. t u Example 9.25. The complex plane does not carry any negative nonconstant subharmonic function (Exercise 9.21). Whether a domain does or does not carry such a function is an important distinguishing characteristic. It classifies domains in b C into three types, and serves as the basis for the next definition. Definition 9.26. The Riemann sphere is called elliptic. A proper domain D C [ f1g is called parabolic if it does not carry a nonconstant bounded from above subharmonic function; otherwise it is called hyperbolic. Remark 9.27. It is unfortunate that at times terminology, including our last definitions, can be confusing. 2

They are, of course, much smoother than that, but this suffices for the discussion that follows.

9.7 Perron Families

249

• It is easy to see that the type of a domain is a conformal invariant in the sense that for all domains D and all conformal maps f , D and f .D/ have the same type. In Sect. 8.4 we gave a different definition of hyperbolic simply connected domains. We will see in Theorem 9.47 that both notions coincide for simply connected plane domains. • However, according to the above definitions, the extended complex plane C [ f1g punctured at a positive number n of points is parabolic. Whereas in the classification of domains in relation to what their simply connected holomorphic cover is, such domains are regarded to be hyperbolic if n 3. Example 9.28. Every bounded domain D C is hyperbolic. Choose any c 62 D andˇR > ˇ0 so that D is contained in U.c; R/, and observe that the function z 7! ˇz c ˇ log ˇ ˇ is nonconstant, negative, and harmonic (hence subharmonic) on D. R

9.7 Perron Families To prove that the Dirichlet problem has a solution (in fact, a unique one) for certain domains (in addition to closed discs), we need to consider families of subharmonic functions that are closed under certain operations. We start by establishing some terminology. Definition 9.29. Let D be a domain in C, u W D ! R be a continuous function, and U be an open disc such that cl U D. The continuous function uU defined in D by being harmonic in U and coinciding with u in D U is called the harmonization of u in U . Lemma 9.30. Let D be a domain in C and U be an open disc such that cl U D. If u is subharmonic in D, then so is uU . Proof. It suffices to show that uU satisfies the mean value inequality (9.15) at every point c in @U . Since u is subharmonic in D, u.z/ uU .z/ for all z in D. But then 1 uU .c/ D u.c/ 2

Z

2

1 u.c C r e / d 2

Z

2

uU .c C r e{ / d

{

0

for all small positive values of r, and the result follows.

0

t u

Remark 9.31. We have shown that the family of subharmonic function on a domain D is a cone (i.e., the family is closed under addition and multiplication by positive constants) in the vector space of continuous functions on D. It is also closed under maximization (by property 6 above) and harmonization (by Lemma 9.30). These last two properties are the key to progress. Definition 9.32. Let D be a domain in C. A Perron family F in D is a nonempty collection of subharmonic functions in D such that

250

9 Harmonic Functions

(a) If u; v are in F , then so is maxfu; vg. (b) If u is in F , then so is uU for every disc U with cl U D. The following result, due to Perron, is useful for constructing harmonic functions. Theorem 9.33 (Perron’s Principle). If F is a uniformly bounded from above Perron family in D, then the function defined for z 2 D by V .z/ D supfu.z/ W u 2 F g

(9.16)

is harmonic in D. Proof. First note that by definition a Perron family is never empty. Since we are assuming that there exists a constant M such that u.z/ < M for all z in D and all u in F , the function V is clearly well defined and real-valued. Let U be any disc such that cl U D. It is enough to show that V is harmonic in U . For any point z0 in U , there exists a sequence fuj W j 2 Ng of functions in F such that lim uj .z0 / D V .z0 /: (9.17) j !1

Without loss of generality, we may assume uj C1 uj for all j in N, since if fuj g is any sequence in F satisfying (9.17), then the new sequence given by v1 D u1 and vj C1 D maxfuj C1; vj g for j 1 is also contained in F , satisfies (9.17) (with uj replaced by vj , of course), and is nondecreasing, as needed. The sequence fwj D .uj /U g of harmonizations of the uj in U consists of subharmonic functions with the following properties: 1. wj uj for all j 2. wj wj C1 < M for all j , since the two inequalities clearly hold outside U and on the boundary of U , from which it follows that they also hold in U . Thus the sequence fwj g lies in F , is nondecreasing, and satisfies lim wj .z0 / D j !1

V .z0 /. It follows from the Harnack’s convergence Theorem 9.14 that the function defined by ˆ.z/ D lim wj .z/ D supfwj .z/ W j 2 Ng j !1

is harmonic in U . We will now show that ˆ D V in U . Let c denote any point in U . As before, we can find a nondecreasing sequence fsj g in F such that V .c/ D lim sj .c/. j !1

By setting t1 D maxfs1 ; w1 g and tj C1 D maxfsj C1 ; wj C1 ; tj g for all j 1, we obtain a nondecreasing sequence ftj g in F such that tj wj for all j , and such that lim tj .z/ D V .z/ for z D c and z D z0 .

j !1

9.8 The Dirichlet Problem (Revisited)

251

The harmonizations of the tj in U give a nondecreasing sequence frj D .tj /U g in F satisfying M > rj tj wj for all j . As before, the function defined by ‰.z/ D supfrj .z/ W j 2 Ng is harmonic in U , and coincides with V at c and z0 . But ‰ ˆ, since rj wj for all j , and hence ‰ ˆ is a nonnegative harmonic function in U . Since it is equal to zero at z0 , by the minimum principle for harmonic functions, it is identically zero in U , and the result follows. t u

9.8 The Dirichlet Problem (Revisited) This section has two parts. The first describes a method for obtaining the solution to the Dirichlet problem, provided it is solvable. In the second part, we offer a solution. Recall that the Dirichlet problem for a bounded region D in C and a function f 2 C.@D/ is to find a continuous function U on the closure of D whose restriction to D is harmonic and which agrees with f on the boundary of D. Under these conditions, let F denote the family of all continuous functions u on cl D such that u is subharmonic in D and u f on @D. Then F is a Perron family of functions uniformly bounded from above. Note that the constant function u D minff .z/ W z 2 @Dg belongs to F , hence F is nonempty. The other conditions for F to be a Perron family are also easily verified. Therefore, by Theorem 9.33, the function V defined by (9.16) is harmonic in D. Now, if we assume that there is a solution U to the Dirichlet problem for D and f , then we can show that U D V . Indeed, for each u in F the function u U is subharmonic in D, and satisfies u U D u f 0 on @D, from where it follows that u U 0 in D, and hence V U in D. But U belongs to F , and it follows that U V , and therefore U D V . The Dirichlet problem does not always have a solution. A very simple example is given by considering the domain D D f0 < jzj < 1g and the function f .z/ D

0; 1;

if jzj D 1; if z D 0:

The corresponding function V given by Theorem 9.33 is harmonic in the punctured disc D. If the Dirichlet problem were solvable in our case, then V would extend to a continuous function on jzj 1 that is harmonic in jzj < 1 (see Exercise 9.17). But then the maximum principle would imply that V is identically zero, a contradiction. To solve the Dirichlet problem, we start with a bounded domain D C, with boundary @D, and the following definition.

252

9 Harmonic Functions

Definition 9.34. A function ˇ is a barrier at z0 2 @D, and z0 is a regular point for the Dirichlet problem provided there exists an open neighborhood N of z0 in C such that (1) (2) (3) (4)

ˇ 2 C.cl D \ N /. ˇ is subharmonic in D \ N . ˇ.z/ > 0 for z 6D z0 , ˇ.z0 / D 0. ˇ.z/ D 1 for z 62 N .

Remark 9.35. A few observations are in order. 1. Condition (4) is easily satisfied by adjusting a function ˇ that satisfies the other three conditions for being a barrier. To see this we may assume that N is relatively compact in C, and choose a smaller neighborhood N0 of z0 with cl N0 N . Then let m D minfˇ.z/I z 2 cl.N N0 / \ cl Dg; note that m > 0, and define ˇ1 .z/ D

minfm; ˇ.z/g m

for z 2 N \ D; for z 2 cl.D N /:

ˇ1 , and observe that ˇ2 satisfies all the conditions for being a Finally set ˇ2 D m barrier at z0 . Thus, to prove the existence of a barrier, it suffices to produce a function that satisfies the first three conditions. 2. The existence of barriers is a local property. If a point z0 2 @D can be reached by an analytic arc (a curve that is the image of Œ0; 1 under an injective analytic map defined in a neighborhood of Œ0; 1) with no points in common with cl D fz0 g, then a barrier exists at this point. To establish this we may, without loss of generality, assume that z0 D 0, that the closure of D lies in the right half plane, and that the analytic arc consists of the negative real axis including the origin. 1 Using polar coordinates z D re{ , we see that ˇ.z/ D r 2 cos 2 , < < , satisfies the first three conditions for a barrier function. Definition 9.36. Let D be a nonempty domain in C. A solution u to the Dirichlet problem for f 2 CR .@D/ is proper provided infff .w/I w 2 @Dg u.z/ supff .w/I w 2 @Dg for all z in D. A far-reaching generalization of Schwartz’s Theorem 9.15 is provided by our next result.

9.8 The Dirichlet Problem (Revisited)

253

Theorem 9.37. Let D be a nonempty domain in C. There exists a proper solution to the Dirichlet problem for D for every bounded continuous real-valued function on @D if and only if every point on @D is a regular point for the Dirichlet problem. Proof. We leave it to the reader to discuss the cases where the boundary of D is empty or consists of a single point. So assume from now on that @D has two or more points. First we assume that there exists a proper solution to the Dirichlet problem for D for every bounded continuous real-valued function on @D; we prove that every z0 2 @D is a regular point for the Dirichlet problem. Define a continuous function f on C as follows: f .z/ D

jz z0 j 1

if jz z0 j 1 if jz z0 j 1

and restrict it to @D. Let u be a proper solution to the Dirichlet problem for f 2 CR .@D/. Thus 0 u 1. Since f is not identically zero, u > 0 by the minimum principle and is thus a barrier at z0 . To prove the converse, let f be a bounded continuous real-valued function on @D, and let F consist of all functions v that satisfy the following conditions: • • • •

v is continuous on cl D v is subharmonic on D v satisfies m D inf f v sup f D M in D v.z/ f .z/ for all z 2 @D.

Now observe that the constant function m belongs to F ; this family is obviously closed under maximization and harmonization, and is thus a Perron family, uniformly bounded from above by M . It follows from Theorem 9.33 that the function V defined by (9.16) is harmonic in D and satisfies m V M there. Let z 2 @D. To establish that lim V .w/ D f .z/ we verify that w!z

(a) lim infw!z V .w/ f .z/ (b) lim supw!z V .w/ f .z/ The arguments that follow are generic for this type of problem: for part (a) we construct a particular function w 2 F that helps, while the proof of (b) involves arguing about all v 2 F . Proof of (a): If f .z/ D m, there is nothing to prove. Assume hence that f .z/ > m. Choose a positive real such that f .z/ > m, and note that can be chosen arbitrarily small. There exists a neighborhood N of z such that f ./ f .z/ for all 2 N \ @D. Let ˇ be a barrier at z that is identically equal to 1 outside of N , and define w./ D .f .z/ m /ˇ./ C f .z/

for 2 cl D:

254

9 Harmonic Functions

We show that the function w 2 F . Observe that w 2 CR .cl D/ is subharmonic on D. For 2 cl D, w./ f .z/ < M; and w./ D mˇ./ C .f .z/ /.1 ˇ.// > mˇ./ C m.1 ˇ.// D m: For 2 @D, w./ D m f ./ if 62 N and w./ f .z/ f ./ if 2 N: We have completed the proof that w 2 F . Hence w./ V ./ for all 2 D, and therefore lim inf V ./ w.z/ D f .z/ : !z

Since is arbitrary, (a) follows. Proof of (b): As before, there is nothing to verify under certain conditions: in this case, if f .z/ D M . So we assume that f .z/ < M and choose > 0 such that f .z/ C < M , and a relatively compact neighborhood N of z such that f ./ f .z/ C for 2 N \ @D. Let, as above, ˇ be a barrier at z. Fix an arbitrary v 2 F . We claim that v./ .M f .z/ /ˇ./ f .z/ C

for all 2 N \ D:

(9.18)

Observe that the function on the LHS of (9.18) is subharmonic; hence it suffices to establish the inequality for 2 @.N \ D/; we consider two cases: (i) 2 @N \ cl D, and (ii) 2 cl N \ @D. In case (i), the LHS of (9.18) satisfies v./ .M f .z/ /ˇ./ D v./ M C f .z/ C f .z/ C I while in case (ii) we estimate it by v./ f ./ f .z/ C : Hence, we obtain, for all 2 N \ D, the estimate on v given by v./ f .z/ C C .M f .z/ /ˇ./I since v 2 F is arbitrary, the same estimate holds for the function V . Thus

9.8 The Dirichlet Problem (Revisited)

255

lim sup V ./ f .z/ C : !z

Since > 0 can be chosen to be arbitrarily small, (b) follows.

t u

As a second application of Perron’s method, we establish the following result. Theorem 9.38. Let D be a hyperbolic domain in C and K a compact subset of D with D K connected and @K regular (for the Dirichlet problem). There exists a unique function ! 2 CR ..D K/ [ @K/ such that • • • •

! is harmonic on D K. ! D 1 on @K. 0 < ! < 1 on D K. If !1 2 CR ..D K/ [ @K/ is a competing function that satisfies the above three properties, then !1 !.

Proof. Let 0 be any nonconstant positive superharmonic function on D, whose existence is guaranteed by the hyperbolicity of D. We now adjust this function to obtain a more suitable one. 0 Let m0 be the minimum of 0 on K, and set 1 D . Then 1 is a positive, m0 nonconstant, superharmonic function on D, with 1 jK 1. There clearly exists a point z 2 K such that 1 .z/ D 1. If z is in the interior of K, then 1 D 1 on the connected component of K containing the point z, thus also on the boundary of that component. In particular, there exists a z 2 @K with . z/ D 1. There also exists a point w 2 D K with 1 .w/ < 1, since otherwise 1 1, and the fact that 1 .z/ D 1 would imply that 1 is constant. Setting D minf1; 1 g we obtain a superharmonic function on D that satisfies 0 < 1, .w/ < 1, and jK D 1. Let F consist of all functions v 2 CR ..D K/ [ @K/ satisfying the following conditions: 1. v is subharmonic on D K. 2. v j.D K/. 3. v has compact support. The family we have defined is closed under maximization and harmonization, and thus a Perron family if not empty; we proceed to establish this next. Choose a small disc in D so that its complement in D contains K. Let v0 be the solution to the Dirichlet problem on K with boundary values 1 on @K and 0 on @; then 0 v0 1 on cl. K/. Extending v0 to be 0 on D yields a subharmonic function on D K. By the maximum principle, v0 0 on D K. (This inequality holds on D since v0 j.D / D 0 and > 0, hence also on @. On @K both functions v0 and have the constant value 1; thus .v0 /j. K/ 0.) Defining !.z/ D supfv.z/g for all z 2 .D K/ [ @K; v2F

256

9 Harmonic Functions

we see that • • • • • • •

v0 ! ! is harmonic on D K ! D 1 on @K ! 2 CR ..D K/ [ @K/ ! is not constant, since !.w/ 0 ! 1 on .D K/ [ @K 0 < ! < 1 on D K

.w/ < 1

Finally let ! 0 be a competing function satisfying the above conditions. Let v 2 F and consider the function ! 0 v. Assume that the compact set K 0 D contains the support of v. Since ! 0 0 and v D 0 on @K 0 , ! 0 v 0 on @.K 0 K/, hence also on .K 0 K/ [ .D K 0 / D D K and on @K. Since v 2 F is arbitrary, ! 0 !. t u Definition 9.39. For D and K as above, the function ! described by the theorem is called the harmonic measure of K.

9.9 Green’s Function and RMT Revisited In this section we introduce a new function and use it to give a second proof of the RMT. Definition 9.40. Let D C be a nonempty domain and let z 2 D. A real-valued function g on D fzg is the Green’s function for D with singularity at z if (1) g is harmonic and positive on D fzg. (2) The function w 7! g.w/ C log jw zj is (extends to be) harmonic on D (3) gQ 2 CR .D fzg/ is another function satisfying the above two conditions, then gQ g. Remark 9.41. (a) The last condition in the definition guarantees that the Green’s function is unique if it exists. (b) The entire plane C does not have a Green’s function with singularity at any point. Assume to the contrary that g is a Green’s function for C with singularity at zero (this latter assumption involves no loss of generality). It follows that z 7! g.z/ C log jzj extends to be a real-valued harmonic function u on C. Then u has a (single-valued) harmonic conjugate v, and thus e.uC{ v/ is an entire function. Let R > 0; we estimate for jzj D R ˇ u.z/{ v.z/ ˇ ˇˇ g.z/logjzj ˇˇ 1 1 ˇ D ˇe ˇe D : ˇ R jzj

9.9 Green’s Function and RMT Revisited

257

and conclude that the inequality holds for all jzj R. Since R may be arbitrarily large, we have arrived at a contradiction, the contradiction that our entire function (an exponential!) vanishes identically. (c) The Green’s function for the unit disc D with singularity at 0 is given by z 7! log jzj. For proper simply connected domains in C, we may use the RMT to construct their Green’s functions (see Exercise 9.24). We handle the reverse direction by giving a second proof of the RMT based on the existence of Green’s functions—which we now proceed to prove. The value of the Green’s function for a (hyperbolic) domain D with singularity at c 2 D at any point d 6D c in D will be denoted by gD .d; c/, and by g.d; c/ when the domain is clear from the context. Theorem 9.42. Let D be a nonempty domain in C. There exists a Green’s function on D with singularity at some point c 2 D if and only if D is hyperbolic. In the latter case D has a Green’s function with a singularity at any arbitrary point of D. Proof. Assume that there exists a Green’s function on D with singularity at some point c 2 D. It involves no loss of generality (by Exercise 9.25) to assume that D contains the unit disc and that c D 0. Let g be the Green’s function for D with singularity at 0. Then lim g.z/ D C1. z!0

Let m > 0. Then f D minfg; mg is positive and superharmonic on D f0g, and f is constant (equal to m) near 0; therefore f is positive and superharmonic on D. By choosing m sufficiently large, we can make certain that f is not constant. Thus D is hyperbolic, since it carries the nonconstant negative subharmonic function f . To establish the converse, we define F to consist of all real-valued functions v satisfying the following conditions: 1. v 0 is subharmonic on D f0g. 2. Kv D fw 2 D f0gI v.w/ 6D 0g [ f0g is compact (“roughly expressed,”v has compact support) 3. w 7! v.w/ C log jwj is (extends to be) a subharmonic function on D. To see that F is nonempty we define the function v.w/ D maxf log jwj ; 0g for w 6D 0. It is subharmonic on the plane punctured at the origin, hence certainly on D f0g, and it clearly belongs to the family F . It is quite obvious that this family is closed under maximization and harmonization; hence it is a Perron family. We show next that the family F is uniformly bounded outside every neighborhood of 0. We have not yet used the hypothesis that D is hyperbolic. It is this last assertion that requires hyperbolicity (in the form of existence of harmonic measures, see Theorem 9.38). Let 0 < r < 1 and let !r be the harmonic measure of fjzj rg. Thus !r is harmonic on D fjzj rg, 0 < !r < 1 on this set, and !r .z/ D 1 for jzj D r. Let r D maxf!r .z/I jzj D 1g; hence 0 < r < 1.

258

9 Harmonic Functions

For u 2 F , let ur D maxfu.z/I jzj D rg. We claim that ur !r .z/ u.z/ 0 for all z 2 D such that jzj r:

(9.19)

Hence, in particular, we record for later use that ur r u.z/ 0 for jzj D 1:

(9.20)

To verify (9.19), note that the function ur !r u is superharmonic on D\fjzj > rg and is nonnegative on fjzj D rg. Let K D Ku be the compact support of u. Since u D 0 on @K, ur !r u 0 there. Thus (9.19) holds for K fjzj rg; this inequality certainly holds on D K, and the claim is verified. Next we use the fact that z 7! u.z/ C log jzj is continuous on fjzj 1g and subharmonic on fjzj < 1g. Thus ur C log r D maxfu.z/ W jzj D rg C log r maxfu.z/ W jzj D 1g C log 1 r u r where the last inequality follows from (9.20). We conclude that log r ; ur 1 r and since u D 0 off the compact set K, we see that maxfu.z/I z 2 D and jzj rg

log r : r 1

By Perron’s principle (Theorem 9.33) g.z/ D sup u.z/; z 2 D f0g u2F

defines a nonnegative harmonic function. This function is actually positive since it would otherwise be constant. Finally we must show that z 7! g.z/ C log jzj is harmonic on fjzj < rg. For any such z and for every u 2 F , u.z/ C log jzj ur C log r

r log r : r 1

Thus the same inequality holds when we replace u by g. To complete the proof that g is the Green’s function of D with singularity at 0, we must show that if gQ is a competing candidate, then gQ g (from this it will follow that either gQ > g or gQ D g). Now if u 2 F , then U D gQ u is superharmonic on

9.9 Green’s Function and RMT Revisited

259

D. If K is the support of u, then U 0 on D K, and by the minimum principle for superharmonic functions also on K. Since u 2 F is arbitrary, gQ g. t u Remark 9.43. The above existence proof for Green’s function generalizes to Riemann surfaces. We present next a second, simpler, argument that is valid only for plane domains and also yields additional information. Theorem 9.44. Let D C be a domain with nonempty @D that is regular for the Dirichlet problem. Then D is hyperbolic. In particular, for every c 2 D, let u 2 C.D [ @D/ be a harmonic function in D with u.z/ D log jz cj for z 2 @D. Then gD .z; c/ D u.z/ log jz cj

for all z 2 D fcg:

Proof. It suffices, of course, to prove the particular claim. Without loss of generality D D fz 2 CI jzj < 1g D and c D 0. Let G.z/ D u.z/ log jzj for z 2 D [ @D f0g. Observe that the hypothesis on D guarantees that u exists and that u.z/ log jzj D 0 for z 2 @D. For jzj D 1, u.z/ > 0 and log jzj D 0. It follows that G.z/ 0 for z 2 D D: For 0 < jzj < 1, u.z/ > 0 and log jzj < 0, hence G.z/ > 0. Hence G is a positive harmonic function on D f0g. Obviously z 7! G.z/ C log jzj defines a harmonic function on D. So G is a candidate for the Green’s function of D with singularity at 0. Let gQ be a competitor, and let d 2 D f0g and > 0. Choose a neighborhood N of @D in cl D such that d 62 N and G.z/ < for all z 2 N . Then gQ G on @.D N / D @N , and thus also on D N ; in particular, g.d Q / G.d / . Since d 2 D is arbitrary and so is > 0, gQ G and G D gD .; 0/. t u The proofs of the next two results are left as exercises for the reader. Theorem 9.45. The complex plane punctured at n 0 points is parabolic. Theorem 9.46. Let D be a hyperbolic domain in C. Then g.z; w/ D g.w; z/

for all z 6D w 2 D:

The final application of our work on subharmonic functions is a second proof of Theorem 9.47 (Riemann Mapping Theorem, Version 2). Every hyperbolic simply connected domain in C is conformally equivalent to the unit disc D. Proof. Let c 2 D. We want to use the Green’s function g.; c/ to produce a Riemann Q map f .; c/. If g had a harmonic conjugate g, Q we could define f D e.gC{ g/ . The problem is that g has a singularity at c. To get around this problem, we note that the function u.; c/ defined on D by u.z; c/ D g.z; c/ C log jz cj is harmonic and real-valued. Since D is simply connected, u.; c/ has a harmonic conjugate v.; c/ in D. It follows that F .z; c/ D e.u.z;c/C{v.z;c// defines (for fixed c) a holomorphic function of z on D. Now

260

9 Harmonic Functions

u.z; c/ D g.z; c/ log jz cj D < log.F .z; c// D log jF .z; c/j or, equivalently, g.z; c/ D log j.z c/F .z; c/j : We define f .z; c/ D .z c/F .z; c/: Obviously f .; c/ is a holomorphic function on D that vanishes (has a simple zero) only at c. Since log jf .z; c/j D g.z; c/ < 0, f .z; c/ 2 D for all z 2 D. For any three points c; d; z 2 D, set '.z/ D '.c; d; z/ D

f .d; c/ f .z; c/ 1 f .d; c/f .z; c/

:

(9.21)

Now '.c; d; / is f .; c/ followed by a M¨obius transformation that leaves D invariant, and hence a holomorphic map of D into D, with '.d / D '.c; d; d / D 0. In the above argument, we fixed c and d and regarded ' as a function of z. We need to relate ' to some f ; for these purposes we may regard ' as a function of one of the three variables c, d , and z for fixed values of the other two. However, only for fixed c and d is ' a holomorphic function of z. Thus the emphasis on the variable z in the left-hand side of (9.21). We study the function './ in a neighborhood of z D d . Let n 1 be the order of vanishing of ' at d ; then we have the power series expansion '.z/ D ˛.z d /n 1 C a1 .z d / C a2 .z d /2 C ; with ˛ 6D 0, for all z with jz d j sufficiently small, and if we set 1 v.z/ D log j'.z/j; n we obtain a positive harmonic function on D punctured at the (isolated) zeros of '. We may regard v to be defined on all of D with value C1 at the zeros of '. We proceed to compare v to g.; d / D log jf .; d /j. Let F be the family of functions used to define g.; d /. Let w 2 F and let K be the compact support of w. By enlarging K if necessary, we may assume that none of the zeros of ' lie on @K. Let K0 be the set K from which we delete small discs about each of the finitely many zeros excluding d of ' in the compact set K. We claim that v.z/ w.z/ 0 for all z 2 K. Let us first establish this claim for z 2 @K0 . The boundary of K0 consists of @K and finitely many circles about zeros of '. The circles do not cause any problems since w is bounded on each of them (by a bound independent of the radius of the circle if we a priori specify that the radii of these circles be less than some fixed number) and v ! C1 as we shrink the circles further (which is clearly permissible). Now on the boundary of @K, v 0 and w D 0. Since the superharmonic function v w on K is 0 on @K0 , by the

9.9 Green’s Function and RMT Revisited

261

minimum principle for superharmonic functions, it is also nonnegative on K0 . It follows that v w on D fcg. Hence for z 2 D fcg we have 1 log j'.z/j D v.z/ sup h.z/ D g.z; d / D log jf .z; d /j n h2F or

1

j'.z/j j'.z/j n jf .z; d /j :

(9.22)

Setting z D c (since by continuity the last set of inequalities are also valid at c), we conclude that jf .d; c/j jf .c; d /j, and since c and d are arbitrary, we may interchange them to obtain jf .c; d /j D jf .d; c/j ; an equality that also follows from Theorem 9.46 (which we did not prove and hence are not using). We consider another holomorphic function from D into the closed unit disc. We define '.z/ h.z/ D ; z 2 D: f .z; d / From (9.22), we conclude that jh.z/j 1; since ˇ ˇ ˇ ˇ ˇ ˇ ˇ '.c/ ˇ ˇ f .d; c/ f .c; c/ 1 ˇˇ ˇˇ f .d; c/ ˇˇ ˇDˇ D D 1; jh.c/j D ˇˇ f .c; d / ˇ ˇ 1 f .d; c/f .c; c/ f .c; d / ˇ ˇ f .c; d / ˇ h is constant, and there exists a 2 C with j j D 1 and such that ' D f .; d /, or, equivalently, f .d; c/ f .z; c/ f .z; d / D for all z 2 D: 1 f .d; c/f .z; d / We conclude that f .d; c/ D f .z; c/ iff f .z; d / D 0 iff z D d I that is, f .; c/ is an injective holomorphic map of D into D that takes c to 0. The proof concludes by showing that f .; c/ is surjective, in a manner quite similar to that used in the earlier proof of the RMT. Assume that we do not have surjectivity, and let D f .; c/.D/ D. The domain is simply connected and contains 0. Let t 2 be any point in D . We construct a function h W ! D by the formula q zt 2 {t 1Nt 2 z q h.z/ D ; z 2 : zt 2 1 C { tN 1N 2 t z The square root is defined (and holomorphic) since p is simply connected and zt 2 D 6 0 for z 2 ; we choose the square root so that t 2 D {t. Straightforward 2 1Nt z

262

9 Harmonic Functions

calculations now show that ˇ ˇ 1 C jtj2 > 1; jh.z/j < 1; h.0/ D 0; ˇh0 .0/ˇ D 2 jtj

for all z 2 :

We now define W D h ı f .; c/ and observe that W is a holomorphic function from D into D, with a simple zero at c and no other zeros. Hence log jW j is a competing function for the Green’s function g.; c/. Thus log jW j g.; c/ D log jf .; c/j or jW j jf .; c/j : We thus see that ˇ ˇ ˇ h.z/ ˇ ˇ ˇ ˇ z ˇ1

ˇ ˇ for small positive jzj and ˇh0 .0/ˇ 1I t u

a contradiction.

Remark 9.48. Let D C be a nonempty simply connected domain and let c 2 D. We have introduced two numerical invariants for the pair .D; c/. The first is jf 0 .c/j, where f is any Riemann map from D onto D with f .c/ D 0. The second is the value at c of gD .; c/ C log j. c/j. The two invariants agree.

Exercises 9.1. Prove that the equivalent forms for the Laplacian given in equations (9.1) and (9.2) are correct. 9.2. Show that a function z 7! f .z/ on a symmetric region is harmonic (analytic) if and only if the function z 7! f .Nz/ is. 9.3. Show that if ! D f dz C g dz is a continuous differential form on a domain D, then ! D { .f dz g dz/: 9.4. We have shown that if u1 and u2 are real-valued harmonic functions on D, then u1 du2 u2 du1 is a closed form on D and have asserted that it also holds for complex-valued harmonic functions. Prove this assertion. 9.5. Prove the maximum and minimum principles for real-valued harmonic functions: 1. As a general result for real-valued functions that satisfy the MVP.

Exercises

263

2. As a consequence of Harnack’s inequalities for positive harmonic functions. 9.6. Let K be a compact subset of a domain D C, and let u be a positive harmonic function on D. Show that there exists a constant c 1 that depends only on K and D, but not on u, such that 1 u.z1 / c; c u.z2 / for all z1 and z2 2 K. 9.7. Complete the proof of Theorem 9.14 and show that both possibilities for its conclusion do occur. 9.8. Let u be a continuous real-valued function on a domain D. @2 u @2 u Suppose that the partial derivatives and exist and satisfy Laplace’s @x 2 @y 2 equation u D 0 in D. Show that u is harmonic on D. Hint: Use the notation in the proof of Theorem 9.20. Let c D a C { b in D. Show first that for all > 0, the function F .z/ D u.z/ v.z/ C .x a/2 satisfies the maximum principle in fjz j r0 g. 9.9. Does the area MVP imply harmonicity for continuous functions? 9.10. If u is real-valued and harmonic { on jzj < 1, continuous on jzj 1, and { u.e / D cos 2 C sin 2, find u . 2 9.11. Suppose that u.0/ D 1, where u is harmonic and positive in a neighborhood of fz 2 CI jzj 1g. Prove that 17 u 34 7. 9.12. Let ˛ be a real number. For D e{ with 2 R, let './ D cos C { ˛ sin : Which of the following assertions are true for all ˛ in R? Which are true for some values of ˛? Z './ 1 d is holomorphic for jzj < 1. (a) The function f .z/ D 2{ jjD1 z (b) There exists a function f holomorphic for jzj < 1, continuous for jzj 1 and satisfying f ./ D './ for jj D 1. (c) There exists a function f holomorphic for jzj < 1 such that

245

Graduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA

Advisory Board: Colin Adams, Williams College, Williamstown, MA, USA Alejandro Adem, University of British Columbia, Vancouver, BC, Canada Ruth Charney, Brandeis University, Waltham, MA, USA Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA Roger E. Howe, Yale University, New Haven, CT, USA David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USA Jill Pipher, Brown University, Providence, RI, USA Fadil Santosa, University of Minnesota, Minneapolis, MN, USA Amie Wilkinson, University of Chicago, Chicago, IL, USA

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.

For further volumes: http://www.springer.com/series/136

Rub´ı E. Rodr´ıguez • Irwin Kra • Jane P. Gilman

Complex Analysis In the Spirit of Lipman Bers Second Edition

123

Rub´ı E. Rodr´ıguez Facultad de Matem´aticas Pontificia Universidad Cat´olica de Chile Santiago, Chile

Irwin Kra Department of Mathematics State University of New York at Stony Brook Stony Brook, NY, USA

Jane P. Gilman Department of Mathematics and Computer Science Rutgers University Newark, NJ, USA

ISSN 0072-5285 ISBN 978-1-4419-7322-1 ISBN 978-1-4419-7323-8 (eBook) DOI 10.1007/978-1-4419-7323-8 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012950351 © Springer Science+Business Media New York 2007, 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

For Victor, Eleanor and Bob and to the memory of Mary and Lipman Bers

Preface to Second Edition

The second edition contains significant new material in several new sections. We have also expanded sections from the first edition (improving, we expect, the exposition throughout) and included new figures and exercises for added clarity, as well as, of course, corrected errors and typos in the previous version. Among the most important changes are: • We have expanded and clarified several sections from the first edition. • We have significantly enlarged the exercise sections. Some of the problems are routine, others challenging, and some require knowledge of other subjects usually covered in various first-year graduate courses. The problems are listed in more or less random order as far as their difficulty. • In both editions of this text, we use the approach to integration based on differential forms. In an alternative approach differential forms are a by-product of work on integration of functions motivated by ideas from standard treatments of integral calculus. That is the approach that Bers took in courses that he taught; it is also the approach used by Ahlfors. Of course, either of the two approaches are equally valid and lead to the same major results. In this second edition, we provide an appendix that outlines this alternative path to the main results. • New sections on Perron’s method for solving the Dirichlet problem, Green’s function, an alternative proof of the Riemann mapping theorem, and a description of the divisor of a bounded analytic function on the disc via infinite Blaschke products are included. • We prove the Bers theorem on isomorphisms between rings of holomorphic functions on plane domains. • A section on historical references prepared by Ranjan Roy has been added. In addition, the following items related to our work might be of interest to those reading this volume. • An answer manual for the exercises prepared by Vamsi Pritham Pingali is available, to instructors using the book for a course, from the publisher. • An electronic version of the book is available from the publisher. vii

viii

Preface to Second Edition

• One of the authors (IK) has created and maintains a section about the book on his web site. It contains, among other things, updated information and errata. http://www.math.sunysb.edu/irwin/bookcxinfo.html The other authors may also have information about the book on their web sites. It is our pleasure to thank Ranjan Roy for producing and allowing us to include in this volume his historical note. We are grateful to our colleagues and students who pointed out places for improvement in the first edition and in drafts of the second one. Among them: Bill Abikoff, Robert Burckel, Eduardo Friedman, Bryna Kra, Peter Landweber, Howard Masur, Sudeb Mitra, Lee Mosher, Robert Sczech, and Jacob Sturm. It is still true, of course, that errors and shortcomings may remain in the final version of this edition and these are entirely our responsibility. Spring 2012 New York, NY, USA Santiago, Chile Newark, NJ, USA

Irwin Kra Rub´ı E. Rodr´ıguez Jane P. Gilman

Preface to First Edition

This book presents fundamental material that should be part of the education of every practicing mathematician. This material will also be of interest to computer scientists, physicists and engineers. Because complex analysis has been used by generations of practicing mathematicians working in a number of different fields, the basic results have been developed and redeveloped from a number of different perspectives. We are not wedded to any one viewpoint. Rather we will try to exploit the richness of the development and explain and interpret standard definitions and results using the most convenient tools from analysis, geometry and algebra. Complex analysis has connections and applications to many other subjects in mathematics, both classical and modern, and to other sciences. It is an area where the classical and the modern techniques meet and benefit from each other. We will try to illustrate this in the applications we give. Complex analysis is the study of complex valued functions of a complex variable and its initial task is to extend the concept of differentiability from real valued functions of a real variable to these functions. A complex valued function of a complex variable that is differentiable is termed analytic, and the first part of this book is a study of the many equivalent ways of understanding the concept of analyticity. The equivalent ways of formulating the concept of an analytic function are summarized in what we term the fundamental theorem for functions of a complex variable. In dedicating the first part of this book to the very precise goal of stating and proving the fundamental theorem we follow a path in the tradition of Lipman Bers from whom we learned the subject. In the second part of the text we then proceed to the leisurely exploration of interesting consequences and applications of the fundamental theorem. We are grateful to Lipman Bers for introducing us to the beauty of the subject. The book is an outgrowth of notes from Bers’s original lectures. Versions of these notes have been used by us at our respective home institutions, some for more than 20 years, as well as by others at various universities. We are grateful to many colleagues and students who read and commented on these notes. Our interaction

ix

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Preface to First Edition

with them helped shape this book. We tried to follow all useful advice and correct, of course, any mistakes or shortcomings they identified. Those that remain are entirely our responsibility. Newark, NJ, USA New York, NY, USA Santiago, Chile

Jane P. Gilman Irwin Kra Rub´ı E. Rodr´ıguez

Acknowledgement

The first author was supported in part by Fondecyt Grant # 1100767. The third author was supported in part by grants from the National Security Agency, from the Rutgers University Research Foundation, and from Yale University while a visiting Fellow.

xi

Contents

1

The Fundamental Theorem in Complex Function Theory . . . . . . . . . . . . 1.1 Some Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Where Do Series Converge? . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 A Problem on Partitions . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.3 Evaluation of Definite Real Integrals . .. . . . . . . . . . . . . . . . . . . . 1.2 The Fundamental Theorem of Complex Function Theory . . . . . . . . . 1.3 The Plan for the Proof.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Outline of Text.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Appendix: Historical Notes by Ranjan Roy . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 3 5 6 6

2

Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction and Preliminaries.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 Properties of Conjugation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Properties of Absolute Value .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.3 Linear Representation of C . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.4 Additional Properties of Absolute Value . . . . . . . . . . . . . . . . . . 2.1.5 Lines, Circles, and Half Planes . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.6 Polar Coordinates .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.7 Coordinates on C . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 More Preliminaries that Rely on Topology, Metrics, and Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Differentiability and Holomorphic Mappings . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Convention .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 The Cauchy-Riemann (CR) Equations . . . . . . . . . . . . . . . . . . . .

15 15 17 18 18 18 19 21 22

Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Complex Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Properties of Limits Superior and Inferior . . . . . . . . . . . . . . . . 3.1.2 The Radius of Convergence .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 More on Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

39 40 45 46 47

3

23 28 30 30

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3.3

The Exponential Function, the Logarithm Function, and Some Complex Trigonometric Functions . .. . . . . . . . . . . . . . . . . . . . 3.3.1 The Exponential Function .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 The Complex Trigonometric Functions . . . . . . . . . . . . . . . . . . . 3.3.3 The Definition of and the Logarithm Function . . . . . . . . . An Identity Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Zeros and Poles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

53 53 56 57 62 67

4

The Cauchy Theory: A Fundamental Theorem . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Line Integrals and Differential Forms . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Reparameterization . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 Subdivision of Interval . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.3 The Line Integral . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 The Precise Difference Between Closed and Exact Forms .. . . . . . . . 4.2.1 Caution .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Existence and Uniqueness.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Integration of Closed Forms and the Winding Number .. . . . . . . . . . . 4.4 Homotopy and Simple Connectivity . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 More on the Winding Number .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Cauchy Theory: Initial Version . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Appendix I: The Exterior Differential Calculus . . . . . . . . . . . . . . . . . . . . 4.8 Appendix II: An Alternative Approach to the Cauchy Theory .. . . . 4.8.1 Integration of Functions .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8.2 The Key Theorem . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

81 82 84 84 85 88 94 94 95 97 100 103 106 107 108 109

5

The Cauchy Theory: Key Consequences . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Consequences of the Cauchy Theory .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Cycles and Homology . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Jordan Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 The Mean Value Property .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Appendix: Cauchy’s Integral Formula for Smooth Functions.. . . . .

119 119 126 129 131 134

6

Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Functions Holomorphic on An Annulus . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Isolated Singularities .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Zeros and Poles of Meromorphic Functions . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Local Properties of Holomorphic Maps . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Evaluation of Definite Integrals . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 Appendix: Cauchy Principal Value .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

139 139 143 147 149 153 156 162

Sequences and Series of Holomorphic Functions . . .. . . . . . . . . . . . . . . . . . . . 7.1 Consequences of Uniform Convergence . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 A Metric on C.D/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.1 Properties of d . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.2 Properties of . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

171 171 174 175 176

3.4 3.5

7

Contents

xv

7.3 7.4 7.5

The Cotangent Function . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Compact Sets in H.D/ . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Runge’s Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5.1 Preliminaries for the Proof of Runge’s Theorem.. . . . . . . . . 7.5.2 Proof of Runge’s Theorem . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5.3 Two Major Lemmas . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1 7.5.4 Approximating . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . zc Conformal Equivalence and Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . 8.1 Fractional Linear (M¨obius) Transformations . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 Fixed Points of M¨obius Transformations .. . . . . . . . . . . . . . . . . 8.1.2 Cross Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Aut.D/ for D D b C, C, D, and H2 . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 The Riemann Mapping Theorem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.1 The Poincar´e Metric .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.2 Upper Half-plane Model . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.3 Unit Disc Model . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.4 Contractions and the Schwarz’s Lemma.. . . . . . . . . . . . . . . . . . 8.5 Finite Blaschke Products .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

179 183 187 189 190 191

Harmonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Harmonic Functions and the Laplacian . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Integral Representation of Harmonic Functions . . . . . . . . . . . . . . . . . . . . 9.3 The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3.1 Geometric Interpretation of the Poisson Formula .. . . . . . . . 9.3.2 Fourier Series Interpretation of the Poisson Formula .. . . . 9.3.3 Classical Reformulation of the Poisson Formula . . . . . . . . . 9.4 The Mean Value Property: A Characterization of Harmonicity .. . . 9.5 The Reflection Principle . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Subharmonic Functions .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.7 Perron Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8 The Dirichlet Problem (Revisited) . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.9 Green’s Function and RMT Revisited . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

229 230 232 235 237 239 240 243 244 245 249 251 256

10 Zeros of Holomorphic Functions . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Infinite Products .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Holomorphic Functions with Prescribed Zeros .. . . . . . . . . . . . . . . . . . . . 10.3 The Ring H.D/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Euler’s -Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.1 Basic Properties.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.2 Estimates for .z/ . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4.3 The Formulae for the Function.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.5 Divisors and the Field of Meromorphic Functions . . . . . . . . . . . . . . . . . 10.6 Infinite Blaschke Products . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

267 268 272 276 280 280 284 287 289 290

8

9

193 199 200 202 202 205 207 211 212 214 218 219 221

xvi

Contents

Bibliographical Notes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 297 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 299 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 301

Standard Notation and Commonly Used Symbols

Standard Terminology

Term

Meaning

LHS RHS Deleted neighborhood of z CR d D MMP MVP CkR or C .D or Œa; b/

Left-hand side Right-hand side Neighborhood with z removed Cauchy–Riemann equations Proper subset Subset, may not be proper Euclidean distance on C Hyperbolic distance on D Maximum modulus property Mean value property k-times differentiable real- (or complex-) valued functions on the domain D (or the interval Œa; b/ If and only if The complement of B in A fa 2 AI a … Bg Piecewise differentiable path Integration over the path .t / D z0 C Re2{t , 0 t 1 The empty set

iff AB pdp R jz0 jDR

;

xvii

xviii

Standard Notation and Commonly Used Symbols

A List of Symbols

Term Z Q R C b C { {R 0 into finitely many (more than one) infinite arithmetic progressions with distinct differences? The answer is no. It is obviously possible to construct such partitions if some differences are allowed to be equal. So assume that the differences are different and to the contrary that Z>0 D S1 [ S2 [ [ Sn ; where n 2 Z>1 , and for 1 i n, Si is an arithmetic progression with initial term ai and difference di , for 1 i < j n, Si \ Sj D ;, and 1 < d1 < d2 < < dn . Then 1 X X X X zi D zi C zi C C zi ; i D1

i 2S1

i 2S2

i 2Sn

and each series converges for jzj < 1. Summing the above geometric series we see that za1 z za2 zan D C C C for all z with jzj < 1: 1z 1 zd1 1 zd2 1 zdn

(1.1)

Choose a sequence of complex numbers2 fzk g of absolute value less than 1 with 2{ lim zk D e dn . Then

k!1

2{

e dn zk lim D 2{ k!1 1 zk 1 e dn

1 In the subsequent parts of this text (except the historical remarks) we usually use z, w, and c, among others, but not x or y (which usually denote the real and imaginary part of z) to denote a complex variable. 2 Notation for the polar form of a complex number is established in Chap. 2.

1.2 The Fundamental Theorem of Complex Function Theory

3

and

lim

k!1

zaki 1 zdki

D

e

2{ai dn

1e

for i D 1; 2; : : : ; n 1;

2{di dn

(all these quantities are finite) while lim

k!1

contradiction to (1.1).

zakn 1 zdkn

does not exist. This is an obvious

1.1.3 Evaluation of Definite Real Integrals Some integrals are difficult, perhaps impossible, to evaluate using methods usually studied in undergraduate calculus courses. Examples are Z

1 1

dx and 1 C x4

Z

2 0

d : 2 C sin

Using the residue theorem (see Sect. 6.6), it is quite easy to evaluate them.

1.2 The Fundamental Theorem of Complex Function Theory Theorem 1.1. Let D C denote a domain (an open connected set), and let f D u C { v W D ! C be a complex-valued function defined on D. The following conditions are equivalent (here u and v are real-valued functions of the complex variable z D x C {y, with x and y real): (1) The complex derivative f 0 .z/ exists for all z 2 D

.Riemann/I

that is, the function f is holomorphic on D. (2) The functions u and v are continuously differentiable and satisfy @v @u @v @u D and D : @x @y @y @x

(Cauchy–Riemann: CR)

Alternatively, the function f is continuously differentiable and satisfies @f D 0: @z

(CR–complex form)

4

1 The Fundamental Theorem in Complex Function Theory

e of D there exists a holomorphic (3) For each simply connected subdomain D e ! C such that F 0 .z/ D f .z/ for all z 2 D. e function F W D (4) The function f is continuous on D, and if is a (piecewise smooth) closed curve in a simply connected subdomain of D, then Z f .z/ dz D 0:

((1) H) (4): Cauchy’s theorem; (4) H) (1): Morera’s theorem) An equivalent formulation of this condition is: The function f is continuous on D and the differential form f .z/ dz is closed on D. (5) If fz 2 C W jz z0 j rg D with r > 0, then f .z/ D

1 2{

Z j z0 jDr

f ./ d z

.Cauchy0s integral formula/

for each z such that jz z0 j < r. (6) The nth complex derivative f .n/ .z/ exists for all z 2 D and for all integers n 0: (7) If fz W jz z0 j rg D with r > 0, then there exists a unique sequence of complex numbers fan g1 nD0 such that f .z/ D

1 X

an .zz0 /n

.Weierstrass/

nD0

for each z such that jz z0 j < r. Furthermore, the series converges uniformly and absolutely on every compact subset of fz W jz z0 j < rg. The coefficients an may be computed as follows. Z f ./ 1 d .Cauchy/ an D 2{ j z0 jDr . z0 /nC1 and f .n/ .z0 / : .Taylor/ nŠ [ (8) Choose a point zi 2 Ki , where Ki is the connected component decompoan D

i 2I

sition of the complement of D in C [ f1g, and let S D fzi I i 2 I g. Then the function f is the limit (uniform on compact subsets of D) of a sequence of rational functions with singularities only in S . .Runge/

1.3 The Plan for the Proof

5

1.3 The Plan for the Proof We prove the fundamental theorem by showing the following implications. .1/ , .2/ ) .3/ ) .4/ ) .5/ ) .6/ ) .1/I .5/ ) .7/ ) .1/ , .8/: It is of course possible to follow other paths through the various claims to obtain our main result. For the convenience of the reader, we describe where the various implications are to be found. At times the reader will need to slightly enhance an argument to obtain the required implication. .1/ ( .2/: Corollary 2.41. .1/ ) .2/: Theorem 2.33 and Corollary 5.8. .1/ ) .3/: Theorem 4.61 and Corollary 4.52. .3/ ) .4/: This is a trivial implication. See Lemma 4.14 and the definitions preceding it. .4/ ) .5/: Theorems 5.12 and 5.2. .5/ ) .6/: The proof of Theorem 5.5. .6/ ) .1/: This is a trivial implication. .5/ ) .7/: The proof of Theorem 5.5. .7/ ) .1/: Theorem 3.19. .1/ ) .8/: Theorem 7.37. .8/ ) .1/: Theorem 7.2. In standard texts, typically each of these implications is stated as a single theorem. The tag words in parentheses in the fundamental theorem are the names or terms that identify the corresponding results. The forward implication .1/ ) .n/ would be the theorem: “If f is a holomorphic function, then condition .n/ holds,” where n 2 f2; 3; 4; 5; 6; 7; 8g. For example, .1/ ) .2/ would be stated as “If f is holomorphic, then the Cauchy–Riemann equations hold”. The organization of all these conditions (potentially 56 theorems—some trivial) into a single unifying theorem is the hallmark of Bers’s mathematical style: clarity and elegance. Here it provides a conceptual framework for results that are highly technical and often computational. The framework comes from insight that, once articulated, will drive the subsequent mathematics and lead to new results. While the organization of the results into one unifying theorem is a distinctive characteristic of Bers’s mathematics, the treatments we give for some of the topics are not necessarily the ones that Bers used when he taught the course. In particular, his approach to integration theory did not normally start with differentials, and his emphasis on homotopy and homology were minimal. Currently, most graduate students will often either have studied some topology and some differential geometry before they take a complex analysis course or will be taking courses that cover these topics concurrently. Such students may skip over these parts of the text or skim them as a review. Additionally, our choice of topics beyond the fundamental theorem and

6

1 The Fundamental Theorem in Complex Function Theory

hyperbolic geometry has been guided by the tastes of the various authors. Some important theorems we have omitted or not treated in detail include the Picard Theorem 6.9, whose proof Bers sometimes included in his courses.

1.4 Outline of Text Chapter 2 contains the basic definitions. It is followed by a study of power series in Chap. 3. Chapter 4 contains the central material, the Cauchy theory, of the subject. We prove that the class of analytic functions is precisely the same as the class of functions having power series expansions, and we establish other parts of the fundamental theorem. Many consequences of the Cauchy theory are established in the next two chapters. Some readers may skip or skim parts of chaps. 2 and 3. In the second part of the text we proceed to the leisurely exploration of interesting ramifications and applications of the fundamental theorem. It starts with an exploration of sequences and series of holomorphic functions in Chap. 7, that also contains Runge’s theorem on approximations of holomorphic functions by rational functions. The Riemann mapping theorem (RMT) and the connection between function theory and hyperbolic geometry are the highlights of Chap. 8. The next chapter deals with harmonic functions, including a discussion of the Dirichlet problem and an alternative proof of the RMT. Zeros of holomorphic functions are discussed in the last chapter, which contains a study of the ring of holomorphic functions on a fixed domain, infinite Blaschke products, and an introduction to special functions. The latter is the beginning of the deep connections to classical and modern number theory.

1.5 Appendix: Historical Notes by Ranjan Roy The statements of the propositions comprising the fundamental theorem of complex function theory, given above, are accompanied by the names of the mathematicians to whom those results are attributed. The history of the discoveries of these and other mathematicians can not only provide interesting footnotes in a complex analysis course but can also complement our understanding of the substance of the topic. The study of the development of complex analysis is complicated but also enhanced by the fact that several results are named for mathematicians whose work was anticipated by earlier mathematicians. For example, Theorem 10.21, credited to the Danish mathematician J. Jensen (1859–1925) who discovered it in 1899, was proved by C. Jacobi (1804–1851) for polynomials with real coefficients in his 1827 paper, Ueber den Ausdruck der verschiedenen Wurzeln einer Gleichung durch bestimmte Integrale, and, as noted by E. Landau, the proof carries over to the general case. We also note that the mathematicians of past centuries often used a notation different from ours. For instance, for Gauss, Cauchy, and Weierstrass,

1.5 Appendix: Historical Notes by Ranjan Roy

7

x or y denoted complex variables; p again, Gauss and Weierstrass employed the symbol i , whereas Cauchy used 1 for “the” square root of 1. In presenting the specific details of their researches, it is often helpful to work within their notational perspective, as in these notes. C.F. Gauss (1777–1855) appears to be the first mathematician to state Cauchy’s (integral) theorem–indeed, to have a clear conception of integration in the complex domain. In his letter of RDecember 18, 1811 to his friend F.W. Bessel, Gauss discussed the meaning of x:dx for x D a C bi . He explained that, just as the realm of real numbers may be conceived of as a line, the set of complex numbers may be viewed as a two-dimensional plane with abscissa a and ordinate b. He then defined the integral over any (rectifiable) curve as the sum of infinitesimals .x/:dx, where R dx was an infinitesimal increment along the curve. Gauss wrote that the value of x:dx remained the same along the two paths as long as x ¤ 1 for all points x in the region between the two paths. To contrast this result R with one for which x D 1 inside the region, he defined log x by the integral x1 dx starting at 1 and ending at x ¤ 0. If the curve from 1 to x circumscribed the origin, then each circuit would add ˙2 i . He noted that this helped explain why log x was a multivalued function. Though Gauss promised to publish these remarkable theorems and their proofs at an appropriate occasion, he never did so; such instances of delayed publication provide further obstacles in our study of the mathematical past. Thus, it remained for A.L. Cauchy (1789–1857) to gradually work out the theory of complex integration. In 1814, he set out to rigorously establish some earlier results on real definite integrals of Euler, Laplace, and Legendre. They had obtained such results by formally replacing real with complex parameters within the integrals. But Cauchy proved a formula, equivalent to Cauchy’s theorem for a rectangular contour, and established a 1781 result of Leonhard Euler (1707–1783) on the Gamma function. Note that Euler’s result went unpublished until 1794. As early as 1729, Euler had discovered the Gamma function as an infinite product: .m C 1/ D

1m 2mC1 2m 3mC1 3m 4mC1 ; mC1 mC2 mC3

m 0I

(1.2)

then, in 1730, he showed that the infinite product (1.2) equaled the Gamma integral: R1 m t 0 . log x/ dx . Soon afterwards, Euler defined e and the exponential function e , and, in fact, when he returned to the study of the Gamma integral late in life, he took x D et and m D n 1 to get Z

1

.n/ D

t n1 et dt:

(1.3)

0

Observe that (1.3) is equivalent to formula (10.17), except that Euler took n > 0 to be real. We note that the notation .n/ was introduced in 1811 by Legendre and the modern notation for the limits of integration was first given by Fourier in 1818.

8

1 The Fundamental Theorem in Complex Function Theory

In his 1781 paper, Euler used (1.3) to get (in modern notation) Z

1

x n1 ekx dx D

0

.n/ ; with k > 0: kn

p He then boldly took k D p C q 1 with p > 0 and obtained Z

1

x n1 epx cos qx dx D

.n/ cos n fn

(1.4)

x n1 epx sin qx dx D

.n/ sin n ; fn

(1.5)

0

Z

1 0 1

where f D .p 2 C q 2 / 2 and tan D pq : He then deduced a number of remarkable and novel integrals as particular and limiting cases. For example: Z Z

1 0 1 0

cos x p dx D x epx

r

; 2

sin qx dx D ; x

Z

r

1

sin x p dx D x

1

sin x dx D : x 2

0

Z

0

; 2

Thus, by assuming k complex, Euler came upon results he regarded as amazing and he expounded on them in some detail, sensing them to be potentially very useful. Cauchy also found Euler’s results on the Gamma function significant, and the first part of his 1814 paper, M´emoire sur les int´egrales d´efinies, contained a proof of a particular case of his integral theorem, therebyRproviding a rigorous foundation p for Euler’s results. Cauchy started with the integral f dy, where y D M C N 1 D p M.x; z/ C N.x; z/ 1 ; x, and z real. He noted that @ @x

Z f .y/ dy D f .y/

@y ; dx

@ @z

Z f .y/ dy D f .y/

@y : @z

(1.6)

p Setting f .y/ D P C Q 1, he obtained p @M @N @N @M p @y D P Q C P CQ 1 S C T 1 ; @x @x @x @x @x (1.7) p @M @N @N @M p @y D P Q C P CQ 1 U C V 1 : f .y/ @z @z @z @z @z (1.8)

f .y/

From (1.6)–(1.8), and taking

@ @ @z @x

R

f .y/ dy D

@ @ @x @z

R

f .y/ dy; he had

1.5 Appendix: Historical Notes by Ranjan Roy

9

@T p @V p @U @S C C 1 D 1 : @z @z @x @x Equating the real and imaginary parts, Cauchy derived @S @U D ; @z @x

@T @V D : @z @x

(1.9)

Note that, by taking y D x C {z, Equations (1.9) become the Cauchy–Riemann equations. We remark that d’Alembert had found these equations in the course of his researches in fluid mechanics; similar equations had appeared in the work of Lagrange and Euler. But Cauchy was the first to understand the significance of these equations in distinguishing analytic from nonanalytic functions, though he came to this insight late in his life. To arrive at the particular case of his integral theorem, Cauchy next integrated the equations in (1.9) with respect to x and z within finite limits to obtain two equations, expressible in modern form as a single formula: Z f .y/ dy D 0;

(1.10)

C

where C represents a rectangular curve (or even a class of curvilinear quadrilaterals), though we note that Cauchy did not employ geometric language in his 1814 paper. Later in this paper, he extended this result to an infinite interval, R 1 and then, using the substitution y D .p C {q/x, he transformed the integral 0 f .y/ dy to an integral along a ray through p C {q. Euler’s results (1.4) and (1.5) followed as a corollary by taking f .y/ D y n1 ey . Cauchy’s 1814 Rbasic approach was Rto conceive of andR express the real and imaginary parts of f .z/ dz as separate: P dx Qdy and Qdx C P dy. But his 1825 paper, M´emoire sur les int´egrales d´efinies prises entre des limites imaginaires, took a much more geometric point of view. He considered points in the complex plane and R discussed contours within the complex plane. He gradually began to regard f .z/ dz as an entity in itself. In his 1825 paper, Cauchy presented a proof of his integral theorem (1.10) for a more general curve C and for functions he called finite and continuous. To his mind, such functions were continuously differentiable. He showed that the value of the integral from a point A to a point B did not change for two neighboring curves. Then by a homotopy type of argument, he extended the result to two non-neighboring curves. This argument is not rigorous by modern standards. Indeed, Cauchy himself may have been unsure of his reasoning; the paper contains his attempt to reformulate it in terms of the calculus of variations. And later, in his 1846 Sur les int´egrales qui s’´etendent a` tous les points d’une courbe ferm´ee, he expressed the integral as Z

Z

Z

f dz D C

.P dx Qdy/ C { C

Qdx C P dy; C

(1.11)

10

1 The Fundamental Theorem in Complex Function Theory

where C was a closed curve within which there was no singularity of f .z/. He applied Green’s theorem, Theorem 4.20, “

Z P dx C Qdy D ˙ C

D

@Q @P @x @y

dx dy

(where the plus sign was chosen if C had a positive orientation and D represented the region within C ) to each of the two real integrals. The Cauchy–Riemann equations in the form @P D @Q and @P D @Q then showed that the two integrals @x @y @y @x were equal to 0. In fact, it appears that Green’s theorem was first stated in Cauchy’s 1846 paper. It is possible that this work was inspired by the 1828 paper by George Green (1793– 1841) on the application of mathematics to electric and magnetic phenomena. Cauchy did not prove Green’s theorem; Riemann presented the first proof in his famous dissertation of 1851. Also in 1851, Cauchy noted that the Cauchy– Riemann equations implied analyticity; he assumed that the partial derivatives were continuous. We also note that Weierstrass was aware of the proof of Cauchy’s integral theorem by means of Green’s theorem as early as 1842. All these proofs of Cauchy’s integral theorem assumed the continuity of f 0 .z/. In 1883, E. Goursat (1858–1936) retained this assumption in his discovery of a new proof of the theorem for the case of a rectangular contour. But he soon realized that he did not require continuity, so that when he republished the proof in 1900, in the first issue of the Transactions of the American Mathematical Society, he removed this condition. To deal with functions with singularities inside the closed curve, Cauchy’s 1814 paper had a result amounting to a particular case of the residue theorem, Theorem 6.17. But it was in his 1826 paper, Sur un nouveau genre de calcul analogue au calcul infinit´esimal, that he formally defined a residue for a function with a pole. Note that the idea of an essential singularity was not then known. He further showed that residues could be applied to the evaluation of definite integrals. In 1830, leaving France for political reasons, Cauchy became professor of theoretical physics at the University of Turin. There he published an important paper on the power series expansions of analytic functions, significant portions of which were reprinted with some changes in his Exercices d’analyse of 1841. He first proved Cauchy’s integral formula f .x/ D

1 2

Z

x f .x/ dp; xx

(1.12)

p

where x D X ep 1 and jxj < X ; note that x did not represent the complex conjugate. He began the proof by showing that Z

f .x/ dp D 2f .0/:

(1.13)

1.5 Appendix: Historical Notes by Ranjan Roy

11

For this purpose, Cauchy observed that DX f .x/ D X p11 Dp f .x/; he then integrated with respect to X from 0 to X and with respect to p from to . The left-hand side simplified to Z

f .x/ f .0/ dp D

Z

f .x/ dp 2 f .0/;

and the right-handRside became zero, yielding (1.13). This in turn implied that when .x/ , taking f .0/ D 0 he had f .x/ dp D 0. He then replaced f .x/ by x f .x/f xx jxj < X , to arrive at Z

xf .x/ dp D xx

Z

xf .x/ dp x x Z x2 x D f .x/ 1 C C 2 C dp D 2f .x/: x x

This proved the integral formula (1.12). To obtain the Maclaurin series (a particular x case of the Taylor series), he expanded .xx/ in (1.12) as a geometric series 1 f .x/ D 2

Z

1 f .x/ x dp D 1 x 2

Z

x2 x f .x/ 1 C C 2 C dp: x x

The general term in the series was given by 1 2

Z

f .x/ xn n x

xn dp D 2

Z

f .x/ dp: xn

We mention that, although he could easily have done so, Cauchy did not establish the existence of the derivatives of f 0 .x/. Nevertheless, Cauchy applied repeated integration by parts to write 1 2

Z

f .x/ 1 dp D xn 2 n

Z

f 0 .x/ 1 dp D x n1 2.1 2 n/ .n/

Z

f .n/ .x/ dp:

Note that, by (1.13), the last integral could be set equal to f nŠ.0/ ; Cauchy therefore argued that the function f .x/ could be expanded as a Maclaurin series for those values of x whose modulus was less than the least value for which f .x/ was not continuously differentiable. More generally, he could have obtained the coefficient .n/ as f nŠ.a/ by expanding the series in powers of x a. This was and is known as the Taylor expansion of f .x/; Brook Taylor (1685–1731) proved this result for real functions in 1712 and published it 3 years later in his Methodus Incrementorum. We should perhaps not be surprised, however, that the first explicit statement of the Taylor series and the particular case of the Maclaurin series were given by

12

1 The Fundamental Theorem in Complex Function Theory

the father of calculus, Isaac Newton (1642–1727). He presented these results in unpublished portions of his De quadratura curvarum of 1691–92, in connection with series solutions of algebraic differential equations. We must also note that in a 1671 letter to John Collins, James Gregory (1638–1675) wrote down power series expansions of functions such as tan x and sec x by computing the derivatives of these functions. This implies that he was aware of the Taylor expansion 40 years before Taylor. The Laurent series given in Theorem 6.1, a generalization of the Taylor series, was derived by P. A. Laurent (1813–1854), a French military engineer. In 1843, he published a short note on the series expansion of a function analytic in an annulus. He generalized Cauchy’s integral formula to the annulus and deduced that the function had a series expansion, provided that infinitely many terms with negative powers of x a were included. Cauchy reported to the French Academy on this work, calling Laurent’s result a new theorem but mentioning that it could be deduced from his own 1840 work on the mean values of a function. Surprisingly, Karl Weierstrass (1815–1897) had already proved Laurent’s theorem in his 1841 paper, Darstellung einer analytischen Function einer complexen Ver¨anderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt. In this paper, Weierstrass expanded a function F .x/, for x inside an annulus around the origin, into a series of the form A0 C A1 x C C A x C C A1 x 1 C C A x C D

1 X

A x :

D1

In addition to the conditions of continuity and finiteness, the function was required to satisfy the condition that the difference of the two quotients F .x C hk/ F .x/ hk

and

F .x C k/ F .x/ ; k

where x was in the annulus and h was any complex number, became infinitely small when k became infinitely small. This condition would be satisfied by the class of continuously differentiable functions. Thus, a function analytic in the annulus had a Laurent series expansion. Weierstrass expressed the coefficient A as an integral around a circle in the annulus with center at the origin. From this he deduced that if jF .x/j M on jxj D a, then jA j M a . This implied that if F remained bounded in a neighborhood of the origin, then jA j M a . He then let a ! 0 to derive A D 0 so that F .x/ D A0 C A1 x C A2 x 2 C : Weierstrass’s paper containing this theorem was not published until 50 years later when it appeared in his collected works. It is clear, however, that he did his work independently, since he became aware of Cauchy’s results much later. It

1.5 Appendix: Historical Notes by Ranjan Roy

13

seems that the Laurent expansion was never mentioned in Weierstrass’s lectures. Weierstrass may have been dissatisfied with his proof of the Laurent expansion because it employed integration, whereas he regarded power series as conceptually more fundamental than differentiation or integration. Weierstrass’s 1876 proof of the Casorati–Weierstrass theorem (Theorem 6.8) did not employ Cauchy’s integral formula; the proof was not too elegant but very interesting. Weierstrass’s valuable insight was to use power series to define the analyticity of a function; this perspective became especially valuable when analysts began to take p-adic or other noncomplex numbers as the underlying field of constants. We note that the Casorati–Weierstrass theorem was first published in 1868 by F. Casorati (1835– 1890), who used integration in an annulus. However, Schwarz reported to Casorati that he had been present when Weierstrass discussed this theorem in his 1863 Berlin lectures. Then again, in 1868, the Russian complex analyst Y. Sokhotskii (1842–1927) employed Cauchy’s integral formula to independently prove the Casorati–Weierstrass theorem in his master’s thesis at St. Petersburg University; thus, the theorem is also named the Casorati–Sokhotskii–Weierstrass theorem. Bernhard Riemann (1826–1866) took the Cauchy–Riemann equations as the basis of his theory of analytic functions. He was primarily interested in the geometric properties of these functions, that is, in their orientation-preserving and conformal (or angle-preserving) properties. In his 1851 dissertation, Riemann defined simple connectivity of a domain and outlined a proof of the proposition that every simply connected proper subdomain of the complex plane could be biholomorphically (conformally) mapped onto the unit disc. This is the RMT, Theorem 8.20. The proof was not quite correct and also depended upon the unproven Dirichlet principle. It required the efforts of such outstanding mathematicians as H. A. Schwarz, H. Poincar´e, D. Hilbert, P. Koebe, and C. Carath´eodory to put Riemann’s ideas and intuitions on a solid basis. Poincar´e famously remarked that Riemann’s method was a method of discovery, whereas that of Weierstrass was a method of proof. Schwarz gave a proof of the RMT in his 1869 paper, Zur Theorie der Abbildung. The paper begins with a proof of Schwarz’s lemma for a biholomorphic function f , mapping the unit disc onto a region U and satisfying f .0/ D 0. This lemma was then forgotten until it was rediscovered in 1905 by Carath´eodory, who applied it to give a short proof of Landau’s extension of Picard’s Theorem 6.9. Landau’s extension states that if f .z/ D a0 C a1 z C az z2 C ; a1 ¤ 0 is holomorphic and does not take the values 0 and 1 in jzj < R, then R is bounded by a number that depends on the first two coefficients, a0 and a1 . The result as given in Theorem 5.34 is due to Carath´eodory. He reported that his initial unpublished proof of this theorem depended on the Poisson integral formula (9.3). But he showed this work to Schmidt, famous for his fundamental contributions to functional analysis, who provided a superior argument. This proof, utilizing the

14

1 The Fundamental Theorem in Complex Function Theory

maximum principle (Corollary 5.31), is presented in this and other modern texts. Carath´eodory discovered other applications of this lemma and in 1912 he named it after Schwarz. Morera’s theorem, contained in the fundamental theorem, was first proved in a paper of 1886, Un teorema fondamentale nella teo rica delle funzioni di una variabile complessa. The Italian mathematician, G. Morera (1856–1909), supplied another way to define analyticity; he proved that any locally integrable complex function was analytic. This is also known as the converse of Cauchy’s theorem. In another paper of 1886, Sulla rappresentazione delle funzioni di una variabile com plessa per mezzo di espressioni analitiche infinite, Morera used his theorem to reprove Weierstrass’s convergence theorem for compactly convergent series, established in his Zur Functionenlehre of 1880.

Chapter 2

Foundations

The first section of this chapter introduces the complex plane, fixes notation, and discusses some useful concepts from real analysis. Some readers may initially choose to skim this section. The second section contains the definition and elementary properties of the class of holomorphic functions—the basic object of our study.

2.1 Introduction and Preliminaries This section is a summary of basic notation, a description of some of the basic algebraic and geometric properties of the complex number system, and a disjoint collection of needed facts from real analysis (advanced calculus). We remind the reader of some of the formalities behind the standard notation which we usually approach quite informally. Not all concepts used as prerequisites are defined (among these are neighborhood, connected, path-connected, arc-wise connected, and compact sets); we assume that the reader has been exposed to them.1 We start with some standard notation: Z>0 Z Q R C b C: Here Z represents the integers, Z>0 the positive integers,2 Q the rationals (the integer n is included in the rationals as the equivalence class of the quotient n1 ), and R the reals. Whether one views the reals as the completion of the rationals or identifies

1 The reader may want to consult J. R. Munkres Topology (Second Edition), Dover, 2000, or J. L. Kelley, General Topology, Springer-Verlag, 1975 as well as definitions in Chap. 4. 2 In general Xcondition and fx 2 XI conditiong will describe the set of all x in X that satisfy the indicated condition.

R.E. Rodr´ıguez et al., Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics 245, DOI 10.1007/978-1-4419-7323-8 2, © Springer Science+Business Media New York 2013

15

16

2 Foundations

them with Dedekind cuts (we will not use these concepts explicitly), their most important property from the perspective of complex variables is the least upper bound property; that is, that every nonempty set of real numbers that has an upper bound has a least upper bound. The inclusion of R into the complex numbers C needs a bit more explanation. It is specified as follows: for z in C, we write z D x C { y with x and y in R, where the symbol { represents a square root of 1; that is, { 2 D 1. With these conventions we can define addition and multiplication of complex numbers using the usual rules for these operations on the reals3 : for all x; y; ; 2 R, .x C {y/ C . C {/ D .x C / C {.y C / and .x C {y/. C {/ D .x y/ C {.x C y/: The real numbers, R, are identified with the subset of C consisting of those numbers z D x C { y with y D 0; the imaginary numbers, {R, are those with x D 0. For z D x C { y in C with x and y in R we write x D C; with B in C¤0 and C real.

2.1 Introduction and Preliminaries

a

21

zw

b w ½

zw w ½ 0

z '

r

µ +'−2¼.

'

µ+' r½

r½

0 µ r

µ

z

Fig. 2.3 Vector multiplication. (a) Sum of arguments smaller than 2. (b) Sum of arguments larger than 2

2.1.6 Polar Coordinates A nonzero vector in C can be described by polar coordinates .r; / as well as by the rectangular coordinates .x; y/ we have been using. If z 2 C and z ¤ 0, then we can write z D x C { y D r .cos C { sin / ; y x where r D jzj and D arg z (an argument of z) D arcsin D arccos . r r Note that the last two identities are needed to define the argument and that arg z is defined up to addition of an integral multiple of 2. This is why we labeled an argument of z as opposed to the argument.4 If w D Œcos ' C { sin ' is another nonzero complex number, then, using the usual addition formulas for the sine and cosine functions, we have zw D .r/Œcos. C '/ C { sin. C '/: This polar form of the multiplication formula shows that the complex multiplication of two (nonzero) complex numbers is equivalent to the real multiplication of their moduli and the addition of their arguments, giving a geometric interpretation of how the operation of multiplication acts on vectors represented in polar coordinates; see Fig. 2.3. It also shows (again) that jz wj D jzj jwj. Polar coordinates also provide another way to view Fig. 2.2. In particular, it follows that if n 2 Z and z D r .cos C { sin / is a nonzero complex number, then zn D r n Œcos n C { sin nI

4

The number will be defined rigorously in Definition 3.34. Trigonometric functions will be introduced in the next chapter where some of their properties, including addition formulae, will be developed. For the moment, polar coordinates should not be used in proofs.

22

2 Foundations

it also proves the famous de Moivre’s formula: .cos C { sin /n D cos.n / C { sin.n / for n 2 Z and 2 R. Therefore, for each nonzero complex number z D r .cos C { sin / and each n in Z>0 , there exist precisely n complex numbers w such that wn D z; they are the n n-th roots of z, and are given by C 2k C 2k 1 C { sin ; r n cos n n with k D 0; 1; : : : ; n 1. Note that these n complex numbers are the vertices of a regular n-gon in the plane.

2.1.7 Coordinates on C We have already seen that we can use three sets of coordinates on C, as follows. 1. Rectangular .x; y/: Each equation x D constant (respectively y D constant) yields a line parallel to the imaginary axis (respectively real axis), while the equation x D y yields the line through the origin with slope equal to 1. 2. Complex .z; z/: Only one of these coordinates is needed to describe a point by the equation z D constant (or z D constant), while the equation z D z yields the real axis. 3. Polar .r; /: The equation r D a with a > 0 is a circle of radius a centered at the origin, whereas the equation D constant is a ray emanating from (but not including) the origin. The equation r D denotes a type of spiral ending at (but not passing through) the origin. The choice of the appropriate one among the various possible coordinates on C may simplify a problem. As an example we solve the following one. Let n be a positive integer, and suppose we want to find the set of points z in C that satisfy the equation zn D zn : (2.8) Using rectangular coordinates would lead us to solve .x C { y/n D .x { y/n ; which is doable but far from pleasant. Instead, we first note that certainly z D 0 satisfies (2.8). For z ¤ 0, we may use the polar coordinates: the equation we are trying to solve is then equivalent to

2.2 Topological Preliminaries

23

r n Œcos n C { sin n D r n Œcos n { sin n; which implies that n D k for some integer k. Thus we immediately see that the complete solution to (2.8) is the set of 2n rays D o from the origin (including the origin) with .2n 1/ 2 ;:::; : o 2 0 ; ; n n n

2.2 More Preliminaries that Rely on Topology, Metrics, and Sequences We collect some facts on sets of complex numbers and functions defined on them, that mostly follow from translating to the complex system the analogous results from real analysis. The formula d.z; w/ D jz wj, for z and w 2 C, defines a metric on C. Thus .C; d / is a metric space, with a metric that agrees with the Euclidean metric on R2 (under the linear representation of the complex plane described earlier). Definition 2.1. We say that a sequence (indexed by n 2 Z>0 ) fzn g of complex numbers converges to ˛ 2 C if given > 0, there exists an N 2 Z>0 such that jzn ˛j < for all n > N ; in this case we write lim zn D ˛:

n!1

A sequence fzn g of complex numbers is called Cauchy if given > 0, there exists an N 2 Z>0 such that jzn zm j < for all n; m > N . Theorem 2.2. If fzn g and fwn g are Cauchy sequences of complex numbers, then (a) fzn C ˛ wn g is Cauchy for all ˛ 2 C. (b) fzn g is Cauchy. (c) f jzn j g R0 is Cauchy. Proof. (a) It suffices to assume that ˛ ¤ 0. Given > 0, choose N1 such that for jzn zm j < 2 for all n; m > N1 and choose N2 such that jwn wm j < 2j˛j all n; m > N2 . Choose N D max fN1 ; N2 g. Then, for all n and m > N , we have j.zn C ˛ wn / .zm C ˛ wm /j jzn zm j C j˛j jwn wm j < : (b) It follows directly from jzn zm j D jzn zm j D jzn zm j. (c) We know that for all z and w in C we have

24

2 Foundations

jjzj jwjj jz wj : Applying this inequality to zn and zm in the sequence, we obtain jjzn j jzm jj jzn zm j ; and the result follows. t u Remark 2.3. The above arguments mimic arguments in real analysis needed to establish the corresponding results for real sequences. We will, in the sequel, leave such routine arguments as exercises for the reader. Corollary 2.4. fzn g is a Cauchy sequence of complex numbers if and only if f 0. The -ball about c, or the open disc with center c and radius , is the set Uc ./ D U.c; / D fz 2 CI jz cj < g; that is, the interior of the circle with center c and radius . Proposition 2.8. A subset A of C is bounded if and only if there exist a complex number c and a positive number R such that A U.c; R/: Remark 2.9. A proof is omitted for one of three reasons (in addition to the reason described in Remark 2.3): either it is trivial or it follows directly from results in real analysis or it appears as an exercise at the end of the corresponding chapter.5 The third possibility is always labeled as such; when standard results in real analysis are needed, there is some indication of what they are or where to find them. For example, the next two theorems are translations to C of standard metric results for R2 . It

5

Exercises can be found at the end of each chapter and are numbered by chapter, so that Exercise 2.7 is to be found at the end of Chap. 2.

2.2 Topological Preliminaries

25

should be clear from the context when the first possibility occurs. It is recommended that the reader ensures that he/she is able to supply an appropriate proof when none is given. Theorem 2.10 (Bolzano–Weierstrass). Every bounded infinite set S in C has at least one limit point; that is, there exists at least one c 2 C such that, for each > 0, the ball U.c; / contains a point z 2 S with z ¤ c. Theorem 2.11. A set K C is compact if and only if it is closed and bounded. We will certainly be using a number of consequences of compactness not discussed in this chapter (e.g., in a compact metric space, every sequence has a convergent subsequence) and also of connectedness, which we will not define here. Definition 2.12. Let f be a function defined on a set S in C. We assume that f is complex-valued, unless otherwise stated. Thus f may be viewed as either a map from S into R2 or into C and also as two real-valued functions defined on the set S . Let c be a limit point of S and let ˛ be a complex number. We say that the limit of f at c is ˛, and we write lim f .z/ D ˛ z!c

if for each > 0 there exists a ı > 0 such that jf .z/ ˛j < whenever z 2 S and 0 < jz cj < ı: Remark 2.13. The condition that c is a limit point of S ensures that there are points z in S arbitrarily close to (but different from) c so that f .z/ is defined there. Note that it is not required that f .c/ be defined. The above definition is again a translation of language from R2 to C. Thus we will be able to adopt many results (the next three theorems, in particular) from real analysis. In addition to the usual algebraic operations on pairs of functions f W S ! f C and g W S ! C familiar from real analysis, such as f Ccg with c 2 C, fg, and g (provided g does not vanish on S ; that is, if g.z/ ¤ 0 for any z 2 S or, equivalently, if no z 2 S is a zero of g), we will consider other functions constructed from a single function f , that are usually not emphasized in real analysis. Among them are the following: . 0 there exists an N 2 Z>0 such that jfn .z/ fm .z/j < for all z 2 S and all n and m > N: Note that in this case the limit function f is uniquely determined; it is the pointwise limit f .z/ D lim fn .z/, for all z 2 S . n!1

Theorem 2.23. Let ffn g be a sequence of functions defined on S C. If: .1/ ffn g converges uniformly on S . .2/ Each fn is continuous on S . Then the function f defined by f .z/ D lim fn .z/; z 2 S n!1

is continuous on S . Proof. Start with two points z and c in S . Then for each natural number n we have jf .z/ f .c/j jf .z/ fn .z/j C jfn .z/ fn .c/j C jfn .c/ f .c/j : Now fix > 0. By (1), the first and third term on the right-hand side are less than for n large. If we now fix c and n, it follows from (2) that the second term is less 3 than as soon as z is close enough to c. Thus f is continuous at c. t u 3 Definition 2.24. A domain or region in C is a subset of C which is open and connected. Remark 2.25. Note that a domain in C could also be defined as an open arcwise connected subset of C. (See also Exercise 2.20.) Also note that each point in a domain D is a limit point of D, and therefore it makes sense to ask, at each point in D, about the limit of any function defined on D.

28

2 Foundations

2.3 Differentiability and Holomorphic Mappings Up to now, the complex numbers were used mainly to supply us with a convenient alternative notation. This is about to change. The definition of the derivative of a complex-valued function of a complex variable mimics that for the derivative of a real-valued function of a real variable. However, we shall see shortly that the properties of the two classes of functions are quite different. Definition 2.26. Let f be a function defined in some disc about c 2 C. We say that f is (complex) differentiable at c provided f .c C h/ f .c/ h exists. In this case the limit is denoted by lim

(2.9)

h!0

f 0 .c/;

ˇ df ˇˇ df .c/; ; or .Df /.c/; dz dz ˇzDc

and is called the derivative of f at c. Remark 2.27. (1) It is important that h be an arbitrary complex number (of small nonzero modulus) in the above definition. (2) Note that f .c C h/ f .c/ f .z/ f .c/ D lim : z!c h zc (3) If f is differentiable at c, then f is continuous at c: The converse is not true in general; see Example 2.32.4. (4) We consider two identities for a function f defined in a neighborhood of c 2C: lim

h!0

f .c C h/ D a0 C .h/ with lim .h/ D 0; h!0

and f .c C h/ D a0 C a1 h C h.h/ with lim .h/ D 0: h!0

As in real analysis, the first of these says that f is continuous at c if and only if f .c/ D a0 ; the second says that f is differentiable at c if and only if f .c/ D a0 and f 0 .c/ D a1 . Whereas in the real case the second statement is sharp with regard to smoothness, we shall see that in the complex case, under appropriate conditions, it can be improved significantly. Notation 2.28. If the function f is differentiable on a domain D (i.e., at each point of D), then it defines a function f 0 W D ! C. Thus for every n 2 Z0 we can define inductively f .n/ , the n-th derivative of f , as follows:

2.3 Differentiability and Holomorphic Mappings

29

0 f .0/ D f , and if f .n/ is defined for n 0, then we set f .nC1/ D f .n/ whenever the appropriate limits exist. It is customary to abbreviate f .2/ and f .3/ by f 00 and f 000 , respectively. Of course, f .1/ D f 0 . Definition 2.29. Let f be a function defined in a neighborhood of c 2 C. Then f is holomorphic or analytic at c if it is differentiable in a neighborhood (perhaps smaller) of c. A function defined on an open set U is holomorphic or analytic on U if it is holomorphic (equivalently, differentiable) at each point of U . It should be emphasized that holomorphicity is always defined on open sets. A function f is called anti-holomorphic if fN is holomorphic. The usual rules of differentiation hold. Let f and g be functions defined in a neighborhood of c 2 C, let F be a function defined in a neighborhood of f .c/, and let a 2 C. Then (recall Remark 2.15): .f C ag/0 .c/ D f 0 .c/ C ag 0 .c/ .fg/0 .c/ D f .c/g 0 .c/ C f 0 .c/g.c/ 0 0 0 .F (the chain rule) ıf0 / .c/ D F 0.f .c//f .c/ f .c/ 1 .c/ D provided f .c/ ¤ 0 (d) f f .c/2 n (e) if f .z/ D z with n 2 Z (and z 2 C¤0 if n 0), then f 0 .z/ D n zn1

(a) (b) (c)

Remark 2.30. About the chain rule (c): If f .z/ D w is a differentiable function of z and if F .w/ D is a differentiable function of w, then we often write the chain rule as d d dw D : dz dw dz A “proof” follows. Let z0 be arbitrary in the domain of f , and set w0 D f .z0 / and 0 D F .w0 /. Note that w D f .z/ ! w0 as z ! z0 . Now .F ı f /0 .z0 / D

0 d .z0 / D lim z!z dz 0 z z0

D lim

z!z0

D

. 0 /.w w0 / 0 w w0 D lim lim w!w0 w w0 z!z0 z z0 .w w0 /.z z0 /

d dw .w0 / .z0 / D F 0 .w0 / f 0 .z0 /: dw dz

This “proof” has an error in it, what is it? Definition 2.31. A function defined on the complex plane is called entire if it is holomorphic on C, that is, if its derivative exists at each point of C.

30

2 Foundations

Example 2.32. We illustrate some of the concepts introduced with more or less familiar examples. 1. Every polynomial (in one complex variable) is entire. These (apparently) simple objects have fairly complicated behavior, that is studied, for example, as part of complex dynamics. P 2. A rational function is a function of the form R D , where P and Q are Q polynomials (in one complex variable), with Q not the zero polynomial. Note that the polynomial Q has only finitely many zeros (the number of zeros, properly counted, equals the degree of Q; see Exercise 3.19). The rational function R is holomorphic on C fzeros of Qg. az C b with a, b, c, and d fixed 3. A special case of Example 2.32.2 is R.z/ D cz C d complex numbers satisfying ad bc ¤ 0. These rational functions are called fractional linear transformations or M¨obius transformations and will be studied in detail in Sect. 8.1. They are the building blocks for much that will follow in this book—automorphisms of domains in the Riemann sphere, and Blaschke products, and as important ingredients for much current research in areas of complex analysis: Riemann surfaces, Fuchsian, and (the more general case of) Kleinian groups. 4. In real analysis it takes work to construct a continuous function on R that is nowhere differentiable. The situation with respect to complex differentiability is much simpler. The functions z 7! z and z 7! jzj are both continuous on C, but they are nowhere (complex) differentiable, since the corresponding limits (2.9) do not exist at any c in C.

2.3.1 Convention Whenever we write z D x C { y for variables and f D u C { v for functions, then we automatically mean that x D 0 such that N D U.c; "/ fcg, and, for any M > 0, there exists a z 2 N such that jg.z/j > M ; that is, such that jf .z/ wj < M1 . Thus w is a limit point of f .N / and the last statement in the theorem is proved. Now it suffices to prove that for all w 2 C and all " > 0, the function g is unbounded in U.c; "/ fcg. If g were bounded in such a neighborhood, it would have a removable singularity at z D c and thus would extend to a holomorphic function on U.c; "/; therefore f would be meromorphic there. t u A much stronger result can be established. We state it without proof.2 Theorem 6.9 (Picard). If f is holomorphic in 0 < jz cj < R2 and has an essential singularity at z D c, then there exists a w0 2 C such that for all w 2 C fw0 g; f .z/ D w has infinitely many solutions in 0 < jz cj < R2 . 1 shows the above theorem is sharp, with c D Example 6.10. The function exp z 0 and w0 D 0.

2

For a proof, see Conway’s book listed in the bibliography.

146

6 Cauchy Theory: Local Behavior and Singularities

We now have a complete description of the behavior of a holomorphic function near an isolated singularity. Theorem 6.11. Assume that f is a holomorphic function in a punctured disc U 0 D U.c; R/ fcg around the isolated singularity c 2 C. Then (1) c is a removable singularity if and only if f is bounded in U 0 if and only if lim f .z/ exists and is finite. z!c

(2) c is a pole of f if and only if lim f .z/ D 1, in the sense of Definition 3.52. z!c

(3) c is an essential singularity of f if and only if f .U 0 / is dense in C. Proof. (1) follows from Theorem 6.7 and the definition of removable singularity. The Casorati–Weierstrass Theorem 6.8 shows that if c is an essential singularity then f .U 0 / is dense in C. We complete the proof by showing (2). If c is a pole of f of order N 1, then we know from the Laurent series expansion for f that there exists 0 < r < R and a holomorphic function g W U.c; r/ ! C such that g.z/ on U.c; r/ fcg and g.c/ ¤ 0. By continuity of g we may f .z/ D .z c/N M , and it assume that jg.z/j M > 0 for all z in U.c; r/. Then jf .z/j jz cjN follows that lim f .z/ D 1. Conversely, if lim f .z/ D 1, then c is not a removable z!c

z!c

singularity of f since for every M > 0 there exists ı > 0 such that jf .z/j M for 0 < jz cj < ı; and the Casorati–Weierstrass theorem implies that c is not an essential singularity of f . Thus it must be a pole. t u Example 6.12. For an entire function f .z/ D

1 X

an zn (we know that its radius of

nD0

convergence D C1), there are two possibilities: (a) Either there exists an N such that an D 0 for all n > N , in which case f is a polynomial of degree P N . If deg f D N 1, then f has a pole of order N at i 1, and f .z/ f .0/ D N i D1 ai z is the principal part of f at 1. If deg f D 0, then f is constant, of course. (b) Or f has an essential singularity at 1. We can now establish the following result. Theorem 6.13. Let f W C [ f1g ! C [ f1g. Then, (a) If f is holomorphic, it is constant, and (b) If f is meromorphic, it is a rational function. Proof. If f is holomorphic on C [ f1g, a compact set, it must be bounded. Since it is also an entire function, it must be constant, by Liouville’s theorem.

6.3 Residues

147

If f is meromorphic on C [ f1g, its set of poles must be finite, being isolated points in a compact set. Denote them by z1 ; : : : ; zk , which may include 1. If this set is empty, we are in the previous case. Otherwise, let Nj > 0 be the order of the pole at zj and let Pj .z/ be the principal part of f at zj . Then Pj .z/ D

8P < Nj

an;j nD1 .zzj /n ;

: PNj

nD1

It follows that f The result follows.

Pk

j D1 Pj

if zj 2 C;

an;j zn ; if zj D 1:

is holomorphic in C [ f1g and therefore constant. t u

6.3 Residues If f is a holomorphic function in a deleted neighborhood U of c in C [ f1g, then ! D f .z/ dz defines a holomorphic differential form on U with an isolated singularity at c. Definition 6.14. If the holomorphic function f has an isolated singularity at c 2 C, with Laurent series expansion (6.1) on jz cj < R2 for some R2 > 0; we define the residue of the differential form ! D f .z/ dz at c by the formula Res.f .z/ dz; c/ D Res.!; c/ D a1 : Remark 6.15. Note that if f has a simple pole at c (or more generally, if c .f / 1), then Res.f .z/ dz; c/ D lim .z c/f .z/: z!c

Our next result will give an alternate way to define the residue of the differential form f .z/ dz, and will show its invariance. It is an analogue of the Cauchy integral formula for the case of an isolated singularity. Theorem 6.16. Let A denote the annulus R1 < jz cj < R2 . If is a closed path C1 X in A and if f is holomorphic in A with Laurent series f .z/ D an .z c/n , nD1

then 1 2{

Z f .z/ dz D I.; c/ a1 :

148

6 Cauchy Theory: Local Behavior and Singularities

In the special case that R1 D 0 and I.; c/ D 1, we have 1 2{

Z f .z/ dz D Res.f .z/ dz; c/:

Proof. Write X

g.z/ D

an .z c/n and f .z/ D

n¤1

a1 C g.z/: zc

The function g has a primitive in the annulus A; namely, X n¤1

1 an .z c/n : nC1

Z g.z/ dz D 0, the result follows.

Since

t u

Theorem 6.17 (Residue Theorem). Let f be holomorphic in a domain D C except for isolated singularities at z1 ; : : : ; zn in D. If is a positively oriented Jordan curve homotopic to a point in D and all zj are in the interior of , then Z f .z/ dz D 2{

n X

Res.f .z/ dz; zj /:

j D1

Proof. Put a small positively oriented circle around each zj and use the extended version of Cauchy’s Theorems 5.26and 6.16. t u Let c and b 2 C. As above assume that c is an isolated singularity of a holomorphic function f defined in a deleted neighborhood U of c. Let h be a holomorphic function defined in a deleted neighborhood of b with values in U , and let be a positively oriented circle centered at b with sufficiently small radius. Then Res..f ı h/.z/ dh.z/; b/ D D

1 2{ 1 2{

Z

f .h.z//h0 .z/ dz

Z

f .w/ dw h. /

D I.h. /; c/ Res.f .w/ dw; c/:

(6.2)

The above invariance property of residues allows us to extend its definition to holomorphic differentials with an isolated singularity at infinity. If 1 is an isolated singularity of the holomorphic function f with Laurent series expansion f .z/ D

6.4 Zeros and Poles of Meromorphic Functions C1 X

149

an zn for jzj > R, then we define the residue of the differential form ! D

nD1

f .z/ dz at 1 by the formula Res.!; 1/ D a1 :

To justify the last definition (its consistency with previously defined concepts), we show that it satisfies the invariance property (6.2). This is in agreement with our earlier convention: to obtain invariants at z D 1 we change the variable from z to 1 w D , and then use the invariants defined for w D 0. z Thus, in this case, we have 1 1 Res.f .z/ dz; 1/ D Res f d ;0 : w w But 1 1 f d D w w D

C1 X nD1 C1 X

n

an w

!

1 dw w2

.ak2 /wk dw:

kD1

6.4 Zeros and Poles of Meromorphic Functions In this section we study several consequences of the residue theorem. Let D be a domain in C and f W D ! b C be a meromorphic function. This means that f is holomorphic except for isolated singularities in D, which are removable or poles (see Sect. 3.5). We have denoted the field of meromorphic functions on D by M.D/. Note that if is a positively oriented Jordan curve in D which is homotopic to a point in D, then the interior i. / of is contained in D. Furthermore, the number of zeros and the number of poles in the interior of of any nonconstant meromorphic function in M.D/ are both finite, since range [ i. / is a compact set. The next result counts the difference between these two numbers. We recall from Definition 3.54 that the meromorphic function f has order or multiplicity c .f / at c 2 D, if there exists a holomorphic function g defined near c such that g.c/ ¤ 0 and f .z/ D .z c/c .f / g.z/ near c.

150

6 Cauchy Theory: Local Behavior and Singularities

Theorem 6.18 (The Argument Principle). Let D be a domain in C and let f 2 M.D/. Suppose is a positively oriented Jordan curve in D which is homotopic to a point in D. Let c 2 C and assume that f .z/ ¤ c and f .z/ ¤ 1 for all z in range . Then Z X 1 f 0 .z/ z .f c/ D Z P; dz D 2{ f .z/ c z2i. /

where Z is the number of zeros of the function f c inside (counting multiplicities) and P is the number of poles of the function f inside (counting multiplicities). f 0 .z/ for z 2 D, then F 2 M.D/, and we claim that F has f .z/ c only simple poles, at the zeros and poles of f c, and that

Proof. If F .z/ D

Res.F .z/ dz; d / D d .f c/ for all d 2 D:

(6.3)

The theorem then follows immediately from (6.3) and the residue theorem. To verify our claim, it suffices to assume that c D 0 D d . If we set n D 0 .f /, then f .z/ D zn g.z/ with g holomorphic near 0 and g.0/ 6D 0. It follows that f 0 .z/ D nzn1 g.z/ C zn g 0 .z/ and hence

n g 0 .z/ f 0 .z/ f 0 .z/ D C near 0. Thus dz has residue n at zero. f .z/ z g.z/ f .z/

t u

Remark 6.19. The name “argument principle” attached to the previous result may be explained in the following way: let D be a domain in C and let f 2 M.D/ be a nonconstant function. Assume is a positively oriented Jordan curve in D which is homotopic to a point in D. Let c 2 C and suppose that f .z/ ¤ c and f .z/ ¤ 1 for all z in range . If Z denotes the number of zeros of the function f c inside (counting multiplicities) and P denotes the number of poles of f inside (counting multiplicities), then the argument of f c increases by 2.Z P / upon traversing f0 . Indeed, note that D .log.f c//0 and recall that f c log.f c/ D log jf cj C { arg.f c/: Therefore, Z

f 0 .z/ dz D f .z/ c

Z

Z d log jf .z/ cj C {

d arg.f .z/ c/:

The first integral on the rightmost side of the equation equals zero because z 7! log jf .z/ cj is a well-defined (single-valued) function on the range of . The second integral on the rightmost side equals the change in the argument of f c as one traverses .

6.4 Zeros and Poles of Meromorphic Functions

151

Corollary 6.20. Let f be a nonconstant holomorphic function on a neighborhood of c 2 C, ˛ D f .c/, and m D c .f ˛/. Then there exist r > 0 and " > 0 such that for all ˇ 2 C with 0 < jˇ ˛j < ", f ˇ has exactly m simple zeros in 0 < jz cj < r. Proof. Observe that m 1. Choose a positively oriented circle around c such that f ˛ vanishes only at c in cl i. / and f 0 .z/ ¤ 0 for all z 2 cl i. / fcg. If we consider the curve 1 D f ı , it follows from Theorem 6.18 that 1 I.1 ; ˛/ D 2{

Z

f 0 .z/ dz D m: f .z/ ˛

Let w D f .z/ and conclude that for every ˇ not in the range of 1 , we have 1 2{

Z

1 f 0 .z/ dz D f .z/ ˇ 2 {

Z 1

1 dw D I.1 ; ˇ/: wˇ

Now, there exists a ı > 0 such that jf .z/ ˛j ı for all z in range . Hence jˇ ˛j < ı implies that, for all z in range , jf .z/ ˇj D j.f .z/ ˛/ .˛ ˇ/j jf .z/ ˛j j˛ ˇj > 0: Thus f ˇ does not vanish on range for such ˇ. Since I.1 ; ˇ/ is constant on each connected component of the complement of the range of 1 in C, there is an " > 0 such that jˇ ˛j < " < ı ) I.1 ; ˇ/ D m: If ˇ ¤ ˛ and jˇ ˛j < ", then all the zeros of f ˇ in i. / are simple (since f 0 is not zero near, but not necessarily at, c), and therefore f ˇ has m simple zeros in i. / fcg D fz 2 CI 0 < jz cj < rg for some positive value of r. t u Corollary 6.21. A nonconstant holomorphic function is an open mapping. Proof. If f W D ! C is holomorphic on a domain D and is not a constant, we obtain from Corollary 6.20 that for any ˛ in f .D/ there exists > 0 such that U.˛; / f .D/, and the result follows. t u Corollary 6.22. An injective holomorphic function is a diffeomorphism from its domain onto its image. Remark 6.23. Corollary 6.20 characterizes when a holomorphic function f is locally injective: f is injective in a neighborhood of c if and only if m D c .f f .c// D 1. Together with Corollaries 6.21 and 6.22, we see that this condition is equivalent to being a local diffeomorphism. In particular, in the case that m D 1, f defines a bi-holomorphic map (a bijective holomorphic function; the local inverse of f is also holomorphic) between neighborhoods of c and f .c/. See also the discussion in the next section.

152

6 Cauchy Theory: Local Behavior and Singularities

Theorem 6.24 (Rouch´e’s Theorem). Let f and g be holomorphic functions on a domain D. Let be a positively oriented Jordan curve with cl i. / contained in D, and assume that jf j > jgj on range . Then Zf Cg D Zf , where Zf denotes the number of zeros of f in i. /. Proof. It follows from Theorem 6.18 that Zf D I.f ı ; 0/. Now apply Theorem 4.60 with 1 D g ı and 2 D f ı . t u Example 6.25. Rouch´e’s theorem is very useful in locating zeros of a holomorphic function, as this example shows. Let h.z/ D z5 C z4 C 6z C 1. Then ˇ4 ˇ ˇ ˇ ˇz C 6z C 1ˇ 29 < 32 D ˇz5 ˇ for all jzj D 2, and

ˇ ˇ5 ˇz C z4 C 1ˇ 3 < 6 D j6zj

for all jzj D 1. Therefore h has its five zeros contained in jzj < 2, and four of them are contained in fz 2 CI 1 < jzj < 2g. Theorem 6.26 (Integral Formula for the Inverse Function). Let R > 0. Suppose f is holomorphic on jzj < R, f .0/ D 0, f 0 .z/ ¤ 0 for jzj < R, and f .z/ ¤ 0 for 0 < jzj < R. For 0 < r < R, let r be the positively oriented circle of radius r about 0, and let m D min jf j on r . Then Z 1 tf 0 .t/ dt g.w/ D 2{ r f .t/ w defines a holomorphic function in jwj < m with f .g.w// D w on jwj < m and g.f .z// D z for z 2 i.r / \ f 1 .jwj < m/: Proof. Observe that m > 0, and fix w0 with jw0 j < m. Then on the circle r we have jf .z/j m > jw0 j : Thus f and .f w0 / have the same number of zeros in i.r /, by Rouch´e’s theorem, and hence f .z/ w0 D 0 has a unique solution z0 in i.r /. Therefore it suffices to show the following: (1) g.w0 / D z0 if f .z0 / D w0 . (2) g is a holomorphic function on the disc jwj < m. To verify (1), note that it follows from the residue theorem that g.w0 / D Res.F .s/ ds; z0 /; where F .s/ D

s f 0 .s/ for jsj < R: f .s/ w0

6.5 Local Properties of Holomorphic Maps

153

Thus g.w0 / D lim .s z0 / s!z0

D lim

s!z0

s f 0 .s/ f .s/ f .z0 /

.2s z0 /f 0 .s/ C .s 2 sz0 /f 00 .s/ f 0 .s/

D z0 ; where the second equality follows from l’Hopital’s rule (see Exercise 3.25). Alternatively, to avoid the use of l’Hopital’s rule, we change the previous series of equalities to g.w0 / D lim .s z0 / s!z0

D lim .s z0 / s!z0

s f 0 .s/ f .s/ f .z0 / sf 0 .s/ 00 .s z0 / f 0 .z0 / C f 2.z0 / .s z0 / C

D z0 : To show (2) we note that jf .t/j > jwj on r and hence 1 D f .t/ w

Thus

ˇZ ˇ Since ˇˇ

1 w n 1 X

D : w f .t/ nD0 f .t/ f .t/ 1 f .t/ 1

Z 1 1 X n t f 0 .t/ g.w/ D w dt: nC1 2{ nD0 r f .t/

ˇ ˇ t f 0 .t/ ˇ M for some constant M that is independent of n, the dt ˇ mnC1 nC1 r f .t/ last power series has radius of convergence m. t u

6.5 Local Properties of Holomorphic Maps In this section we describe the behavior of an analytic function f near any point z0 in its domain of definition D, using results from the previous section. We use the following standard notation: z D x C { y; w D s C { t D f .z/ D u.x; y/ C { v.x; y/ for z 2 D:

154

6 Cauchy Theory: Local Behavior and Singularities

Proposition 6.27. Let D be a domain in C, z0 a point in D, and f a function holomorphic on D. Then the following properties hold: (1) If f 0 .z0 / ¤ 0, then f defines a homeomorphism of some neighborhood of z0 onto some neighborhood of f .z0 /. (2) If there exists n 2 Z1 such that 0 D f 0 .z0 / D D f .n/ .z0 / and f .nC1/ .z0 / ¤ 0; then f is n C 1 to 1 near z0 . (3) If f 0 .z0 / ¤ 0, then angles between tangent vectors to curves at z0 are preserved, and infinitesimal lengths at z0 are multiplied by jf 0 .z0 /j.3 More generally, if 0 D f 0 .z0 / D D f n .z0 / and f .nC1/.z0 / ¤ 0 for some n in Z1 , then angles between tangent vectors to curves at z0 are multiplied by n C 1. (4) Conversely, if g 2 C1 .D/ preserves angles, then g 2 H.D/. (5) The change in infinitesimal areas is given by multiplication by jf 0 .z0 /j2 . Proof. Let w0 D f .z0 /. We proceed to establish the various parts of the theorem. (1) The condition implies that z0 .f .z/ w0 / D 1, and it follows from Corollary 6.20 that there exist r > 0 and " > 0 such that for all w 2 C with 0 < jw w0 j < ", f w0 has exactly one simple zero in 0 < jz z0 j < r. In other words, f is injective near z0 . Now use Corollary 6.22 to conclude. (2) Let g.z/ D f .z/ w0 . It is enough to prove that g is n C 1 to 1 near z0 . But g.z0 / D 0 D g 0 .z0 / D D g .n/ .z0 /; and g .nC1/ .z0 / ¤ 0; and therefore, for jz z0 j small, we may write g.z/ D

X

am .z z0 /m .where anC1 ¤ 0/

mnC1 1 X

D .z z0 /nC1

akCnC1 .z z0 /k

kD0

D .z z0 /

nC1

.h.z//nC1

D .g1 .z//nC1 ; where h and g1 are holomorphic functions near z0 , g1 .z0 / D 0, and .g1 /0 .z0 / D h.z0 / ¤ 0, since .h.z0 //nC1 D anC1 . The existence of h is a consequence of

3

Both claims are stated in standard shorthand form. The first statement is reformulated in the proof that follows; the second can be reformulated as jf .z/ f .z0 /j D jf 0 .z0 /j jz z0 j C .z/, where limz!z0 .z/ D 0.

6.5 Local Properties of Holomorphic Maps

155

Exercise 5.1. By (1), g1 is a homeomorphism from a neighborhood of z0 to a neighborhood of 0. Since p.z/ D znC1 is clearly nC1 to 1 from a neighborhood of 0 to a neighborhood of 0, and since g D p ı g1 , it follows that g is n C 1 to 1 from a neighborhood of z0 to a neighborhood of g.z0 / D 0, as claimed. (3) Let us write f .z/ f .z0 / w : D lim f 0 .z0 / D lim z!z0 z!z0 z z z0 Assume first that f 0 .z0 / ¤ 0. Then f 0 .z0 / D e{ ; for some > 0; 2 R: If z W Œ0; 1 ! D is a C1 -curve with z.0/ D z0 and z0 .0/ ¤ 0, then w D f ı z is a C1 -curve with w.0/ D w0 , w0 .0/ ¤ 0. Furthermore, if we denote z D z z0 w (for z close to but different from z0 ) and w D f .z/ w0 , then w D z z implies that w arg w D arg z C arg ; z which together with lim arg

z!z0

imply that

w D z

arg w0 .0/ D arg z0 .0/ C arg f 0 .z0 /:

All uses of the multi-valued arg function need to be interpreted appropriately; we leave it to the reader to do so. The assertion about lengths means that the ratio of the length of w to the length of z tends to jf 0 .z0 /j as z tends to z0 . This follows immediately from ˇ ˇ ˇ w ˇ ˇ ˇ D : lim z!z0 ˇ z ˇ The argument for the case with vanishing derivative is almost identical to the one used to establish (2) and is hence left to the reader. (4) Let z W Œ0; 1 ! D be a C1 -curve with z0 .t/ ¤ 0 for all t. Then w D g ı z W Œ0; 1 ! g.D/ is also a C1 -curve and w0 .t/ D gz z0 .t/ C gz z0 .t/: Since g preserves angles, arg

w0 .t/ must be independent of arg z0 .t/. But z0 .t/

156

6 Cauchy Theory: Local Behavior and Singularities

w0 .t/ z0 .t/ D gz C gz 0 ; 0 z .t/ z .t/ and therefore gz 0. (5) We compute the Jacobian of the holomorphic map f at z0 : ˇ ˇ ˇ ux .z0 / vy .z0 / ˇ ˇ ˇ D ux .z0 /vy .z0 / uy .z0 /vx .z0 / J.f /.z0 / D ˇ vx .z0 / vy .z0 / ˇ ˇ ˇ2 D u2x .z0 / C v2x .z0 / D ˇf 0 .z0 /ˇ : t u Remark 6.28. The above property (2) of holomorphic mappings is also a consequence of Corollary 6.20. Much of the above discussion for (1) and (2), as well as the next corollary, are slight amplifications of the material in the previous section. We have collected them in Proposition 6.27 together with (3), (4), and (5) so as to have the complete local behavior description in one place. Corollary 6.29. A holomorphic function f is injective near a point z0 in its domain if and only if f 0 .z0 / ¤ 0 if and only if f is a homeomorphism near z0 .

6.6 Evaluation of Definite Integrals The residue theorem is a powerful tool for the evaluation of many definite integrals. We illustrate this with a few examples. For the third example, the reader might want to look at the appendix to the chapter. 1. The first integral to be evaluated is Z

1 1

x4

1 dx: C1 Z

This method will work for the evaluation of integrals of the form

C1

F .x/

1 satisfying 1 .F /

dx, where F is a rational function with no singularities on R 2. 1 dz. To apply the residue We will obviously want to integrate F .z/ dz D 4 z C1 theorem we must carefully choose the path of integration. Let R > 1; choose R to be the portion on R from R to CR followed by the upper half of the circle jzj D R, as in Fig. 6.1. Since {

z4 C 1 D .z e 4 /.z e

3{ 4

{

/.z C e 4 /.z C e

3{ 4

/;

6.6 Evaluation of Definite Integrals

157 γR

−R

R

Fig. 6.1 The path of integration for first example

the function F has (possibly) nonzero residues only at these four roots of unity, and we conclude from the residue theorem that Z 1 { 3{ 4 4 C Res F; e : dz D 2{ Res F; e 4 R z C 1 The residues are easy to compute: ˇ ˇ ˇ Res.F; e / D { ˇ 2 .z C {/.z C e 4 / 1

{ 4

and Res.F; e

3{ 4

{ zDe 4

D

1 p p 2. 2 { 2/

ˇ ˇ 1 ˇ /D D p p : 3{ ˇ 2. 2 C { 2/ .z2 {/.z C e 4 / ˇzDe 3{ 4 1

Next we estimate the absolute value of the integral over the semicircle fzI =z 0; jzj D Rg: ˇZ ˇ ˇ ˇ

0

ˇ ˇ R R { e{ d ˇˇ 4 ! 0 as R ! C1: 4 4 { R e C1 R 1

p 1 2 dx D . 4C1 x 2 1 2. A second class of integrals that can be evaluated by the residue theorem consists of those of the form Z 2 I D Q.cos ; sin / d; Z

1

We conclude that

0

where Q is a rational function of two variables with no singularities on the unit circle S 1 D fzI jzj D 1g. To apply the residue theorem, we express I as an integral of a holomorphic function over the unit circle. We use the change of variables z D ei ; from where dz D e{ { d D { z d

158

6 Cauchy Theory: Local Behavior and Singularities

and cos D

z C z1 e{ e{ z z1 e{ C e{ D ; sin D D : 2 2 2{ 2{

Example 6.30. Let 0 < b < a and evaluate Z 2 Z 1 I D d D a C b cos 0 jzjD1 Z

1 dz 1 1 .{ z/ a C b z C z 2

2 { dz 2 C 2az C b bz jzjD1 X 2 { D 2 { ; z : Res bz2 C 2az C b

D

jzj0 . It is this discontinuity that is the key to the method used here. • By the residue theorem 1 2{

Z F .z/dz D Res .F .z/dz; {/ C Res .F .z/dz; {/

e{ 2 ˛ 1 e{ 2 ˛ 3 C D { e{ 2 ˛ e{ 2 ˛ : 2{ 2{ 2

D

3

162

6 Cauchy Theory: Local Behavior and Singularities

• The above integral is a sum: Z

Z F .z/dz D

R

r ˛ e{0 ˛ d.re{0 / C 1 C r 2 e2{0

Z

20 0

R˛ e{˛ d.Re{ / 1 C R2 e2{

Z 0 r e ˛ e{˛ {0 d.re / C d.e{ / C 2 2{0 2 2{ 20 1 C e R 1Cr e Z R Z 20 r˛ e{.˛C1/ 1C˛ De{0 .˛C1/ dr C {R d 2 2{0 1 C R2 e2{ 0 1Cr e Z R r˛ dr e{.20 /.1C˛/ 2 2{0 1Cr e Z 20 {.1C˛/ e 1C˛ d: { 1 C 2 e2{ 0 Z

˛ {.20 /˛

• We need some estimates: ˇ ˇ Z 20 ˇ ˇ e{.˛C1/ 1 ˇ ˇ 1C˛ 2 ! 0 as R ! 1; d ˇ R1C˛ 2 ˇR 2 e2{ ˇ ˇ 1 C R R 1 0 ˇ ˇ Z 20 {.˛C1/ ˇ ˇ e 1 ˇ 1C˛ ˇ d 2 ! 0 as ! 0: ˇ ˇ 1C˛ 2 2{ ˇ ˇ 1C e 1 2 0 R • The value of F .z/dz is independent of the choices of , 0 , and R. We can hence let the first two of these constants approach zero and the last approach infinity to obtain Z 1 Z 1 r ˛ r˛ 1 {˛ { 3 ˛ 2{.1C˛/ 2 2 D e dr e dr: 2{ { e 2 1 C r2 1 C r2 0 0 • The last identity yields at once that Z

1 0

e{ 2 ˛ x˛ dx D ; 1 C x2 1 C e{ ˛

a real number, as expected.

6.7 Appendix: Cauchy Principal Value We have studied integration of holomorphic forms ! along paths . For some applications it is useful to allow ! to have some singularities, simple poles in our case, on . We describe a path toward this goal of evaluating integrals of differential forms with some singularities.

6.7 Appendix: Cauchy Principal Value

163

Fig. 6.4 A modification of a Jordan path

Definition 6.31. We consider an oriented pdp C, and a differential form ! D f .z/ dz that is holomorphic in a neighborhood D of Rexcept for isolated singularities on the range of that are simple poles. Because ! is not defined, Rwe want to R introduce two paths ı and d;ı , the second is disconnected, for which ! and ı d;ı ! are defined and finite. Let w1 ; w2 ; : : : ; wm be the set of singularities of ! on . Let 2 R>0 be the minimum of the finite set consisting of one half the distances between the various wk and the radii of the largest discs about these points contained in D. Let 0 < ı < . For each k, let Ck be the positively oriented circle of radius ı and center wk . This circle intersects in a finite set of points. Let aı;k be the last point before wk in this intersection and bı;k the first point after wk . These two points define two arcs on Ck . Choose one of these arcs; call it ˛ı;k , the arc subtended at wk , and give it the orientation consistent with that of . Let 2 < ˛ı;k < 2 be the angle of this arc measured from aı;k . The ı-modification ı of the curve is obtained by replacing for each k the segment of between aı;k and bı;k by the arc ˛ı;k . The disconnected ı-modification d;ı of the curve is obtained by removing for each k the segment of between aı;k and bı;k . See Figs. 6.4 and 6.5. We define the principal value Z Z pr. v. ! D lim !;

e

e

ı!0 d;ı

provided the limit exists. Remark 6.32. Each of the sets faı;k g, fbı;k g, and f˛ı;k g is bounded. Hence we can construct a sequence fın g that converges to zero with the property that all three sequences faın ;k g, fbın ;k g, and f˛ın ;k g have limits denoted by ak , bk ˛k , respectively. Once these limits are known to exist, it is easy to see that the three nets4 faı;k g, fbı;k g, and f˛ı;k g also converge to the appropriate limits.

4

This topological concept and its properties are not discussed here.

164

6 Cauchy Theory: Local Behavior and Singularities

Fig. 6.5 The modification localized (after a translation and rotation)

Lemma 6.33. Under the hypothesis of the last definition, there exists a constant M > 0 independent of ı such that Z

Z !

! D {

ı

d;ı

m X

˛ı;k Res.!; wk / C ı.ı/;

kD1

where j.ı/j < M . Hence !

lim

ı!0

!

Z

Z ı

! d;ı

D {

m X

˛k Res.!; wk /:

(6.4)

kD1

Proof. Only the first identity needs verification. Fix k. By a translation and rotation, we may assume that wk D 0 and ak 2 R0 . How many zeros does the function g.z/ D Œf .z/ eŒf .z/ have? 6.10. Evaluate

Z jzjD3

f 0 .z/ dz; f .z/ 1

z3 where f .z/ D 2 2z C z2 C . 81 6.11. Suppose f is holomorphic for jzj < 1 and f What can be said about f 000 .0/?

7 1 D 3 for n D 2; 3; : : : n n

Exercises

167 23

6.12. Let f be an entire function such that jf .z/j jzj 3 for all jzj > 10. Compute f .8/ .10:001/. 6.13. Evaluate the following real integrals using residues: Z

1

1

Z

cos x dx ; 1 C x2

1 1

6.14. Find all Laurent series of the form

1 X

sin x dx: 1 C x2

an zn for the functions

1

• f .z/ D • f .z/ D

z2 ; .1 z/2 .1 C z/

1 ; .z 1/.z 2/.z 3/

and • f .z/ D

2 z2 : z.1 z/.2 z/

6.15. If f is entire and satisfies jf 00 .z/ 3j 0:001 for all z 2 C, f .0/ D 0, f .1/ D 2, f .1/ D 4, what is f .{/? 6.16. If f is an entire function such that 2 for all z 2 C and f .{/ D {C2, what is f .{/? 6.17. If f is holomorphic on 0 < jzj < 2 and satisfies f . n1 / D n2 and f . n1 / D n3 for all n 2 Z>0 , what kind of singularity does f have at 0? 6.18. Let D be a bounded domain in C with smooth boundary. Assume f is a nonconstant holomorphic function in a neighborhood of the closure of D such that jf j is constant on @D, say, jf j D c on @D. Show that f takes on each value d such that jd j < c at least once in D. 6.19. Suppose f is holomorphic in a neighborhood of the closure of the unit disc. Show that for jzj 1 f .z/.1 jzj2 / D

1 2{

Z j jD1

1 zN

f . / d ;

z

and conclude that the following inequality holds: jf .z/j .1 jzj2 /

1 2

Z 0

2

ˇ ˇ ˇf .exp{ /ˇ d:

168

6 Cauchy Theory: Local Behavior and Singularities

6.20. Let f be an entire function. Suppose there exist positive constants A and B such that jf .z/j A C B jzj10 for all z 2 C. Show that f is a polynomial. What is its degree? 6.21. Suppose f is meromorphic in a neighborhood of the closed unit disc and that jf .z/j D 1 for jzj D 1. Find the most general such function. 6.22. Let C denote the positively oriented unit circle. Consider the function 21 2 z26 1 3 1 2 f .z/ D C exp z z z : 81 2 3 Evaluate the following integrals: Z

Z

f 0 .z/ dz ;

f .z/ dz ; C

C

Z C

f 0 .z/ dz: f .z/

1 1 2 Dn ,f D 6.23. If f is holomorphic for 0 < jzj < 1 and satisfies f n n 2 2n for n D 2; 3; 4; : : :, what can you say about f ? 6.24. Suppose f is entire and f .z/ ¤ t 2 for all z 2 C and for all t 2 R. Show that f is constant. 1 6.25. If f is holomorphic for 0 < jzj < 1 and satisfies f D n2 and n 1 D 2n2 , n D 2; 3; 4; : : :, find lim inf jf .z/ 2j. f z!0 2 Z 1 sin2 2x dx using residues. 6.26. Find x2 0 6.27. Prove the following extension of the maximum modulus principle. Let f be holomorphic ˇ andˇ bounded on jzj < 1, and continuous on jzj 1 except maybe at z D 1. If ˇf .e{ /ˇ A for 0 < < 2 then jf .z/j A for all jzj < 1. 6.28. Let D denote the unit disc and let ffn g be a sequence of holomorphic functions in D such that lim fn D f uniformly on compact subsets of D. Suppose that each n!1

fn takes on the value 0 at most seven times on D (counted with multiplicity). Prove that either f 0 or f takes on the value 0 at most seven times on D (counted with multiplicity). 6.29. Show that the function f .z/ D z C 2z2 C 3z3 C 4z4 C is injective in the unit disc D D fz 2 CI jzj < 1g. Find f .D/. 6.30. Suppose f is a nonconstant function holomorphic on the unit disc fz 2 CI jzj < 1g and continuous on fz 2 CI jzj 1g such that for all 2 R, the

Exercises

169

value f .e{ / is on the boundary of the triangle with vertices 0, 1, and {. Is there a 1 z0 with jz0 j < 1 such that f .z0 / D 10 .1 C {/? Is there a z0 with jz0 j < 1 such that 1 f .z0 / D 2 .1 C {/? 6.31. Is there a function f holomorphic for jzj < 1 and continuous for jzj 1 that satisfies f .e{ / D cos C 2{ sin ; for all 2 R‹ R1 2 6.32. The integral I D 0 ex dx can be evaluated using both real and complex analytic methods. R1 R1 2 2 1. By working with 0 ex dx 0 ey dy and Fubini’s theorem, evaluate I . 2. Evaluate I using residues by considering a change of variables t D x 2 and R1 2 evaluating 0 ex cos 2mx dx by integrating along a proper closed curve. 6.33. Formulate and prove forms of the residue theorem and the argument principle for domains in the sphere C [ f1g. 6.34. Describe the type of singularity at 1 for the following functions f W C¤0 !C: 1 (a) f .z/ D exp z n (b) f .z/ D z , n 2 Z sin.z/ (c) f .z/ D z 6.35. Prove that the inverse function of an injective entire function f cannot be entire unless f is a polynomial of degree one.

Chapter 7

Sequences and Series of Holomorphic Functions

We now turn from the study of a single holomorphic function to the study of collections of holomorphic functions. In the first section we will see that under the appropriate notion of convergence of a sequence of holomorphic functions, the limit function inherits several properties that the approximating functions have, such as being holomorphic. In the second section we show that the space of holomorphic functions on a domain can be given the structure of a complete metric space. We then apply these ideas and results to obtain, as an illustrative example, a series expansion for the cotangent function. In the fourth section we characterize the compact subsets of the space of holomorphic functions on a domain. This powerful characterization is used in Sect. 7.5, to study approximations of holomorphic functions and, in particular, to prove Runge’s theorem, which describes conditions under which a holomorphic function can be approximated by rational functions with prescribed poles. The characterization will also be used in Chap. 8 to prove the Riemann mapping theorem.

7.1 Consequences of Uniform Convergence on Compact Sets We begin by recalling some notation and introducing some new symbols. Let D be a domain in C; we denote by C.D/ the vector space of continuous complex-valued functions on D, and recall that H.D/ C.D/ is the vector space of holomorphic functions on D (see Definition 3.57). We say that a compact disc cl Uz .r/ has rational center if z D x C { y with x and y in Q. Proposition 7.1. A necessary and sufficient condition for a sequence of functions ffn g C.D/ to converge uniformly on all compact subsets of D is for the sequence to converge uniformly on all compact discs with rational centers and rational radii contained in D.

R.E. Rodr´ıguez et al., Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics 245, DOI 10.1007/978-1-4419-7323-8 7, © Springer Science+Business Media New York 2013

171

172

7 Sequences and Series of Holomorphic Functions

Proof. Every compact set contained in D can be covered by finitely many such discs. t u It is clear that if a sequence of functions ffn g C.D/ converges uniformly to a function f on all compact subsets of D, then for all z in D we have pointwise convergence: lim fn .z/ D f .z/. The converse is not true: uniform convergence on n!1 all compact subsets of D is stronger than pointwise convergence. To see this observe that we know from Theorem 2.23 that if a sequence of functions ffn g C.D/ converges uniformly to a function f on all compact subsets of D, then f 2 C.D/. On the other hand, it is easy to construct an example of a sequence of continuous functions converging at every point of the domain to a discontinuous function (see Exercise 7.1). We proceed to describe some consequences of uniform convergence on all compact subsets of D, also called locally uniform convergence, for H.D/. The first of these is that H.D/ is closed under locally uniform convergence. Theorem 7.2. If ffn g H.D/ and ffn g converges uniformly on all compact subsets of D, then the limit function f is holomorphic on D. Proof. We already know that f 2 C.D/. Let be any closed curve homotopic to a point in D. Then, by Cauchy’s theorem, Z fn .z/ dz D 0:

By uniform convergence it follows that Z

Z f .z/ dz D lim

N !1

fn .z/ dz D 0;

and then, by Morera’s theorem, f is holomorphic on D. Corollary 7.3. If ffn g H.D/ and

1 X

t u

fn converges uniformly on all compact

nD1

subsets of D, then the limit function (also denoted by

1 X

fn / is holomorphic on D.

nD1

Theorem 7.2 has no analog in real variables: it is easy to see (at least pictorially) that the absolute value function on R, which has no derivative at 0, can be uniformly approximated by differentiable functions (see Exercise 7.1). A more extreme example was constructed by Weierstrass, that of a continuous function defined on Œ0; 1 which is nowhere differentiable and uniformly approximated by polynomials. We will shortly see that uniform convergence of a sequence of holomorphic functions on all compact subsets of their common domain of definition implies uniform convergence of the derivatives on the same sets. This is another feature

7.1 Consequences of Uniform Convergence

173

of holomorphic functions not shared by real differentiable functions: it is easy to construct a sequence of differentiable functions converging uniformly on a closed interval with the property that the sequence of derivatives does not converge uniformly there. We leave this construction to the reader as Exercise 7.2. Theorem 7.4. If ffn g H.D/ and fn ! f uniformly on all compact subsets of D, then fn0 ! f 0 uniformly on all compact subsets of D. Proof. Since f 2 H.D/, it is enough to check uniform convergence of the derivatives on all compact subdiscs R D, with @R D positively oriented. For z 2 i. /, we have 1 f .z/ D 2{ 0

Z

f .w/ 1 dw D lim n!1 2{ .w z/2

Z

fn .w/ dw D lim fn0 .z/I n!1 .w z/2

this convergence is uniform in any smaller compact subdisc, such as e D fz 2 i. /I inffjz wj I w 2 @Rg ı > 0g; R t u

with ı sufficiently small.

Theorem 7.5. Let ffn g be a sequence of holomorphic functions on D such that fn ! f uniformly on all compact subsets of D. If fn .z/ ¤ 0 for all z 2 D and all n 2 Z>0 , then either (a) f is identically zero, or (b) f .z/ ¤ 0 for all z 2 D. Proof. Assume that there is c 2 D with f .c/ D 0 and that f is not identically zero. Then there exists a circle with center c such that cl i. / D and f .z/ ¤ 0 for all z 2 cl i. / fcg. By the argument principle, the number N of zeros of f in i. / is given by Z 1 f 0 .z/ N D dz 1: 2{ f .z/ But Z

f 0 .z/ dz D lim n!1 f .z/

Z

fn0 .z/ dz D 0: fn .z/

t u

Remark 7.6. An equivalent formulation for this theorem is the following, sometimes referred to as Hurwitz’s theorem. Theorem 7.7. Let ffn g be a sequence of holomorphic functions on D such that fn ! f uniformly on all compact subsets of D, and assume f is not identically zero on D. For every disc U such that cl U D with the property that f ¤ 0 on @U , there exists N 2 Z>0 such that f and fn have the same number of zeros in U for all n N .

174

7 Sequences and Series of Holomorphic Functions

Definition 7.8. Let f 2 H.D/. We call f simple, univalent, or schlicht if it is one-to-one (injective) on D; thus a homeomorphism onto f .D/. Theorem 7.9 (Hurwitz). Assume D is a domain in C. If ffn g is a sequence in H.D/ with fn ! f uniformly on all compact subsets of D and fn is schlicht for each n, then either f is constant or schlicht. Proof. Assume that f is neither constant nor schlicht; thus, in particular, there exist z1 and z2 in D with z1 ¤ z2 and f .z1 / D f .z2 /. For each n 2 Z>0 , set gn .z/ D fn .z/ fn .z2 / on the domain D 0 D D fz2 g. Then gn 2 H.D 0 /, gn never vanishes in D 0 and gn ! g D f f .z2 / uniformly on all compact subsets of D 0 . But g is not identically zero and vanishes at z1 ; we have thus obtained a contradiction to Theorem 7.5. t u

7.2 A Metric on C.D/ We introduce, for use in the proof of the compactness theorem of this chapter and in the proof of the Riemann mapping theorem in Chap. 8, a metric on C.D/ for any domain D in C. The metric on C.D/ will have the property that convergence in the -metric is equivalent to uniform convergence on all compact subsets of D. For K compact in D and f 2 C.D/, set jjf jjK D maxfjf .z/j I z 2 Kg: Consider the set of compact (closed) discs contained in D with rational radii and rational centers. There are countably many such discs and they cover D. Call this collection of discs fDi gi 2Z>0 . For n 2 Z>0 , let (1) Kn D

[

Di ;

i n

then fKn g is an exhaustion of D; that is, (2) Each Kn is compact; (3) Kn KnC1 for all n 2 Z>0 ; and (4)

[

int Kn D D;

n2Z>0

where int K denotes the interior of the set K.

7.2 A Metric on C.D/

175

From now on, we shall use only properties (2), (3), and (4) of our exhaustion and not how these sets were constructed. Remark 7.10. A crucial consequence of these properties that we will use often is that given an exhaustion fKn g of D, each compact subset K of D is contained in Kn for some n. For f 2 C.D/ and i 2 Z>0 , we set Mi .f / D jjf jjKi ; and note that Mi C1 Mi : We define d.f / D

1 X

2i min.1; Mi .f //

i D1

1 X

2i D 1:

(7.1)

i D1

7.2.1 Properties of d For all f and g 2 C.D/: (1) (2) (3) (4)

d.f / 0, and d.f / D 0 if and only if f 0 d.f C g/ d.f / C d.g/ For each i , 2i min.1; Mi .f // d.f / For each i , d.f / Mi .f / C 2i

Proof. Properties (1) and (3) are immediate from the definition of d . To prove (2), observe that d.f C g/ D

1 X

2i min.1; Mi .f C g//

i D1

1 X

2i min.1; Mi .f / C Mi .g//

i D1

1 X

2i Œmin.1; Mi .f // C min.1; Mi .g//

i D1

D d.f / C d.g/: For property (4) d.f / D

X j i

2j min.1; Mj .f // C

X j >i

2j min.1; Mj .f //

176

7 Sequences and Series of Holomorphic Functions

X

2j Mj .f / C

j i

0 @

X

1

X

2j

j >i

2j A Mi .f / C 2i

j i

Mi .f / C 2i :

t u

Finally, we define the metric on C.D/ we have been seeking: .f; g/ D d.f g/:

(7.2)

We list some immediate properties of ; only one of them requires proof.

7.2.2 Properties of For all f , g, and h in C.D/, the following hold: (1) (2) (3) (4) (5)

.f; g/ 0 and .f; g/ D 0 if and only if f D g .f; g/ D .g; f / .f; g/ .f; h/ C .h; g/ .f C h; g C h/ D .f; g/; that is, is translation invariant .f; g/ 1; that is, is a bounded metric

Proof of property (3) .f; g/ D d.f g/ D d.f h C h g/ d.f h/ C d.h g/: t u Note that properties (1)–(3) say that is a metric on C.D/. Theorem 7.11. Convergence in the -metric in C.D/ is equivalent to uniform convergence on all compact subsets of D. Proof. Let ffn g C.D/ and assume that ffn g is -convergent. Since for every compact set K D there is an i in Z>0 such that K Ki , it suffices to show uniform convergence on Ki for each i . Given 0 < < 1, we can choose N 2 Z>0 large so that .fm ; fn / D d.fm fn / d.fm f / C d.f fn / < 2i for all m; n N .

7.2 A Metric on C.D/

177

Now 2i min.1; Mi .fm fn // d.fm fn / < 2i ; and thus Mi .fm fn / < < 1I that is, jjfm fn jjKi < : The above inequality implies that the sequence ffn g converges uniformly on Ki . If 1, then use 0 D 34 and proceed as above. We have actually shown more than claimed: if ffn g is a -Cauchy sequence in C.D/, then there exists an f 2 C.D/ such that fn ! f uniformly on all compact subsets of D. Conversely, assume that fn ! f uniformly on Ki for all i . Thus lim Mi .f fn / D 0 for all i:

n!1

Given > 0, first choose i such that 2i < and next choose N such that Mi .f 2 fn / < for all n N . Then 2 d.f fn / Mi .f fn / C 2i

0 of D. Corollary 7.13. is a complete metric on C.D/. Because of Theorem 7.11, we can reformulate the results of the previous section in terms of the metric . In particular, Theorems 7.2 and 7.4 can now be phrased as in the following corollary. We already remarked that H.D/ C.D/. We let jH.D/ denote the restriction of the metric to H.D/. Corollary 7.14. H.D/ is a closed subspace of .C.D/; /. As such, .H.D/; jH.D/ / is a complete metric space. Furthermore, f 7! f 0 is a continuous linear operator from H.D/ to itself. There is an alternate description of the topology induced by , that is, the topology of the metric space .C.D/; /. Namely, Definition 7.15. Given f 2 C.D/, K compact D and > 0, we define Nf ./ D fg 2 C.D/I .g; f / < g

178

7 Sequences and Series of Holomorphic Functions

and Vf .K; / D fg 2 C.D/I jjg f jjK < g: Remark 7.16. For any f 2 C.D/, a basis for the neighborhood system at f (with respect to the topology induced by on C.D/) is given by the sets Nf ./, with > 0. That is, the open sets U C.D/ that contain f are precisely those for which there exists > 0 such that Nf ./ U . We will now show that the same is true for the collection fVf .K; /I K compact D and > 0g: Theorem 7.17. For any f 2 C.D/, a basis for the neighborhood system at f (with respect to the topology induced by on C.D/) is given by the sets Vf .K; /. Proof. It is enough to show that (1) Given Vf .K; /, there exists an Nf .ı/ Vf .K; /, and (2) Given Nf .ı/, there exists a Vf .K; / Nf .ı/. To show (1), we assume without loss of generality that 0 < < 1. Choose i such that K Ki and set ı D 2i . If g 2 Nf .ı/, then d.g f / < 2i . Thus 2i min.1; Mi .g f // < 2i and then Mi .g f / D jjg f jjKi < : But jjg f jjK jjg f jjKi I that is, g 2 Vf .K; /. To show (2), choose i such that 2

i

ı < . For g 2 Vf 2

Mi .g f /

0, there exists N D N.K; / in Z>0 such that jjf fn jjK < for all n > N . We can apply these concepts to convergence of series of meromorphic functions.

7.3 The Cotangent Function

179

Definition 7.19.PLet ffn g be a sequence in M.D/, the meromorphic functions on D. We say that fn converges uniformly (absolutely) on a subset A of D if there 1 X fn exists an integer N such that fn is holomorphic on A for all n > N and N C1

converges uniformly (absolutely) on A.

P Theorem 7.20. Let ffn g M.D/. fn converges uniformly on all compact P If P subsets of D, then the series f D fn is a meromorphic function on D, and fn0 converges uniformly on all compact subsets to f 0 . t u

Proof. The proof is trivial.

7.3 The Cotangent Function As an application of the ideas developed in the last two sections, we establish a series expansion formula for the cotangent function. Theorem 7.21. For all z in C Z the following equalities hold: cot z D

1 cos z D C sin z z

X n2Z; n6D0

1 1 C zn n

1 X 1 1 D C 2z : z z2 n2 nD1

(7.3)

We first observe that the meromorphic function F .z/ D

cos z sin z

has its poles at the integers, and that each of these poles is simple with residue equal 1 X 1 , but this one does to 1. It would seem more natural to sum the series zn nD1 not converge (Exercise 7.5). X X 1 1 z C D converges absolutely and We claim that zn n n.z n/ n2Z; n6D0

n¤0

uniformly on all compact subsets of C. To verify this claim, assume that jzj R with R > 0. Then X jnj2R

X R jzj jnj jn zj jnj .jnj R/ jnj2R

180

7 Sequences and Series of Holomorphic Functions

(N+ 12 ) (1+ı) z

(N+ 12 ) (−1−ı)

Fig. 7.1 The square CN

X jnj2R

X 1 R ˇ n ˇ 2R < C1: ˇ ˇ jnj 2 jnj2 n¤0

We can now verify the expansion (7.3) for cot z. Proof of Theorem 7.21. For >0 , let CN be the positively oriented boundary N 2Z of the square with vertices N C 12 .˙1 ˙ {/; see Fig. 7.1. Then Z X 1 cot t cot t dt D ;t : Res 2{ CN t z t z t 2 i.CN /

Here z 2 C is fixed: we take z 2 i.CN / and z … Z. The poles of the function H.t/ D cot t occur at t D z and at t D n 2 Z, and they are all simple. Furthermore, we t z see that cot t Res ; z D cot z t z and

1 cot t cos t 1 ; n D lim .t n/ D : Res t !n t z sin t t z .n z/

Thus we have 1 2{

Z CN

N 1 X 1 cot t dt D cot z C t z nDN n z " # N 1 X 1 1 1 D cot z C ; nDN n z n z n¤0

7.3 The Cotangent Function

181

where the last equality holds because N X 1 D 0: n nDN n¤0

Hence it suffices to prove Lemma 7.22. We have

Z lim

N !1 CN

cot t dt D 0: t z

Remark 7.23. Once the lemma is verified, we will also have obtained " # 1 1 1 X 1 1 1 X 2z cot z D ; D z nD1 n z n C z z nD1 n2 z2 where the last series converges uniformly and absolutely on all compact subsets of C Z. Proof of Lemma 7.22. We proceed in stages: (1) 1 2{ As usual, for G.t/ D 1 2{

Z CN

cot t dt D 0: t

cot t , we have t

Z CN

X cot t dt D Res.G.t/dt; t/ t t 2i.CN /

D Res.G.t/dt; 0/ C

N X nDN n¤0

1 : n

The last sum is clearly zero, and the residue of G.t/ dt at zero is 0 because G is an even function. # # " " Z Z Z 1 cot t 1 z (2) dt D dt D dt, cot t cot t t z t t.t z/ CN t z CN CN where the first equality holds by (1). (3) There exists an M > 0 (independent of N ) such that jcot tj M for all t 2 CN : For t D u C {v,

182

7 Sequences and Series of Holomorphic Functions

jcos tj2 D cos2 u C sinh2 v; jsin tj2 D sin2 u C sinh2 v; and thus jcot tj2 D

cos2 u C sinh2 v sin2 u C sinh2 v

On the vertical sides of CN we have u D ˙N ˙

1 2

:

and hence

1 cos2 u D cos2 ˙N ˙ D 0; 2 1 D 1; and sin2 u D sin2 ˙N ˙ 2 jcot tj2 D

sinh2 v 1 C sinh2 v

On the horizontal sides of CN , v D ˙N ˙

1: 1 2

and hence

1 C sinh2 ˙N ˙ 12 ! 1 as N ! 1: jcot tj sinh2 ˙N ˙ 12 2

Thus there exists an M > 0 such that jcot tj M for t on the horizontal sides of CN , and the claim is proved. (4) If we denote by L.CN / the length of CN , then ˇZ ˇ ˇ ˇ

CN

ˇ ˇZ ˇ ˇ cot t ˇˇ ˇˇ z dt ˇ D ˇ dt ˇˇ cot t t z t.t z/ CN Z M jzj jdtj CN jtj jt zj M jzj ˇ ˇ ˇ L.CN / ˇ ˇN C 1 ˇ ˇN C 1 ˇ jzj 2 2 M jzj ˇ ˇ ˇ 4 .2N C 1/ ! 0 as N ! 1: Dˇ ˇN C 1 ˇ ˇN C 1 ˇ jzj 2 2 t u

We have thus established the formula (7.3) for the cotangent function. Differentiating the series (7.3) term by term, we obtain the following expansion.

7.4 Compact Sets in H.D/

183

Corollary 7.24. For all z 2 C Z, 2 2

sin z

D

1 X

1 : .z n/2 nD1

In particular, setting z D 12 , 1 X 2 1 D : 4 .2n 1/2 nD1

7.4 Compact Sets in H.D/ We return to the study of C.D/ with the -metric. A metric space X is compact if and only if every sequence in X has a subsequence which converges to a point in X , and a subset X of Rn is compact if and only if it is closed and bounded, a result we generalize to H.D/. Definition 7.25. Let A C.D/. We say that A is bounded in the strong sense or strongly bounded if for all compact K D and all > 0 there exists a > 0 such that A V0 .K; / D fg 2 C.D/I g D f with jjf jjK < g: Remark 7.26. For a subset A in a metric space .X; /, one defines diam A D supf.f; g/I f and g 2 Ag: Usually one says that A is bounded if diam A < C1. For a bounded metric (as in our case), this concept is not very useful. Hence we have introduced the concept of “strongly bounded sets.” Lemma 7.27. A set A C.D/ is strongly bounded if and only if for each compact K D, there exists an M.K/ > 0 such that jjf jjK M.K/ for all f 2 A; that is, A is strongly bounded if and only if the functions in A are uniformly bounded on each compact subset of D. Proof. We leave the proof as Exercise 7.6.

t u

Theorem 7.28. A compact subset A C.D/ is closed and strongly bounded. Proof. Since A is a compact subset of the metric space C.D/, it is closed. If K D is compact, then the function f 7! jjf jjK is continuous on the compact set A. Thus it is a bounded function, and hence A is strongly bounded. t u

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7 Sequences and Series of Holomorphic Functions

Lemma 7.29. Let c 2 C and D D U.c; R/ for some R > 0. Assume A H.D/ is strongly bounded, and let ffk gk2Z> 0 A. The sequence ffk g converges uniformly .n/ on all compact subsets of D if and only if lim fk .c/ exists (in C) for all integers k!1

n 0.

Proof. If fk ! f uniformly on all compact subsets of D, then for every .n/ nonnegative integer n, fk ! f .n/ uniformly on all compact subsets of D; in .n/ particular, fk .c/ ! f .n/ .c/, as a set consisting of one point is certainly compact. Conversely, it suffices to show that ffk g converges uniformly on cl U.c; r/ with 0 < r < R. Choose r0 such that r < r0 < R. Since A is strongly bounded, there exists an M D M.r0 / such that jfk .z/j M for jz cj r0 and all k > 0: Write fk .z/ D

X

an;k .z c/n for jz cj < RI

n0

then Cauchy’s inequalities (5.3) tell us that jan;k j

M for all n and k: r0n

Assume jz cj r. Then ˇ ˇ1 1 ˇ ˇX X ˇ n nˇ an;k .z c/ an;m .z c/ ˇ jfk .z/ fm .z/j ˇ ˇ ˇ nD0 nD0 ˇ ˇN n 1 ˇ ˇX X r ˇ nˇ ˇ .an;k an;m /.z c/ ˇ C 2M ˇ ˇ r0 nD0 nDN C1 for all N; k; m. Let > 0 and choose N0 2 Z>0 such that 2M

n 1 X r for N > N0 : < r0 2 nDN C1

Finally, choose N1 such that k; m N1 implies ˇN ˇ 0 ˇX ˇ ˇ ˇ ˇ .an;k an;m /.z c/n ˇ < : ˇ ˇ 2 nD0

7.4 Compact Sets in H.D/

185

This will be achieved by requiring, for example, that jan;k an;m j

0 whose closures are contained in D. For each i 2 Z>0 and each n 2 Z0 , consider the mapping ni W H.D/ ! C; ni .f / D f .n/ .zi /I the maps fni g are C-linear and continuous. Given a sequence ffk gk2Z>0 in A, we consider the set of numbers .n/

ni .fk / D fk .zi /: We show that there exists B Z>0 , jBj D 1, such that .n/

lim fk .zi / exists for all n and i:

k2B k!C1

(7.4)

Assertion (7.4) suffices to prove the theorem; for then, by Lemma 7.29, the sequence ffk gk2B converges uniformly on the closed disc cl U.zi ; ri / for each i , which implies that the same sequence converges uniformly on all compact subsets of D. Since A is closed, lim fk 2 A. Thus every sequence in A has a subsequence k2B k!C1

converging to a point of A, and A is hence compact. To establish (7.4), we use the “Cantor diagonalization” method.1

1

This method is often used in analysis.

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7 Sequences and Series of Holomorphic Functions

Since A is strongly bounded, for each i , there exists an M.i / 2 R>0 such that jf .z/j M.i / for all z in cl U.zi ; ri / and all f in A. Thus ˇ M.i / ˇ ˇ ˇ .n/ nŠ: ˇfk .zi /ˇ rin Now (as i and n vary) f 7! ni .f / form a countable set of mappings. Renumber these mappings as f1 ; 2 ; : : : ; m ; : : :g: For m D 1, f1 .fk /gk2Z>0 is a bounded sequence of complex numbers, and therefore there exists a subsequence B1 of Z>0 such that lim 1 .fk / exists: k2B1 k!C1

For m D 2, f2 .fk /gk2B1 f2 .fk /gk2Z>0 is a bounded sequence of complex numbers. Therefore there exists a subsequence B2 of B1 such that lim 2 .fk / exists: k2B2 k!C1

Continue to obtain a nested sequence of sets B1 B2 Bm with the property that lim m .fk / exists:

k2Bm k!C1

At last, diagonalize (justifying the name of the procedure); that is, let B D fn1 ; n2 ; : : : ; nm ; : : : g; where nm is the m-th term of Bm . Then lim m .fk / exists for all m 2 Z>0

k2B k!C1

7.5 Runge’s Theorem

187

because fnm ; nmC1 ; : : :g Bm :

t u

Corollary 7.31. A set A H.D/ is compact if and only if it is closed and bounded in the strong sense. Definition 7.32. A set A H.D/ is relatively compact if cl A is compact. This definition clearly makes sense in much more general settings. Corollary 7.33 (Montel’s Theorem). Every strongly bounded subset of H.D/ is relatively compact. Note that the converse to Montel’s theorem also holds. Definition 7.34. Let A be a strongly bounded set in H.D/ and let ffk gk2Z>0 be a sequence in A. We say that f 2 H.D/ is adherent to ffk gk2Z>0 if it is a limit point of this sequence; that is, if for every " > 0 there exists k 2 Z>0 such that 0 < .f; fk / < ". Remark 7.35. If fxk gk2Z>0 is a sequence in a compact metric space X , then there exists a convergent subsequence of fxk gk2Z>0 ; furthermore, if every subsequence of fxk gk2Z>0 that converges has the same limit, then the sequence fxk gk2Z>0 converges. Theorem 7.36 (Vitali’s Theorem). Let D be a domain in C, and assume that the elements in a sequence ffk gk2Z>0 H.D/ are uniformly bounded on compact subsets of D. Let S D and assume that S has a limit point in D. If lim fk .z/ k!1

exists (pointwise) for all z 2 S , then the sequence ffk g converges uniformly on compact subsets of D. Proof. The assumptions imply that the set A D ffk gk2Z>0 H.D/ is strongly bounded. It follows from Montel’s theorem that its closure is compact, and therefore every subsequence of the sequence ffk g has a converging sub-subsequence. Thus there exists an f in H.D/ adherent to A. Say that f and g are both adherent to ffk g; then f .z/ D lim fk .z/ D g.z/ for z 2 S k!1

and thus f D g on D. By our previous remark, ffk g converges to f in the -metric. t u

7.5 Runge’s Theorem We consider the problem of approximating holomorphic functions by rational functions. We regard a nonconstant polynomial as a rational function whose only pole is at infinity. The ability to uniformly approximate a holomorphic function

188

7 Sequences and Series of Holomorphic Functions

depends on the region where the function is being approximated, as well as upon the function itself. The strongest statement about uniform approximation of holomorphic functions that we prove is Runge’s approximation theorem. A number of proofs appear in the literature; ours is a variant of these.2 We have already proved a form of Runge’s theorem for an open disc : a holomorphic function on has a power series expansion at the center of the disc; for every positive integer n, we obtain a polynomial of degree n by discarding all the higher order terms in the series. These polynomials converge to the function uniformly on any compact subset of the disc. On the other hand, we also know that uniform polynomial approximation does not hold in general. For instance, consider a punctured disc D fz 2 CI 0 < jz cj < Rg, with R > 0 and c arbitrary, and the analytic function on defined by 1 f .z/ D (we take advantage of the fact that it is a rational function whose only zc pole is at c — not in , of course); if f were uniformly approximated by a sequence of polynomials fpn g in the closed annulus K D f0 < r jz cj < Rg, r C exp.2{t/ for 0 t 2 we would obtain the then by taking .t/ D 2 contradiction that Z Z 0 D lim pn .z/ dz D f .z/ dz D 2{: n!1

However, truncation of the Laurent series expansion for f on shows that f is indeed uniformly approximated on K by rational functions whose poles lie outside . This fact is generalized to arbitrary open sets by the next. Theorem 7.37 (Runge). Let K be a compact subset of C and let S be a subset of b C K that intersects nontrivially each connected component of b C K. If f is a holomorphic function on an open set containing K, then it can be uniformly approximated on K by rational functions with simple poles lying on S ; that is, for every > 0 there exists a rational function R with possibly simple poles only in S such that jf .z/ R.z/j < for all z 2 K: Runge’s theorem is the implication .1/ H) .8/ of the fundamental theorem. The converse follows from Theorem 7.2. We can always choose for S a smallest set consisting of one point from each connected component of b C K. For the important special case where b C K is connected and S is chosen as S D f1g, Runge’s theorem asserts that each function that is analytic in an open neighborhood of K can be uniformly approximated in K by a sequence of polynomials. 2

We follow a course outlined by S. Grabiner, A short proof of Runge’s theorem, Am. Math. Monthly 83 (1976), 807–808, and rely on arguments appearing in Conway’s book listed in our bibliography.

7.5 Runge’s Theorem

189

An outline for the proof of Runge’s theorem is given in Sect. 7.5.2, after some needed preliminaries from real analysis and topology given in the next subsection. The proof of the theorem depends on three major lemmas that are stated and proved in the subsequent subsections; the first two are given in Sect. 7.5.3 and the third in Sect. 7.5.4.

7.5.1 Preliminaries for the Proof of Runge’s Theorem We rely on some new terminology and notation. We also call the reader’s attention to some elementary topological concepts that we will need. If F and K are subsets of C with F closed and K compact, then the distance between these two sets is the nonnegative real number d.F; K/ D inffjz wj I z 2 F and w 2 Kg; that is easily seen to satisfy d.F; K/ D 0 if and only if F \ K ¤ ;: In particular, if F consists of only one complex number c, we set d.c; K/ D d.fcg; K/ D inffjc wj I w 2 Kg: If A and B are connected subsets of C that are not disjoint, then A [ B is connected. If C is a connected component of the set A, then C is an open subset of A. Let K and S be the sets described in the hypothesis of Runge’s theorem, and define B.K; S / to be the set of continuous complex-valued functions on K that are uniform limits of sequences of rational functions with poles only in S . Sums and products of elements of B.K; S / are obviously also elements of B.K; S /, as are products of constants by elements of B.K; S /. We summarize the (additional) properties of this algebra in the following lemma. The proof is left as an exercise. Lemma 7.38. The algebra B.K; S / contains the rational functions with poles in S , and is closed under uniform limits in K. Runge’s theorem asserts that every holomorphic function on a neighborhood of K belongs to B.K; S /. To establish this, we will also need the following topological result. Lemma 7.39. Let U and V be open subsets of C with V U and @V \ U D ;. If H is a connected component of U and H \ V ¤ ;, then H V .

190

7 Sequences and Series of Holomorphic Functions

Proof. Let c be a point in H \ V and let G be the connected component of V containing c. It is enough to show that G D H . Now H [G is connected, contained in U and contains c. Since H is the component of U , containing c, G H . Furthermore, @G @V and so @G \ H D ;. This implies that H G D H cl G and, therefore, that H G is open in H . Since G is also open in H , the conclusion that G D H follows. t u

7.5.2 Proof of Runge’s Theorem We outline the steps in the proof of Runge’s theorem. The details needed to fill in the outline will be completed in the next two subsections. Let , K, S , and f be as in the statement of Runge’s theorem. The proof of the theorem (that f 2 B.K; S /) consists of four steps. We establish: (1) (Lemma 7.40 of the next section) There exists a finite collection of oriented line segments 1 , 2 , . . . , n in K such that f .z/ D

n Z 1 X f .w/ dw for all z 2 K: 2{ j D1 j w z

Z

f .w/ dw can be w z j 1 approximated uniformly on K by finite sums of rational functions z 7! P zc , where P is a polynomial and c is a point in S . Wehence drop the subscript j 1 from the notation. It will be convenient to regard P z1 as a polynomial in z. The observation that every uniformly convergent series in h on K (in particular P .h/) belongs to B.K; S / whenever h 2 B.K; S / will be used repeatedly in our arguments that follow. Z f .w/ (3) (Lemma 7.41) The line integral dw can be uniformly approximated on wz K by Riemann sums of the form (2) It suffices to prove that each of the above integrals

X k

ak ; with ak 2 C and bk 2 range : bk z

ak can be approximated uniformly on K by bk z appropriate finite sums, and thus belongs to B.K; S /.

(4) (Lemma 7.42) Each summand

7.5 Runge’s Theorem

191

7.5.3 Two Major Lemmas We establish two results: the first gives an extension of Cauchy’s integral formula, and the second provides an approximation by a rational function to a function defined by an integral. Lemma 7.40. Let K be a compact subset of C and let be an open set containing K. Then there exists a finite collection of oriented line segments 1 , 2 , . . . , n in K such that for every holomorphic function f on , n Z 1 X f .w/ f .z/ D dw 2{ i D1 i w z

(7.5)

for all z 2 K. Proof. After enlarging K if necessary, we may assume that K D cl.int.K//: For example, we enlarge K if it consists of a single point. To simplify statements we adopt the standard convention that a curve W Œa; b ! b C and its range f.t/I t 2 Œa; bg are both called . For any positive real number ı we consider a rectangular grid of horizontal and vertical lines in the plane C so that consecutive parallel lines are at distance ı apart. We let R1 ; R2 ; : : : ; Rm be the rectangles in the grid that have nonempty intersection with K. Since K is compact, there are only a finite number of such rectangles. We can (and from now on do) choose ı such that Rj for all j ; if D C any ı > 0 suffices, and otherwise it is enough to consider any 0 < ı < 12 d.K; C /, since z 2 Rj implies that p d.z; K/ < 2ı. As usual, the boundary of Rj is denoted by @Rj and is oriented in the counterclockwise direction. The integrals of a continuous form along the common boundaries of any pair of contiguous Ri and Rj cancel out (as in the proof of Goursat’s Theorem 4.61). This last observation implies S that we can choose a set S of curves whose ranges are a subset of the sides in m j D1 @Rj , and such that the set S D fi I 1 i ng satisfies (1) If i is in S, it lies on a side of only one Rj . (2) If i is in S, then it is disjoint fromSK. (3) For any continuous function g on m j D1 @Rj , we have m Z X j D1 @Rj

g.z/dz D

n Z X i D1

g.z/dz:

(7.6)

i

Each i is an oriented line segment in K. It remains to prove that equation (7.5) holds with these i . If z 2 K and z is not on the boundary of any of the rectangles, then the function

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7 Sequences and Series of Holomorphic Functions

w 7! g.w/ D

m [ 1 f .w/ @Rj ; ; w2 2{ w z j D1

is continuous. Thus, we have, by (7.6), m Z n Z 1 X 1 X f .w/ f .w/ dw D dw: 2{ j D1 @Rj w z 2{ i D1 i w z

Assume that z belongs to the interior of exactly one of the Rj , call it Rt . If j ¤ t, then z … Rj and Z 1 f .w/ dw D 0I 2{ @Rj w z also, since z 2 Rt , by Cauchy’s integral formula, we have Z 1 f .w/ dw D f .z/: 2{ @Rt w z Thus (7.5) holds for all z 2 Rt . Since range i does not intersect K, both sides of this equation are continuous functions of z on K, and they agree on the set of points z in K that are not on the boundary of any rectangle Rj , a dense subset of K. Thus they agree for all z 2 K. t u Lemma 7.41. Let be a pdp and let K be a compact set disjoint from the range of . If f is a continuous function on and is any positive real number, then there exists a rational function R, with only simple poles, all lying on range , such that ˇZ ˇ ˇ f .w/ ˇ ˇ ˇ ˇ w z dw R.z/ˇ < for all z 2 K: Proof. We assume that is not a constant, and thus that its length is positive. Since K and the image of are disjoint, d.K; range / > 0, and we can choose a number r with 0 < r < d.K; range /: If is parameterized by Œ0; 1, then for all 0 s; t 1 and all z 2 K we have ˇ ˇ ˇ f ..t// f ..s// ˇˇ ˇ ˇ .t/ z .s/ z ˇ 1 jf ..t//.s/ f ..s//.t/ z .f ..t// f ..s///j r2 1 2 jf ..t//j j.s/ .t/j C j.t/j jf ..s// f ..t//j r C jzj jf ..s// f ..t//j :

7.5 Runge’s Theorem

193

Since and f are continuous functions and K is a compact set, there is a constant C > 0 such that jzj < C2 for all z 2 K, j.t/j C2 and jf ..t//j C for all t 2 Œ0; 1. Thus for all s and t in Œ0; 1 and all z 2 K we have ˇ ˇ ˇ f ..t// f ..s// ˇˇ C ˇ : .t/j C ..s// f ..t//j j.s/ jf ˇ .t/ z .s/ z ˇ r 2 Since both and f ı are uniformly continuous on Œ0; 1, there is a partition of Œ0; 1 with 0 D t0 < t1 < < tn D 1 such that ˇ ˇ ˇ f ..t// f ..tj // ˇˇ ˇ ˇ .t/ z .t / z ˇ < L. / j

(7.7)

for tj 1 t tj , 1 j n, and all z 2 K, where L. / denotes the length of (recall Definition 4.62). Define the function R as follows. For z ¤ .tj /, j D 1; : : : ; n, R.z/ D

n X j D1

f ..tj //

.tj / .tj 1 / : .tj / z

Then R is a rational function whose poles are simple and contained in the set f.t1 /; .t2 /; : : : ; .1/gI in particular, they are contained in range . Now inequality (7.7) gives ˇ ˇ ˇ ˇX ˇZ ˇ Z tj ˇ ˇ ˇ n ˇ f .w/ f ..t // f ..t// j 0 ˇ ˇ ˇ .t/ˇˇ dt ˇ w z dw R.z/ˇ D ˇ .t/ z .t / z j ˇj D1 tj 1 ˇ

n Z X tj ˇˇ 0 ˇˇ .t/ dt D L. / j D1 tj 1

for all z 2 K.

t u

7.5.4 Approximating

1 zc

Of central importance in the proof of Runge’s theorem is Lemma 7.42. For every c 2 C K, the rational function g W z 7! .z c/1 2 B.K; S /:

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7 Sequences and Series of Holomorphic Functions

Proof. The proof consists of several claims that will need verification. • Let us choose R 2 R>0 so that K U.0; R/. Let z0 2 C be arbitrary with jz0 j > R and let h be a rational function with a pole only at z0 (thus if z0 2 C, h is finite at infinity). Then h can be approximated uniformly on K by polynomials; in particular, h 2 B.K; f1g/. This assertion is obvious if z0 D 1. In general,3 the 1 X an zn . This series converges uniformly Taylor series for h at 0 is of the form nD0

on every compact sub-disc of fjzj < jz0 jg centered at 0, hence certainly on K. Since its individual terms an zn are in B.K; f1g/ and this space is closed under uniform convergence on K, h 2 B.K; f1g/. • Assume that 1 2 S . We claim that B.K; .S f1g/ [ fz0 g/ B.K; S /: Indeed, if f 2 B.K; .S f1g/ [ fz0 g/ and fRj g is a sequence of rational functions with poles in .S f1g/ [ fz0 g uniformly approximating f on K, we can write f D lim Rj D lim .R1;j C R2;j /; j !1

j !1

where R1;j has all its poles (if any) in S f1g and R2;j has a unique pole (if any) at z0 . Here and below limits are understood in the sense of “uniformly on K”, and estimates on absolute values of functions are on the set K. We have shown in the previous assertion that R2;j 2 B.K; f1g/ and that there exist polynomials Pi;j (they belong to B.K; f1g/) such that lim Pi;j D R2;j :

i !1

For each j , choose i sufficiently large so that jR2;j Pi;j j

0. Choose j sufficiently large so that j >

3

ˇ ˇ 2 and ˇf .R1;j C R2;j /ˇ < : 2

The argument that follows also applies for z0 D 1.

7.5 Runge’s Theorem

195

Then ˇ ˇ ˇ ˇ ˇf .R1;j C Pi;j /ˇ D ˇf .R1;j C R2;j / C .R2;j Pi;j /ˇ ˇ ˇ ˇ ˇ ˇf .R1;j C R2;j /ˇ C ˇR2;j Pi;j ˇ

0g. Note that both D and H2 should be regarded as discs in e C, since they are bounded by circles in b C: the unit circle S 1 and the extended real line b R D R [ f1g, respectively. The next result shows that these two discs in b C are conformally equivalent. Corollary 8.15. If w.z/ D onto D.

z{ for z 2 H2 , then w is a conformal map of H2 zC{

Proof. All M¨obius transformations, in particular w, are conformal. A calculation shows that w maps b R D R [ f1g onto S 1 (the unit circle centered at 0) and w.{/ D 0. By connectivity considerations, it follows that w.H2 / D D. t u

8.2 Aut.D/ for D D b C, C, D, and H2

205

b C, D, and H2 8.2 Aut.D/ for D D C, Theorem 8.16. A function f W C ! C belongs to Aut.C/ if and only if there exist a and b in C, a ¤ 0, such that f .z/ D az C b for all z 2 C. Proof. The if part is trivial. For the only if part, note that f is an entire function, and we can use its Taylor series at zero to conclude that f .z/ D

1 X

an zn for all z 2 C:

nD0

If 1 were an essential singularity of f , then f .jzj > 1/ would be dense in C. But f .jzj > 1/ \ f .jzj < 1/ is empty since f is injective. Thus 1 is either a removable singularity or a pole of f ; in any case, there is a nonnegative integer N such that an D 0 for all n > N and aN ¤ 0; that is, f is a polynomial of degree N . If N were bigger than one or equal to zero, then f would not be injective. t u Theorem 8.17. Aut.b C/ Š PSL.2; C/. Thus the last arrow in the exact sequence (8.3) corresponds to a surjective map. Proof. We need only show that Aut.b C/ is contained in the M¨obius group. Let f be an element of Aut.b C/. If f .1/ D 1, then f is a M¨obius transformation by Theorem 8.16. If f .1/ D c ¤ 1, then consider the M¨obius transformation A.z/ D 1 and conclude that B D A ı f in Aut.b C/ and fixes 1; therefore B is a M¨obius zc 1 transformation. But then so is f D A ı B. t u We now provide a characterization of the elements of Aut.D/; it shows that they form a subgroup of Aut.b C/. Another useful characterization is given in Exercise 8.5. Theorem 8.18. A function B defined on D is in Aut.D/ if and only if there exist a and b in C such that jaj2 jbj2 D 1 and B.z/ D for all z 2 D.

az C b bz C a

206

8 Conformal Equivalence and Hyperbolic Geometry

Proof. The if part: Assume that B is of the above form and observe that a is different from zero. We show that B 2 Aut.D/. This follows from the following easy to prove facts: (1) Mappings B of the given form constitute a group under composition. In az b az C b with jaj2 jbj2 D 1, then B 1 .z/ D has particular, if B.z/ D bz C a bz C a the same form as B. (2) jzj D 1 if and only if jB.z/j D 1. ˇ ˇ ˇb ˇ (3) jB.0/j D ˇˇ ˇˇ < 1. a (4) B.D/ is connected. Thus, from (2), either B.D/ is contained in D or B.D/\D is empty. From (3) we see that B.D/ D. It follows from (1) that B 1 .D/ D. (5) Obviously D D B ı B 1 .D/, which implies that B.D/ D D. The only if part: Let f 2 Aut.D/ and w D f .z/. Then f 1 2 Aut.D/ and z D f 1 .w/. (6) If f .0/ D 0, then it follows by the Schwarz’ lemma applied first to f 1 and then to f that ˇ ˇ jzj D ˇf 1 .w/ˇ jwj D jf .z/j jzj for all z 2 D: The same lemma implies that there exists a 2 R such that f .z/ D e{ z for all z 2 D. So we can take a D e{ 2 and b D 0 to conclude that f has the required form. zc . (7) If f .0/ D c ¤ 0, then 0 < jcj < 1 and we set C.z/ D 1 cz It follows from Exercise 8.5 (see also Exercise 2.2) that the M¨obius transformation C belongs to Aut.D/. Since C ı f fixes the origin, it follows from (6) that C ıf is of the required form, and therefore so is f D C 1 ı.C ıf / by (1). t u Just as in Sect. 8.1 we defined PSL.2; C/ as the quotient of SL.2; C/ by ˙I and then proved that it is isomorphic to the group Aut.b C/, we can define the group PSL.2; R/ D SL.2; R/=f˙I g of appropriate matrices with real coefficients modulo plus or minus the identity matrix and obtain the following description: Theorem 8.19. Aut.H2 / Š PSL.2; R/. Proof. Consider the conformal map w W H2 ! D given in Corollary 8.15. Then Aut.H2 / D w1 Aut.D/ w:

8.3 The Riemann Mapping Theorem

207

By the preceding theorem, any element f of Aut.D/ may be written as f .z/ D

az C b bz C a

with jaj2 jbj2 D 1. Denote a D a1 C { a2 , b D b1 C { b2 . Then .w1 ı f ı w/.z/ D

.a1 C b1 /z C b2 a2 .a2 C b2 /z C b1 a1

with .a1 C b1 /.b1 a1 / C .a2 C b2 /.a2 b2 / D jaj2 jbj2 D 1; thus we have 1 2 associated to any element w ı f ı w of Aut.H / the image in PSL.2; R/ of the .a1 C b1 / b2 a2 in SL.2; R/. matrix a2 C b2 b1 a1 ab Conversely, every matrix in SL.2; R/ induces a M¨obius transformation cd az C b given by S.z/ D . Since S preserves the circle b R and cz C d =S.{/ D

ad bc 1 D 2 2 2 c Cd c C d2

is positive, we conclude that S belongs to Aut.H2 /.

t u

8.3 The Riemann Mapping Theorem We now combine the results about M¨obius transformations of the previous two sections with results from Chap. 7 about compact and bounded families of holomorphic functions to show that every simply connected domain D in C, other than C itself, is conformally equivalent to the unit disc; any conformal map from D onto the unit disc D will be called a Riemann map. Recall that a set A is a proper subset of a set B if A B (thus, in particular, A ¤ B). Theorem 8.20 (Riemann Mapping Theorem). Let D be a nonempty proper simply connected open subset of C, and let c 2 D. Then there exists a unique conformal map f W D ! D with f .c/ D 0, f 0 .c/ > 0, and f .D/ D D. Proof. We are looking for a map for the pair .D; c/. The argument has two parts, existence and uniqueness. Existence. We first reduce to a special case.

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8 Conformal Equivalence and Hyperbolic Geometry

First Reduction: It suffices to assume that D is bounded. Proof. Since D ¤ C, we can choose b 2 C D; since D is simply connected there is a branch g.z/ of log.z b/ on D. Thus eg.z/ D z b for all z 2 D: The function g is injective: for if g.w/ D g.z/, then w b D z b. Furthermore, if d 2 D, then g.z/ g.d / ¤ 2{ for all z 2 D. Otherwise z b D eg.z/ D eg.d /C2{ D eg.d / D d b: Since g.D/ is an open set, we can choose d 2 D and ı > 0 such that jw g.d /j < ı ) w 2 g.D/: Thus jw g.d / 2{j < ı ) w … g.D/; since otherwise w D g.z0 / for some z0 in D, and jw g.d / 2{j D jŒg.z0 / 2{ g.d /j < ı implies that g.z0 / 2{ D g.z1 / for some z1 in D, a contradiction. Now 1 F .z/ D g.z/ g.d / 2{ is a conformal map from D onto F.D/, and the simply connected domain F .D/ is 1 . t u contained in the bounded set cl U 0; ı We are reduced to solving the mapping problem for .F .D/; F .c//, and may thus assume that D is bounded. Second Reduction: We may also assume that c D 0. Proof. The map G W z 7! z c takes the bounded domain D onto its (bounded) translate D 0 . If F is a solution of our problem for .D 0 ; 0/, then F ı G solves the problem for .D; 0/. t u Thus we now assume that D is bounded, simply connected and c D 0 2 D. Proof of the Theorem Under These Assumptions. We define F D ff 2 H.D/I f is either conformal or identically zero; f .0/ D 0; f 0 .0/ 2 R; f 0 .0/ 0 and jf .z/j < 1 for all z 2 Dg:

8.3 The Riemann Mapping Theorem

209

Our first observation is that F is nonempty. Of course, f 0 is in F . This is not good enough for much. Since D is bounded, there exists an M > 0 such that jzj M for all z 2 D:

(8.4)

z Hence, for every a 2 R such that a > M , the function f .z/ D for z in D belongs a to F . Next we show that F is compact using Corollary 7.31.Each element f of F satisfies jjf jjK 1 for all compact sets K D; hence F is bounded in the strong sense. To show that F is closed, let ffn g F be a sequence such that fn ! f uniformly on all compact subsets of D. Then f 2 H.D/, by Theorem 7.2, and since each fn vanishes at 0, so does f . It is now convenient to consider two cases: 1. fn 0 for infinitely many distinct n. In this case f 0 and hence certainly f 2 F. 2. fn 0 for only finitely many n. In this case we may assume that each fn is a conformal map; then fn0 .0/ > 0 for all n, and thus f 0 .0/ 0. Hurwitz’s Theorem 7.9 says that f is either constant (hence identically zero) or univalent (that is, one-to-one). Since jfn .z/j < 1 for all z 2 D, we conclude that jf .z/j 1 for all z 2 D. If jf .z0 /j D 1 for some z0 2 D, then jf j 1 by the maximum modulus principle; this is a contradiction to f .0/ D 0. Thus jf .z/j < 1 for all z 2 D, and we conclude that f 2 F ; thus F is closed, and therefore compact. We now complete the proof of the existence part. If S D ff 0 .0/I f 2 F g; then S R0 . We claim that S is bounded from above. Indeed, choose > 0 so that cl U.0; / D. If ./ D e{ for 0 2 is the circle centered at 0 with radius , then for any f 2 F we have Z 1 f .z/ 0 dz f .0/ D 2{ z2 and thus ˇ 0 ˇ ˇf .0/ˇ 1 2 D 1 2 2 1 2 S for all a > M , a as we saw above. Also, there exists a sequence ffn g F such that If D sup S , then

1 M

> 0, with M as in (8.4), because lim fn0 .0/ D :

n!1

Since F is compact, there exists a convergent subsequence ffnk g with lim fnk D f 2 F . Since f 0 .0/ D , f is a conformal map.

k!1

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8 Conformal Equivalence and Hyperbolic Geometry

Since f .D/ D, to show that f .D/ D D, we assume for contradiction that f .D/ ¤ D and construct an h 2 F with h0 .0/ > , thus contradicting the fact that D sup S . Namely, if f .D/ ¤ D, then there exists w0 D re{ with 0 < r < 1 such that w0 2 @f .D/. We now construct h as follows: 1. Let g1 .z/ D e{ f .z/. The map g1 is the map f followed by a rotation through the angle ; g1 sends w0 to r. r g1 .z/ 2. Let p.z/ D . The map p is g1 followed by an automorphism of D that 1 rg1 .z/ sends r to 0 (see Exercise 8.5). Note that p.z/ ¤ 0 for all z 2 D, since p.z/ D 0 if and only if g1 .z/ D r if and only if f .z/ D w0 . 1 3. Let q.z/ D p.z/ 2 , where we choose the branch of the square root2 with q.0/ D 1 r 2 > 0. The map q is injective because q.z1 / D q.z2 / if and only if p.z1 / D p.z2 /. Furthermore, jq.z/j < 1 for all z 2 D. 1 r 2 q.z/ . The map g2 is q followed by an automorphism of D 4. Let g2 .z/ D 1 1 r 2 q.z/ 1 that sends r 2 to 0. 5. Let h.z/ D e{ g2 .z/. The map h is g2 followed by a rotation through the angle . Conclusion: h is a univalent mapping of D into D. We calculate h.0/ and h0 .0/. In order to use the chain rule we need to see what happens to zero under all the maps used to construct h. It is easily checked that 1

g1 .0/ D 0; p.0/ D r; q.0/ D r 2 ; g2 .0/ D 0; and h.0/ D 0: ˛z C ˇ Aside: Let ˛; ˇ; ; and ı be complex numbers. If A.z/ D for all z 2 C z C ı ı ˛ı ˇ , then A0 .z/ D . . z C ı/2 The calculation of the derivative of h at zero proceeds as follows: h0 .0/ D e{ g20 .0/ D e{

r 1 1 2

q 0 .0/ D e{

r 1 0 q .0/ .1 r/2

.1 r q.0//2 e{ 1 1 1 1 C r 2 e{ 12 0 D p.0/ p .0/ D r 2 g 0 .0/ r 1 2 r 1 2 .1 rg1 .0//2 1

D

r C1 1 { 1 1 r 2 1 { e e D : 1 2 r 1 r 12 1 2r 2

By Exercise 5.1 there certainly exists a holomorphic function q whose square is p. Hence q is also such a function. These are the two branches of the square root of p.

2

8.4 Hyperbolic Geometry

211

Finally r C1

> 1 if and only if 0 < r < 1; 1 2r 2 arriving at the contradiction h0 .0/ > that finishes the existence proof. Uniqueness. This is a straightforward argument using the Schwarz’s lemma.

t u

Corollary 8.21. If D is a nonempty simply connected domain in b C, then D is conformally equivalent to one and only one of the following domains: (i) b C, (ii) C, or (iii) D. The case (i), (ii), or (iii) occurs when the boundary of D consists of no points, one point, or more than one point, respectively. In the last case the boundary of D contains a continuum (a homeomorphic image of a closed interval containing more than one point). Proof. Existence: If D ¤ b C, we may first reduce to the case D C by observing 1 that if D contains 1, we can choose c 2 C D, and setting F .z/ D we zc have that F .D/ C is a nonempty simply connected domain not containing 1 and conformally equivalent to D. If the result holds for F .D/, then it also holds for D. If D is a proper subset of C, the result follows from the RMT. Uniqueness: We need to prove that no two of the simply connected domains b C, b C, and D are conformally equivalent. But C is compact, and hence cannot be conformally equivalent to either C or D. On the other hand, a conformal map from C onto D would be a nonconstant entire bounded function, a contradiction to Liouville’s theorem. t u

8.4 Hyperbolic Geometry Let D be a simply connected domain in the extended complex plane with two or more boundary points. In this section we establish that such a domain3 carries a conformally invariant metric, known as the Poincar´e or hyperbolic metric. These domains are called hyperbolic; they are all conformally equivalent to the unit disc, by the RMT. We show that conformal equivalences between these domains preserve the hyperbolic metric; that is, they are isometries (distance preserving maps) with respect to the hyperbolic metrics on the respective domains. Endowed with these equivalent metrics, the upper half-plane H2 and the unit disc D become models for non-Euclidean (also known as hyperbolic or Lobachevsky) geometry. As we

The metric may be defined on all domains in b C with two or more boundary points, not necessarily simply connected; to prove this takes us beyond the scope of this book.

3

212

8 Conformal Equivalence and Hyperbolic Geometry

have shown, the groups Aut.H2 / and Aut.D/ of conformal automorphisms of these domains consist of M¨obius transformations, a class of maps much easier to study than the group of conformal automorphisms of an arbitrary D. It is a remarkable fact that these M¨obius functions constitute the full group of orientation-preserving isometries of H2 and D with their respective hyperbolic metrics. We conclude this section using Schwarz’s lemma and the hyperbolic metric to establish a deep connection between complex analysis and geometry. Namely, holomorphic maps between hyperbolic domains are either isometries or contractions with respect to their hyperbolic metrics. We first define the Poincar´e metric in a general setting; that is, on an arbitrary simply connected domain D with two or more boundary points (Sect. 8.4.1). We subsequently study it in more detail on H2 and D, where specific computations are most easily carried out (Sects. 8.4.2 and 8.4.3). The results that follow from these computations transfer to the general setting because of the conformal equivalence established in the RMT. Finally in Sect. 8.4.4 we establish the result about contractions.

8.4.1 The Poincar´e Metric Definition 8.22. Let D be a simply connected domain in the extended complex plane with two or more boundary points. We define the (infinitesimal form of the) Poincar´e metric on D D .z/ jdzj as follows. First, in the unit disc, set D .z/ D

2 1 jzj2

; z 2 D:

(8.5)

Next, for arbitrary D, choose a Riemann map W D ! D and define D by ˇ ˇ D .w/ D D ..w// ˇ 0 .w/ˇ ; w 2 D:

(8.6)

Our first task is to show that D .w/ is well defined for all simply connected domains4 D and all w 2 D. Toward this end, let A be a conformal automorphism of D. Recall that there exist complex numbers a and b with jaj2 jbj2 D 1 such that A.z/ D

4

With two or more boundary points.

az C b bz C a

; z 2 D:

8.4 Hyperbolic Geometry

213

An easy calculation now shows that ˇ ˇ D .A.z// ˇA0 .z/ˇ D D .z/; z 2 D:

(8.7)

Let w0 2 D be arbitrary and suppose that and are two Riemann maps of D onto D, with .w0 / D z0 and .w0 / D t0 for two points z0 and t0 2 D. We need to show that ˇ ˇ ˇ ˇ D .z0 / ˇ 0 .w0 /ˇ D D .t0 / ˇ 0 .w0 /ˇ : But A D ı 1 is in Aut.D/ and A.z0 / D t0 ; it now follows from (8.7) that ˇ ˇ ˇ ˇˇ ˇ D .z0 / ˇ 0 .w0 /ˇ D D .A.z0 // ˇA0 .z0 /ˇ ˇ 0 .w0 /ˇ ˇ ˇˇ ˇ D D .A.z0 // ˇ 0 . 1 .z0 //. 1 /0 .z0 /ˇ ˇ 0 .w0 /ˇ ˇ ˇ D D .t0 / ˇ 0 .w0 /ˇ : Remark 8.23. (1) If w 2 D is arbitrary and we choose the Riemann map to satisfy .w/ D 0, then ˇ ˇ D .w/ D 2 ˇ 0 .w/ˇ : (2) It is easy to see, using (8.6) and Corollary 8.15, that H2 .z/ D

1 for all z 2 H2 : =z

(8.8)

The important invariance property of our metric is described in our next result. Proposition 8.24. For every conformal map f defined on D, ˇ ˇ f .D/ .f .z// ˇf 0 .z/ˇ D D .z/ for all z 2 D: Proof. If is a Riemann map (for D), so is ı f 1 (for f .D/).

t u

Any infinitesimal metric on D allows us to define lengths of paths in D, and hence a distance function on the domain. We work, of course, with the length element ds D D .z/ jdzj : Definition 8.25. We define the hyperbolic length of a piecewise differentiable curve in D by Z lD . / D D .z/ jdzj I

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8 Conformal Equivalence and Hyperbolic Geometry

and if z1 and z2 are any two points in D, the hyperbolic (or Poincar´e) distance between them by D .z1 ; z2 / D infflD . /I is a pdp in D from z1 to z2 g:

(8.9)

We leave to the reader (Exercise 8.13) the verification that D defines a metric on D. An isometry from one metric space to another is a distance preserving map between them. It follows from Proposition 8.24 that for every conformal map f defined on D and every pdp in D, lf .D/ .f ı / D lD . / and f .D/ .f .z1 /; f .z2 // D D .z1 ; z2 / for all z1 and z2 2 DI that is, is conformally invariant and f is an isometry between D and f .D/ with respect to the appropriate hyperbolic metrics. In particular, every element of Aut.D/ is an isometry for the hyperbolic metric on D.

8.4.2 Upper Half-plane Model jdzj We know from Remark 8.23 that in H2 we have ds D =.z/ . The hyperbolic length 2 of an arbitrary curve in H and the hyperbolic distance between two points in H2 may be hard to calculate directly from their definitions; an indirect approach is technically less complicated. We show that given any two distinct points in H2 , they lie on either a unique Euclidean circle centered on the real axis or on a unique straight line perpendicular to the real axis. The corresponding portion of the circle or straight line lying in H2 is called a hyperbolic line or geodesic; the unique portion of the geodesic between the two points is called a geodesic path or geodesic segment. The name is justified by showing that the hyperbolic length of a geodesic segment realizes the hyperbolic distance between its two end points. A straight line in C is a circle in C [ f1g passing through infinity (see Exercise 3.21). It is not useful, in general, to assign centers to these circles. However, if such a line intersects R in one point and is perpendicular to R at that point, we consider that point to be the center of the circle. In the current context, we shall be interested only in lines perpendicular to R and use the related fact that a Euclidean circle with center on the real axis is perpendicular to the real axis.

Definition 8.26. For a circle C in C [ f1g centered on the real axis, the part of C lying in the upper half plane is called a hyperbolic line or a geodesic in H2 . The reason for the terminology will shortly become clear. The following lemma establishes the existence of a geodesic path between two points; the proof of its uniqueness follows.

8.4 Hyperbolic Geometry

215

O perpendicular to R through two pairs of points in the upper Fig. 8.2 Unique circles (in C) half plane

Lemma 8.27. For every pair z and w of distinct points in H2 , there exists a unique circle centered at the real axis passing through them, and a unique geodesic in H2 passing through them. e to be the Euclidean line through z and w. Otherwise, Proof. If 0, we can choose an r such that ˇB 1 .w/ˇ 1 for all w of absolute value r. Without loss of generality we can choose r jzj. Hence, by the maximum principle for jwj r, it follows that jF .w/j D

1 jf .w/j .B/1 : 1 r 0 .1 / jB .w/j

r 0 .B/1

Since is arbitrary and the inequality holds for all jzj r < 1, the claim is proved. By Schwarz’s lemma we obtain jF .z/j jzj, or, equivalently, ˇ ˇˇ ˇ jf .z/j jzj ˇz 0 .B/1 ˇ ˇB 1 .z/ˇ D jB.z/j for all z 2 D, proving the first inequality in (a). For the inequality about derivatives, note that by the first inequality, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ f .z/ f .ai / ˇ ˇ f .z/ ˇ ˇ B.z/ ˇ ˇ B.z/ B.ai / ˇ ˇ ˇDˇ ˇˇ ˇDˇ ˇ: ˇ ˇ ˇz a ˇ ˇz a ˇ ˇ ˇ z ai z ai i i Taking limits as z approaches ai yields the second inequality. f is analytic on D, BA and its modulus is at most 1 on D. If jf .a/j D jB.a/j for some a 2 D A, then the f modulus of at a is 1; therefore this function is constant and (b) follows. BA To prove (b), note that it follows from (a) that the function

224

8 Conformal Equivalence and Hyperbolic Geometry

To prove (c), we may assume that ai D 0. By l’Hopital’s rule, ˇ ˇ ˇˇ . / ˇˇ ˇf ˇ ˇ f .0/ ˇ ˇ ˇ ˇ B .0/ˇ D ˇˇ . / ˇˇ D 1; A BA .0/ t u

and (8.16) follows.

The particular case when A consists of one point leads to the following interesting result, a generalization of Schwarz’s lemma that the reader may have encountered before as an exercise. Corollary 8.48 (Schwarz-Pick Lemma). Let f be a holomorphic self-map of the open unit disc D. If a 2 D, then ˇ ˇ ˇ ˇ ˇ f .z/ f .a/ ˇ ˇ z a ˇ ˇ ˇˇ ˇ ˇ ˇ 1 az ˇ N ˇ 1 f .a/f .z/

ˇ ˇ 1 jf .a/j2 for all z 2 D and ˇf 0 .a/ˇ : 1 jaj2

Furthermore, equality for one z 6D a in the first inequality or equality in the second inequality implies that f is an automorphism of D. Proof. Apply the previous theorem to the function Bf .a/ ı f .

t u

This corollary may also be deduced by applying Schwarz’s lemma to the function Bf .a/ ıf ıBa1 (see Exercise 5.4) or, more interesting, is equivalent to Theorem 8.43 (Exercise 8.22).

Exercises

ab 8.1. A matrix A D with a; b; c; d 2 C, ad bc ¤ 0, acts on b C by cd azCb . Show that for each t 2 C6D0 , A and tA induce the same action. z 7! czCd C are permuted. What effect 8.2. Suppose the four distinct points z1 ; z2 ; z3 ; z4 in b will this have on the cross ratio .z1 ; z2 ; z3 ; z4 /? 8.3. Prove in detail that the angles at z1 and z2 in Fig. 8.1 are equal precisely when the four points lie on a circle in b C. 8.4. In the text we established, using a geometric argument, that the cross ratio of four distinct points fz1 ; z2 ; z3 ; z4 g on the Riemann sphere C [ f1g is real if and only if the four points lie on a circle. This exercise presents an alternate, purely analytic, way of establishing this result. 1. Show that every M¨obius transformation maps the extended real line R [ f1g onto a circle in C [ f1g.

Exercises

225

2. Use the last assertion to show that .z1 ; z2 ; z3 ; z4 / is real if and only if the four distinct points fz1 ; z2 ; z3 ; z4 g lie on a circle in C [ f1g. 8.5. Show that f 2 Aut.D/ if and only if there exist in R and a 2 D such that f .z/ D e{

za for all z 2 D: 1az

8.6. This exercise explores the transitivity properties of groups of M¨obius transformations and provides alternate formulations and proofs for many of the properties of M¨obius transformations developed in this chapter by other methods. 1. We begin with the study of PSL.2; C/ viewed as the group of conformal automorphisms of b C, and let a; b; c be three distinct points in b C. • Assume that all the points are finite. Show that the M¨obius transformation A.z/ D

c.a b/z C a.b c/ ; z2b C; .a b/z C .b c/

maps the ordered triple 0; 1; 1 in b C to the ordered triple a; b; c in C. • Show that by appropriate limiting processes applied to the formula for A, we can extend the above result to ordered triples of distinct points a; b; c in b C. What are the resulting formulae? • Show by direct computation that the identity is the only M¨obius transformation that fixes 0, 1, and 1. • Determine the analogues of the above formulae for mappings of an arbitrary triple in b C to another such triple. Thus establishing once again the unique triple transitivity of PSL.2; C/ which can be summarized as: given two ordered triples of distinct points .a0 ; b0 ; c0 / and .a; b; c/ in b C, there exists a unique A 2 PSL.2; C/ mapping the first ordered triple to the second one. • We now allow appropriate repetitions. Say our triples are .a0 ; a0 ; c0 / and .a; b; c/. There are certain limitations on b that should be described. For a0 and a finite, (in this case b 6D 0) the requirement for A should be that A.a0 / D a, A0 .a0 / D b, and A.c0 / D c, of course. Show that there exists a unique such A and determine its formula. Extend the discussion to allow a0 to be infinite. • Extend the previous analysis to triples .a0 ; a0 ; a0 / and .a; b; c/. 2. We can view PSL.2; R/ as a group of automorphisms of b R. Formulate and establish the analogues of the previous part to this setting. 3. More relevant for our work is to study PSL.2; R/ as a group of automorphisms R. Toward this aim, let a0 and a 2 H2 , and b0 and b 2 b R. of H2 [ b • Show that there is a unique A 2 PSL.2; R/ mapping the ordered pair .a0 ; b0 / to the ordered pair .a; b/. What are the appropriate formulae for A?

226

8 Conformal Equivalence and Hyperbolic Geometry

• Let .a0 ; b0 / and .a; b/ be pairs of distinct points in H2 . Show that there is a unique A 2 PSL.2; R/ mapping the ordered pair .a0 ; b0 / to the ordered pair .a; b/ iff H2 .a0 ; b0 / D H2 .a; b/: Determine the formula for A. • Extend the last claim to allow repetitions. 4. Translate the previous set of exercises on automorphisms of H2 to automorphisms of D. 8.7. Formulate and prove (as a consequence of Theorem 8.20) the RMT for appropriate simply connected domains D b C. Include the possibility that c D 1. 8.8. This exercise deals with the construction of specific Riemann maps. 1. Find a conformal map from the disc fz 2 CI jzj < 1g onto fz 2 CI jzj < 1; =z > 0g. 2. Let D be the domain in the extended complex plane b C exterior to the circles jz 1j D 1 and jz C 1j D 1. Find a Riemann map of D onto the strip S D fz 2 CI 0 < =z < 2g. 3. Find a conformal map from the domain in b C defined by fz 2 CI jz 1j > 1; jz C 1j > 1g [ f1g onto the upper half plane. 4. For each n D 1; 2; 3; : : :, find a conformal map from the infinite angular sector 0 < Arg z < onto the unit disc. n 5. Find { the Riemann0 map f from the strip 0 < =z < 1 onto the unit disc satisfying f 2 D 0 and f 2{ > 0. 6. Find a conformal map from the domain fz 2 CI jzj < 1 and z ¤ t for 0 t < 1g onto fw 2 CI jwj < 1g. 7. Find a conformal map from the upper half plane onto the unit disc minus the nonnegative real numbers. 8.9. If f .z/ D w is a Riemann map from the domain jArg zj < jwj < 1 and if f .1/ D 0 and f 0 .1/ > 0, find f .2/.

100

onto the domain

8.10. If w D g.z/ maps the quadrant fz D x C {y 2 CI x > 0; y > 0g conformally onto jwj < 1 with g.1/ D 1, g.{/ D 1 and g.0/ D {, find jg 0 .1 C {/j. 8.11. If f is holomorphic for jzj < 1 and satisfies jf .z/j < 1 for jzj < 1 and f .0/ D f 12 D 0, show that

Exercises

227

ˇ ˇ ˇ 2z 1 ˇ ˇ ˇ jf .z/j ˇz 2z ˇ

for all jzj < 1:

8.12. Suppose ffn g is a sequence of holomorphic functions in jzj < 1 that satisfy 0 and jfn .0/ {j

0 and all jzj < 1:

Show that ffn g contains a subsequence that converges uniformly on compact subsets of the unit disc. 8.13. Supply the details of the proof that H2 defines a metric in H2 that is invariant under the M¨obius group PSL.2; R/. 8.14. Let z and w be points in H2 with z and w the corresponding points as in Definition 8.31; set D.z; w/ D j log.z ; w ; w; z/j. Show that D defines a metric in the upper half plane. 8.15. Let z and w be points in the upper half plane. Prove that D.z; w/ D jz wj C jz wj defines a metric in the upper half plane. log jz wj jz wj 8.16. Prove that D .0; z/ D log

1 C jzj for all z 2 D: 1 jzj

(8.17)

8.17. Show that for all z1 D x1 C { y1 and z2 D x2 C { y2 2 H2 , H2 .z1 ; z2 / D arccosh

.x2 x1 /2 C .y2 y1 /2 : 1C 2y1 y2

8.18. This exercise establishes some of the basic facts about geodesics in H2 with respect to the hyperbolic metric in slightly different form than developed in the text. It should be established using the formula (8.8) for the infinitesimal form for the metric, but ignoring all facts about geodesics that we already established. 1. Show that {R>0 is a geodesic in H2 . 2. Let be a pdp in H2 . Prove that is a geodesic in H2 iff for all c and d 2 range , the length of the piece of between c and d equals H2 .c; d /. 3. Show that an arbitrary geodesic in H2 is a semi-circle (this includes infinite line segments) perpendicular to R. 4. Let c and d be two distinct points in H2 . Show that there exists a unique geodesic passing through them. 5. Formulate and prove the corresponding fact about a given single point and a given direction from it.

228

8 Conformal Equivalence and Hyperbolic Geometry

8.19. This exercise contrasts the transitivity of PSL.2; C/ acting on C [ f1g (established in the text) with that of PSL.2; R/ acting on H2 (to be established here). It should be compared with Exercise 8.6. ab 1. Show that A 2 PSL.2; R/ fixes { iff it is of the form ˙ , with a, b 2 R b a and a2 C b 2 D 1. 2. Let 2 R. Show that there exists a unique A 2 PSL.2; R/ with A.{/ D { and 2 arg A0 .{/. 3. Let z and w 2 H2 and 2 R. Show that there exists a unique A 2 PSL.2; R/ with A.z/ D w and 2 arg A0 .z/. 4. Let fa; bg and fz; wg be two pairs of distinct points in H2 . Show that there exists a unique A 2 PSL.2; R/ with A.a/ D z and A.b/ D w iff H2 .a; b/ D H2 .z; w/.

{ 8.20. Assume f is a bounded holomorphic function on the unit disc and f D 2

{ D 0. Show that there exists a bounded holomorphic function G on D f 2 such that z 2{ z C 2{ f .z/ D G.z/; for all z 2 D: 1 C 2{ z 1 2{ z 8.21. Let D b C be a disc. Show that Aut.D/ consists of M¨obius transformations. 8.22. Establish the equivalence of Theorem 8.43 and Corollary 8.48.

Chapter 9

Harmonic Functions

This chapter is devoted to the study of harmonic functions. They are closely connected to holomorphic maps, since these as well as their real and imaginary parts are harmonic. The study of harmonic functions is important in physics and engineering, and there are many results in the theory of harmonic functions that are not directly connected with complex analysis. In the early parts of this chapter we consider that part of the theory of harmonic functions that grows out of the Cauchy theory. Mathematically this is quite pleasing. One of the most important aspects of harmonic functions is that they solve a boundary value problem, known as the Dirichlet problem. An example is the problem of finding a function that is continuous on a closed disc, that assumes prescribed values on the boundary of the disc, and is harmonic in the interior of the disc. An important tool in the solution is the Poisson formula. In the first section we define harmonic functions and the Laplacian of a function. In the second we obtain integral representations for harmonic functions that are analogous to the Cauchy integral formula, including the Poisson formula; in the third we use these integral representations to solve the Dirichlet problem. The third section includes three interpretations of the Poisson formula: a geometric interpretation, a Fourier series interpretation, and a classical one that involves differential forms. In the fourth section we characterize harmonic functions by their mean value property (MVP). The fifth section deals with the reflection principle for holomorphic and real-valued harmonic functions, a simple but useful extension tool. In the second part of this chapter, starting with the sixth section, we study the class of subharmonic functions and use them to give a more complete solution to the Dirichlet problem, introduce the Green’s function, and provide an alternate proof of the Riemann mapping theorem (RMT). The first two sections in this part introduce the new class of functions and study their basic properties. Perron families of subharmonic functions are studied in the next section, followed by a section in which the Dirichlet problem is revisited. The final section is devoted to the main aim of the second part of this chapter—existence proofs for Green’s function and an alternate proof of the RMT. R.E. Rodr´ıguez et al., Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics 245, DOI 10.1007/978-1-4419-7323-8 9, © Springer Science+Business Media New York 2013

229

230

9 Harmonic Functions

9.1 Harmonic Functions and the Laplacian We begin with Definition 9.1. Let D be a domain in C and g 2 C2 .D/. We define g, the Laplacian of g, by g D

@2 g @2 g C 2I 2 @x @y

(Laplace)

is also called the Laplacian or the Laplace operator. If D is a domain in C and g 2 C2 .D/, we say that g is harmonic in D if it satisfies Laplace’s equation g D 0 in D. Remark 9.2. The analogous of harmonic functions in one real variable are linear and constant functions. The last two definitions have a number of immediate consequences: (1) It is obvious from the definition of the Laplacian as a linear operator on C2 complex-valued functions that it preserves real-valued functions. It is useful to have equivalent formulae for it (Exercise 9.1): D

@2 @2 @2 @2 D 4 ; C D 4 @x 2 @y 2 @z@z @z@z

(9.1)

and in polar coordinates .r; / D

1 r2

@ @2 @ r C 2 : r @r @r @

(2) Recall that for f 2 C1 .D/, f is holomorphic on D if and only if

(9.2) @f D 0 in @z

@g is holomorphic. @z In particular, holomorphic functions are harmonic and (2) gives an easy way to construct analytic functions from harmonic ones. f is harmonic if and only if f is. f is harmonic if and only if 0 such that ˇ ˇ jP Œu1 .z/j < for ˇz e{ 0 ˇ < ı. It follows that 2 jP Œu.z/j jP Œu1 .z/j C jP Œu2 .z/j < ˇ ˇ as long as jzj < 1 and ˇz e{ 0 ˇ < ı. This is the required continuity statement.

t u

9.3.1 Geometric Interpretation of the Poisson Formula This interpretation is due to Schwarz; the presentation follows Ahlfors.1

Recipe. To find P Œu.z/ we replace u.e{ /—the value of the function u at the point e{ —by its value u.e{ / at the point e{ on the unit circle opposite to the point to e{ with respect to z (i.e., on the intersection of the line through e{ and z with the unit circle), and average these values over the unit circle (see Fig. 9.2). This conclusion comes from reinterpreting the second formula (9.8) defining P Œu.z/ as follows: fix a point z inside the unit circle and a point e{ on the unit circle. Let e{ be the unique point on the unit circle which also lies on the straight line through z and e{ . High school geometry (similar triangles, see Fig. 9.2) or a calculation (i.e., using the law of cosines) yields ˇ ˇ ˇ ˇˇ ˇ 1 jzj2 D ˇe{ zˇ ˇe{ zˇ :

1

See [1].

238

9 Harmonic Functions

eıθ

*

Fig. 9.2 The similar triangles

Since the three points z, e{ , and e{ lie on a straight line and z is between the e{ z is a negative real number; it follows from this other two points, the ratio { e z observation that

1 jzj2 D .e{ z/ .e{ z/:

(9.9)

To verify this last equality note that the real number ˇ2 e{ z ˇˇ { e{ z ˇ .e{ z/ .e{ z/ z D ˇe ˇ { { e z e z

is negative and has the same absolute value as .e{ z/ .e{ z/. An alternate argument to obtain (9.9) follows. By definition there is a real number t such that e{ D t e{ C .1 t/ zI jzj2 1 it is easy to compute that t D ˇ ˇ . ˇe{ zˇ2 It now follows from e{ z D t .e{ z/ that ˇ ˇ2 .e{ z/ .e{ z/ D .e{ z/ t .e{ z/ D t ˇe{ zˇ D jzj2 1: We now regard as a function of , with z fixed, and differentiate equality (9.9) logarithmically to obtain

9.3 The Dirichlet Problem

239

e{ e{ d D d : e{ z e{ z Hence we see that (because is an increasing function of ) that ˇ ˇ ˇ e{ z ˇ d ˇ ˇ D ˇ { ˇD ˇ e z ˇ d

1 jzj2 ˇ ˇ : ˇe{ zˇ2

We have thus shown that 1 P Œu.z/ D 2

Z

2

1 u.e / d D 2

{

0

Z

2

u.e{ / d: 0

The recipe follows.

9.3.2 Fourier Series Interpretation of the Poisson Formula We again consider the case of the unit disc and proceed to compute the power series expansion of P Œu.z/ at the origin. Noting that P Œu.z/ D

1 2

Z

2 0

1 zz u.e{ / d for jzj < 1; .e{ z/ .e{ z/

we start with an expansion of the Poisson kernel

.e{

1 zz 1 zz D { { z/ .e z/ .1 e z/ .1 e{ z/ X D .1 zz/ e{ n zn e{ m zm n;m0

D .1 zz/

X

e{ .mn/ zn zm

n;m0

D1C

1 X

e{ n zn C

nD1

1 X

e{ m zm

mD1

and since the last two series converge uniformly and absolutely on all compact subsets of the unit disc, we conclude P Œu.z/ D a0 C

1 X nD1

an zn C

1 X mD1

bm zm ;

240

9 Harmonic Functions

where, for n 2 Z0 and m 2 Z>0 , 1 an D 2

Z

2

e 0

{ n

1 u.e / d; bm D 2

Z

2

e{ m u.e{ / d:

{

(9.10)

0

We thus have the following procedure for extending a given continuous function u on the unit circle to a continuous function on the closed unit disc that is harmonic on the interior of the disc. First compute the Fourier series of u: u.e{ / D

1 X

an e{n C

nD0

1 X

bm e{m ;

mD1

where the Fourier coefficients an and bm are given by (9.10). In this series replace e{n by zn , for each n 2 Z0 , and e{m by zm , for each m 2 Z>0 .

9.3.3 Classical Reformulation of the Poisson Formula For the next reformulation, we start with Definition 9.16. If ! D P dx C Q dy is a differential form, we define !, the conjugate differential of !, by

! D Q dx C P dy:

If D is a simply connected domain in C and u 2 C2 .D/ is real-valued, we know that u is harmonic on D if and only if u is the real part of an analytic function f on D. In this case, df D f 0 .z/ dz D .ux {uy /.dx C { dy/ D .ux dx C uy dy/ C { .uy dx C ux dy/ D du C { du is an exact differential on D and du D dv, where v is a harmonic conjugate of u on D. Thus du and du are exact differential forms on D whenever u is a real-valued harmonic function on a simply connected domain D. In what follows we work with cycles rather than curves. The definitions of cycles and cycles homologous to zero can be found in Sect. 5.2. In the general case, for a harmonic (including complex-valued) function u on an arbitrary (not necessarily simply connected) domain D, the form du D ux dx C uy dy is always exact on D, and its conjugate differential du D uy dx C ux dy is closed since .u/yy D uxx . We conclude that

9.3 The Dirichlet Problem

241

Z

du D 0

(9.11)

for all harmonic functions u on D and all cycles in D that are homologous to zero on that domain. We can now turn to the classical reformulation of the Poisson formula. Assume that is a regular curve with equation z D z.t/ (regular means that z0 .t/ 6D 0 for all t). The direction of the tangent line to the curve at z.t/ is determined by the angle ˛ D Arg z0 .t/ and dx D jdzj cos ˛; dy D jdzj sin ˛: The normal line at z.t/, which points to the right of the tangent line, has direction ˇ D ˛ 2 . The normal derivative of u is the directional derivative of u in the direction ˇ: @u D ux cos ˇ C uy sin ˇ D ux sin ˛ uy cos ˛: @n Thus we see that du D

@u jdzj, and (9.11) can be rewritten as @n Z @u jdzj D 0: @n

@u @u D . It is important to realize that if is the circle fjzj D rg, then @n @r We prove an important generalization of (9.11). Theorem 9.17. If u1 and u2 are harmonic functions on D, then u1 du2 u2 du1 is a closed form on D. Proof. To establish this assertion, it involves no loss of generality to assume that the functions are real-valued (see Exercise 9.4), and hence we may also assume (because the issue is local) that each function uj has a single-valued harmonic conjugate vj ; thus u1 du2 u2 du1 D u1 dv2 u2 dv1 D u1 dv2 C v1 du2 d.u2 v1 /: The last expression d.u2 v1 / is, of course, exact, and u1 dv2 C v1 du2 D = ..u1 C { v1 /.du2 C { dv2 // :

242

9 Harmonic Functions

Now u1 C { v1 is an analytic function and du2 C { dv2 is the total differential of an analytic function. By Cauchy’s theorem, their product is a closed form, and hence we have shown that Z u1 du2 u2 du1 D 0

for all cycles which are homologous to zero in D. In classical language the above formula reads Z @u2 @u1 u2 u1 jdzj D 0: @n @n t u Let us take for D the annulus fz 2 CI R1 < jzj < R2 g and apply the above formula to the functions z 7! u1 .z/ D log r (in polar coordinates) and an arbitrary harmonic function u2 D u on D. We take for the cycle C1 C2 where Cj is the circle fjzj D rj g oriented counter clockwise; here R1 < r1 < r2 < R2 . On any @u d. Hence we also see circle fjzj D rg, with R1 < r < R2 , we have du D r @r that Z Z Z Z @u @u log r1 d d r1 u d D log r2 r2 u d @r @r C1 C1 C2 C2 Z

or

r

log r jzjDr

@u d @r

Z u d D B jzjDr

is a constant (independent of r). Applying the same argument to the functions u1 D 1 (constant function) and u2 D u, we obtain that Z @u d D A r @r jzjDr is constant over the annulus D, and hence is equal to zero (let r ! 0) if u is harmonic in the disc fz 2 CI jzj < R2 g. Thus, for a function u harmonic in an annulus, the arithmetic mean over concentric circles jzj D r is a linear function of log r 1 2

Z u d D A log r C B I jzjDr

if u is harmonic in the disc jzj < R2 or bounded in the punctured disc 0 < jzj < R2 , then A D 0 and the arithmetic mean is constant. In the latter case, if u is harmonic in the disc, then B D u.0/ by continuity (the reader should know other proofs of this fact.) The fact that A may be 0 in the above discussion has consequences:

9.4 The Mean Value Property: A Characterization of Harmonicity

243

Theorem 9.18. If u is a bounded harmonic function on the punctured disc 0 < jz aj < R, then (a) u extends to be harmonic on the disc jz aj < R (b) For real-valued u, the extension has a harmonic conjugate. We leave the proof to the reader, see Exercise 9.17. Changing the origin, we see that if u is harmonic in U.z0 ; R/, then for 0 < r < R, u.z0 / D

1 2

Z

2

u.z0 C r e{ / dI

(9.12)

0

this is the MVP for harmonic functions, that was already established in Corollary 9.7 as a consequence of the fact that real-valued harmonic functions are locally real parts of analytic functions. From it one also obtains the area MVP u.z0 / D

1 2{r 2

“ u.z/ dz dz:

(9.13)

jzz0 jr

Remark 9.19. If u W S 1 ! S 1 is a homeomorphism, then P Œu is also a homeomorphism, from fzI jzj < 1g onto itself. This useful observation is not at all obvious, and not established here.

9.4 The Mean Value Property: A Characterization of Harmonicity Harmonic functions satisfy the MVP, as we have seen in Corollary 9.7. As a matter of fact this property characterizes harmonic functions. The proof below is based on the solution to the Dirichlet problem. Theorem 9.20. A continuous complex-valued function that satisfies the MVP is harmonic. Proof. Let f be a continuous function on a domain D, let c 2 D and let r0 > 0 be sufficiently small so that cl U.c; r0 / D and f satisfies (5.7) for all r r0 . It suffices to assume that f is real-valued. Let v be the continuous function on fjz cj r0 g that is harmonic on fjz cj < r0 g and agrees with f on fjz cj D r0 g. Then f v has the MVP in fjz cj < r0 g, and thus attains its maximum and minimum on fjz cj D r0 g. Since f D v on fjz cj D r0 g, we conclude that f D v on fjz cj r0 g and thus that f is harmonic there. t u

244

9 Harmonic Functions

9.5 The Reflection Principle We start with the simplest form of the general principle we are to establish. Let be a nonempty region in the complex plane which is symmetric about the real axis; that is, zN 2 if and only if z 2 (see Fig. 9.3). Such a region must intersect the real axis nontrivially, and it is a disjoint union of three sets: D C [ [ ; where C D fz 2 I =z > 0g; D \ R; and D fz 2 I =z < 0g: Remark 9.21. A function z 7! f .z/ on a symmetric region is harmonic (analytic) if and only if the function z 7! f .Nz/ is (see Exercise 9.2). We concentrate on the holomorphic case. Assume that f 2 H. / and f is real on at least one segment of ; then f .z/ D f .Nz/ for all z 2 . Proof. The function z 7! g.z/ D f .z/ f .Nz/ is analytic on and vanishes on a subset of with a limit point in (namely, on ); g is thus identically zero on . t u The same conclusion holds if we merely assume that f 2 C. C [ /, is analytic on C and real on , since in this case the extension of f to defined by f .z/ D f .Nz/ for z 2 satisfies the previous hypothesis. We now strengthen this statement considerably. Theorem 9.22. Let be a nonempty region in the complex plane that is symmetric about the real axis. If v is a real-valued and continuous function on C [ , and it is harmonic on C and zero on , then v has a harmonic extension to that satisfies the symmetry condition v.z/ D v.Nz/.

Ω+ σ Ω−

Fig. 9.3 A symmetric region

9.6 Subharmonic Functions

245

Moreover, if v is the imaginary part of an analytic function f in H. C /, then f has an analytic extension to that satisfies the symmetry condition f .z/ D f .Nz/. Proof. We use the symmetry to extend v to all of . We show that the resulting extension (also called v) is continuous on , harmonic on C [ , and vanishes on . Harmonicity of v in is a local property; therefore we only need to show that v is harmonic in a neighborhood of each point x 2 . For this, consider an open disc D with center at x whose closure is contained in . Let V be the unique function that is continuous on cl D, harmonic on its interior, and agrees with v on the boundary of D. Since v restricted to @D satisfies the symmetry condition v.z/ D v.Nz/, so does the function V (on cl D). Hence V vanishes on cl D C \ R. The function V v is continuous on cl D C , harmonic on D C , and vanishes on its boundary; hence it is identically zero on cl D C . Similarly, V v D 0 on cl D . We conclude that V D v on D, and we have shown that v is harmonic on . The function f has a symmetric extension to C [ (that satisfies f .z/ D f .Nz/). We only know that its imaginary part can be extended to all of (and that the extension vanishes on ). We must use information on the imaginary part of f to draw conclusions about its real part. Again, the problem is local, and we work with the disc D defined above. The real-valued harmonic function v on D has a harmonic conjugate u on this disc. The fact that harmonic conjugates are unique up to addition of real constants allows us to normalize so that u D 0 such that cl U.z0 ; r0 / D, u satisfies equality (9.12). Subharmonic functions are similarly characterized: If u is real-valued and continuous in D, then u is subharmonic in D if and only if for all z0 2 D and all r0 > 0 such that cl U.z0 ; r0 / D, u satisfies the mean value inequality u.z0 /

1 2

Z

2

u.z0 C r e{ / d for all 0 r < r0 :

(9.15)

0

Proof. First assume u is subharmonic in D, and consider z0 and r0 as above. Let ' denote the function (solution to the Dirichlet problem) that is continuous in cl U.z0 ; r0 /, harmonic in U.z0 ; r0 /, and coincides with u on @U.z0 ; r0 /. Then u.z/ '.z/ for all z with jz z0 j < r0 . In particular, u.z0 / '.z0 /, and '.z0 / is precisely the RHS of (9.15) by the MVP for harmonic functions. We have established the “only if ” part of the claim. To establish the “if ” part, assume u is real-valued and continuous in D, and for all z0 and r0 as above, u satisfies (9.15). Let G be a bounded subdomain of D such that @G D, and let ' denote a continuous real-valued function on G [ @G such that ' is harmonic in G, and u.z/ '.z/ for all z in @G. We need to show that u.z/ '.z/ for all z in G. If this were not so, then the function v D u ', being continuous in cl G, would attain a positive maximum M at some point in G, and the set S D fz 2 cl G W v.z/ D M g would be a nonempty closed set contained in G. Since both S and @G are compact, there exists a point c in S minimizing the distance from S to @G. Furthermore, on every circumference fjz cj D rg, for small positive values of r, there would exist an arc ˛r such that v < M on ˛r . Therefore

9.6 Subharmonic Functions

1 2

Z

247

2

u.c C r e{ / d '.c/ D 0

1 2

Z

2

v.c C r e{ / d 0

< M D u.c/ '.c/I from this it follows that 1 2

Z

2

u.c C r e{ / d < u.c/ 0

for all small positive values of r, a contradiction.

t u

3. It follows from the mean value inequality (9.15) that a subharmonic function satisfies the maximum principle. It also follows from 2 that subharmonicity is a local property. 4. Let u be a subharmonic function in D with G and ' as in Definition 9.23. Condition (ii) of the definition can now be strengthened to: (ii)’ If u.z/ '.z/ for all z in @G, then either u.z/ D '.z/ or u.z/ < '.z/ for all z in G: For suppose there exists c in G such that u.c/ D '.c/. Then '.c/ D u.c/

1 2

1 2

Z

2

u.c C r e{ / d Z

0 2

'.c C r e{ / d D '.c/ 0

for all small positive values of r, and hence u D ' near c. Thus fz 2 G W u.z/ D '.z/g is an open set, that we are assuming to be nonempty. On the other hand, u and ' are continuous functions on cl G, and hence fz 2 G W u.z/ < '.z/g is an open set, which must then be empty. 5. Let D be a domain in C. If fun gn2N is a sequence of subharmonic functions on D that converges to a function u on D uniformly on all compact subsets of D, then u is subharmonic in D. We leave the proof as an exercise for the reader. 6. If u1 and u2 are subharmonic functions in D, then the function u D maxfu1 ; u2 g is also subharmonic in D. Proof. It is clear that u is continuous in D. Let c 2 D. Without loss of generality we assume u.c/ D u1 .c/. Then

248

9 Harmonic Functions

u.c/ D u1 .c/

1 2

1 2

Z

2

u1 .c C r e{ / d Z

0 2

u.c C r e{ / d 0

for all small positive values of r, and the result follows.

t u

This result provides many examples of subharmonic functions (that are not harmonic): If h is a real-valued harmonic function, then so is h, and therefore hC D maxfh; 0g and jhj D maxfh; hg are subharmonic. In particular, u.z/ D jzj2 is an example of a subharmonic function in C (that is not harmonic). 7. Harmonic functions are of class2 C2 , and they are characterized by the vanishing of their Laplacian. We have seen above that subharmonic functions need not be differentiable, but we now prove that a real-valued function u of class C2 in a domain D is subharmonic if and only if u 0 in D. Proof. First assume that u 2 C2R .D/ and u.z/ > 0 for all z in D. Let G be a bounded subdomain of D such that cl G D, and let ' be a continuous realvalued function in cl G such that ' is harmonic in G, and u.z/ '.z/ for all z in @G. Assume u.z/ — '.z/ for all z in G. Then v D u ' is a continuous function in cl G, and it attains its maximum at a point c in G. But v is of class C2 in G, and hence v.c/ 0; but v D u ' D u and we have obtained a contradiction. Therefore u is subharmonic. In the general case u 2 C2R .D/ and u.z/ 0 for all z in D, set u .z/ D u.z/ C jzj2 for each positive and all z in D. Then u is of class C2 in D, with u > 0 in D, and hence subharmonic in D for each positive . It is also clear that u converges uniformly to u on compact subsets of D as goes to zero, and therefore u is subharmonic in D. For the converse assume u 2 C2R .D/ and u is subharmonic in D. Suppose that there exists c in D such that u.c/ < 0. Then u is superharmonic in a neighborhood of c, and hence harmonic in that neighborhood, from where it follows that u.c/ D 0, a contradiction. t u Example 9.25. The complex plane does not carry any negative nonconstant subharmonic function (Exercise 9.21). Whether a domain does or does not carry such a function is an important distinguishing characteristic. It classifies domains in b C into three types, and serves as the basis for the next definition. Definition 9.26. The Riemann sphere is called elliptic. A proper domain D C [ f1g is called parabolic if it does not carry a nonconstant bounded from above subharmonic function; otherwise it is called hyperbolic. Remark 9.27. It is unfortunate that at times terminology, including our last definitions, can be confusing. 2

They are, of course, much smoother than that, but this suffices for the discussion that follows.

9.7 Perron Families

249

• It is easy to see that the type of a domain is a conformal invariant in the sense that for all domains D and all conformal maps f , D and f .D/ have the same type. In Sect. 8.4 we gave a different definition of hyperbolic simply connected domains. We will see in Theorem 9.47 that both notions coincide for simply connected plane domains. • However, according to the above definitions, the extended complex plane C [ f1g punctured at a positive number n of points is parabolic. Whereas in the classification of domains in relation to what their simply connected holomorphic cover is, such domains are regarded to be hyperbolic if n 3. Example 9.28. Every bounded domain D C is hyperbolic. Choose any c 62 D andˇR > ˇ0 so that D is contained in U.c; R/, and observe that the function z 7! ˇz c ˇ log ˇ ˇ is nonconstant, negative, and harmonic (hence subharmonic) on D. R

9.7 Perron Families To prove that the Dirichlet problem has a solution (in fact, a unique one) for certain domains (in addition to closed discs), we need to consider families of subharmonic functions that are closed under certain operations. We start by establishing some terminology. Definition 9.29. Let D be a domain in C, u W D ! R be a continuous function, and U be an open disc such that cl U D. The continuous function uU defined in D by being harmonic in U and coinciding with u in D U is called the harmonization of u in U . Lemma 9.30. Let D be a domain in C and U be an open disc such that cl U D. If u is subharmonic in D, then so is uU . Proof. It suffices to show that uU satisfies the mean value inequality (9.15) at every point c in @U . Since u is subharmonic in D, u.z/ uU .z/ for all z in D. But then 1 uU .c/ D u.c/ 2

Z

2

1 u.c C r e / d 2

Z

2

uU .c C r e{ / d

{

0

for all small positive values of r, and the result follows.

0

t u

Remark 9.31. We have shown that the family of subharmonic function on a domain D is a cone (i.e., the family is closed under addition and multiplication by positive constants) in the vector space of continuous functions on D. It is also closed under maximization (by property 6 above) and harmonization (by Lemma 9.30). These last two properties are the key to progress. Definition 9.32. Let D be a domain in C. A Perron family F in D is a nonempty collection of subharmonic functions in D such that

250

9 Harmonic Functions

(a) If u; v are in F , then so is maxfu; vg. (b) If u is in F , then so is uU for every disc U with cl U D. The following result, due to Perron, is useful for constructing harmonic functions. Theorem 9.33 (Perron’s Principle). If F is a uniformly bounded from above Perron family in D, then the function defined for z 2 D by V .z/ D supfu.z/ W u 2 F g

(9.16)

is harmonic in D. Proof. First note that by definition a Perron family is never empty. Since we are assuming that there exists a constant M such that u.z/ < M for all z in D and all u in F , the function V is clearly well defined and real-valued. Let U be any disc such that cl U D. It is enough to show that V is harmonic in U . For any point z0 in U , there exists a sequence fuj W j 2 Ng of functions in F such that lim uj .z0 / D V .z0 /: (9.17) j !1

Without loss of generality, we may assume uj C1 uj for all j in N, since if fuj g is any sequence in F satisfying (9.17), then the new sequence given by v1 D u1 and vj C1 D maxfuj C1; vj g for j 1 is also contained in F , satisfies (9.17) (with uj replaced by vj , of course), and is nondecreasing, as needed. The sequence fwj D .uj /U g of harmonizations of the uj in U consists of subharmonic functions with the following properties: 1. wj uj for all j 2. wj wj C1 < M for all j , since the two inequalities clearly hold outside U and on the boundary of U , from which it follows that they also hold in U . Thus the sequence fwj g lies in F , is nondecreasing, and satisfies lim wj .z0 / D j !1

V .z0 /. It follows from the Harnack’s convergence Theorem 9.14 that the function defined by ˆ.z/ D lim wj .z/ D supfwj .z/ W j 2 Ng j !1

is harmonic in U . We will now show that ˆ D V in U . Let c denote any point in U . As before, we can find a nondecreasing sequence fsj g in F such that V .c/ D lim sj .c/. j !1

By setting t1 D maxfs1 ; w1 g and tj C1 D maxfsj C1 ; wj C1 ; tj g for all j 1, we obtain a nondecreasing sequence ftj g in F such that tj wj for all j , and such that lim tj .z/ D V .z/ for z D c and z D z0 .

j !1

9.8 The Dirichlet Problem (Revisited)

251

The harmonizations of the tj in U give a nondecreasing sequence frj D .tj /U g in F satisfying M > rj tj wj for all j . As before, the function defined by ‰.z/ D supfrj .z/ W j 2 Ng is harmonic in U , and coincides with V at c and z0 . But ‰ ˆ, since rj wj for all j , and hence ‰ ˆ is a nonnegative harmonic function in U . Since it is equal to zero at z0 , by the minimum principle for harmonic functions, it is identically zero in U , and the result follows. t u

9.8 The Dirichlet Problem (Revisited) This section has two parts. The first describes a method for obtaining the solution to the Dirichlet problem, provided it is solvable. In the second part, we offer a solution. Recall that the Dirichlet problem for a bounded region D in C and a function f 2 C.@D/ is to find a continuous function U on the closure of D whose restriction to D is harmonic and which agrees with f on the boundary of D. Under these conditions, let F denote the family of all continuous functions u on cl D such that u is subharmonic in D and u f on @D. Then F is a Perron family of functions uniformly bounded from above. Note that the constant function u D minff .z/ W z 2 @Dg belongs to F , hence F is nonempty. The other conditions for F to be a Perron family are also easily verified. Therefore, by Theorem 9.33, the function V defined by (9.16) is harmonic in D. Now, if we assume that there is a solution U to the Dirichlet problem for D and f , then we can show that U D V . Indeed, for each u in F the function u U is subharmonic in D, and satisfies u U D u f 0 on @D, from where it follows that u U 0 in D, and hence V U in D. But U belongs to F , and it follows that U V , and therefore U D V . The Dirichlet problem does not always have a solution. A very simple example is given by considering the domain D D f0 < jzj < 1g and the function f .z/ D

0; 1;

if jzj D 1; if z D 0:

The corresponding function V given by Theorem 9.33 is harmonic in the punctured disc D. If the Dirichlet problem were solvable in our case, then V would extend to a continuous function on jzj 1 that is harmonic in jzj < 1 (see Exercise 9.17). But then the maximum principle would imply that V is identically zero, a contradiction. To solve the Dirichlet problem, we start with a bounded domain D C, with boundary @D, and the following definition.

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9 Harmonic Functions

Definition 9.34. A function ˇ is a barrier at z0 2 @D, and z0 is a regular point for the Dirichlet problem provided there exists an open neighborhood N of z0 in C such that (1) (2) (3) (4)

ˇ 2 C.cl D \ N /. ˇ is subharmonic in D \ N . ˇ.z/ > 0 for z 6D z0 , ˇ.z0 / D 0. ˇ.z/ D 1 for z 62 N .

Remark 9.35. A few observations are in order. 1. Condition (4) is easily satisfied by adjusting a function ˇ that satisfies the other three conditions for being a barrier. To see this we may assume that N is relatively compact in C, and choose a smaller neighborhood N0 of z0 with cl N0 N . Then let m D minfˇ.z/I z 2 cl.N N0 / \ cl Dg; note that m > 0, and define ˇ1 .z/ D

minfm; ˇ.z/g m

for z 2 N \ D; for z 2 cl.D N /:

ˇ1 , and observe that ˇ2 satisfies all the conditions for being a Finally set ˇ2 D m barrier at z0 . Thus, to prove the existence of a barrier, it suffices to produce a function that satisfies the first three conditions. 2. The existence of barriers is a local property. If a point z0 2 @D can be reached by an analytic arc (a curve that is the image of Œ0; 1 under an injective analytic map defined in a neighborhood of Œ0; 1) with no points in common with cl D fz0 g, then a barrier exists at this point. To establish this we may, without loss of generality, assume that z0 D 0, that the closure of D lies in the right half plane, and that the analytic arc consists of the negative real axis including the origin. 1 Using polar coordinates z D re{ , we see that ˇ.z/ D r 2 cos 2 , < < , satisfies the first three conditions for a barrier function. Definition 9.36. Let D be a nonempty domain in C. A solution u to the Dirichlet problem for f 2 CR .@D/ is proper provided infff .w/I w 2 @Dg u.z/ supff .w/I w 2 @Dg for all z in D. A far-reaching generalization of Schwartz’s Theorem 9.15 is provided by our next result.

9.8 The Dirichlet Problem (Revisited)

253

Theorem 9.37. Let D be a nonempty domain in C. There exists a proper solution to the Dirichlet problem for D for every bounded continuous real-valued function on @D if and only if every point on @D is a regular point for the Dirichlet problem. Proof. We leave it to the reader to discuss the cases where the boundary of D is empty or consists of a single point. So assume from now on that @D has two or more points. First we assume that there exists a proper solution to the Dirichlet problem for D for every bounded continuous real-valued function on @D; we prove that every z0 2 @D is a regular point for the Dirichlet problem. Define a continuous function f on C as follows: f .z/ D

jz z0 j 1

if jz z0 j 1 if jz z0 j 1

and restrict it to @D. Let u be a proper solution to the Dirichlet problem for f 2 CR .@D/. Thus 0 u 1. Since f is not identically zero, u > 0 by the minimum principle and is thus a barrier at z0 . To prove the converse, let f be a bounded continuous real-valued function on @D, and let F consist of all functions v that satisfy the following conditions: • • • •

v is continuous on cl D v is subharmonic on D v satisfies m D inf f v sup f D M in D v.z/ f .z/ for all z 2 @D.

Now observe that the constant function m belongs to F ; this family is obviously closed under maximization and harmonization, and is thus a Perron family, uniformly bounded from above by M . It follows from Theorem 9.33 that the function V defined by (9.16) is harmonic in D and satisfies m V M there. Let z 2 @D. To establish that lim V .w/ D f .z/ we verify that w!z

(a) lim infw!z V .w/ f .z/ (b) lim supw!z V .w/ f .z/ The arguments that follow are generic for this type of problem: for part (a) we construct a particular function w 2 F that helps, while the proof of (b) involves arguing about all v 2 F . Proof of (a): If f .z/ D m, there is nothing to prove. Assume hence that f .z/ > m. Choose a positive real such that f .z/ > m, and note that can be chosen arbitrarily small. There exists a neighborhood N of z such that f ./ f .z/ for all 2 N \ @D. Let ˇ be a barrier at z that is identically equal to 1 outside of N , and define w./ D .f .z/ m /ˇ./ C f .z/

for 2 cl D:

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9 Harmonic Functions

We show that the function w 2 F . Observe that w 2 CR .cl D/ is subharmonic on D. For 2 cl D, w./ f .z/ < M; and w./ D mˇ./ C .f .z/ /.1 ˇ.// > mˇ./ C m.1 ˇ.// D m: For 2 @D, w./ D m f ./ if 62 N and w./ f .z/ f ./ if 2 N: We have completed the proof that w 2 F . Hence w./ V ./ for all 2 D, and therefore lim inf V ./ w.z/ D f .z/ : !z

Since is arbitrary, (a) follows. Proof of (b): As before, there is nothing to verify under certain conditions: in this case, if f .z/ D M . So we assume that f .z/ < M and choose > 0 such that f .z/ C < M , and a relatively compact neighborhood N of z such that f ./ f .z/ C for 2 N \ @D. Let, as above, ˇ be a barrier at z. Fix an arbitrary v 2 F . We claim that v./ .M f .z/ /ˇ./ f .z/ C

for all 2 N \ D:

(9.18)

Observe that the function on the LHS of (9.18) is subharmonic; hence it suffices to establish the inequality for 2 @.N \ D/; we consider two cases: (i) 2 @N \ cl D, and (ii) 2 cl N \ @D. In case (i), the LHS of (9.18) satisfies v./ .M f .z/ /ˇ./ D v./ M C f .z/ C f .z/ C I while in case (ii) we estimate it by v./ f ./ f .z/ C : Hence, we obtain, for all 2 N \ D, the estimate on v given by v./ f .z/ C C .M f .z/ /ˇ./I since v 2 F is arbitrary, the same estimate holds for the function V . Thus

9.8 The Dirichlet Problem (Revisited)

255

lim sup V ./ f .z/ C : !z

Since > 0 can be chosen to be arbitrarily small, (b) follows.

t u

As a second application of Perron’s method, we establish the following result. Theorem 9.38. Let D be a hyperbolic domain in C and K a compact subset of D with D K connected and @K regular (for the Dirichlet problem). There exists a unique function ! 2 CR ..D K/ [ @K/ such that • • • •

! is harmonic on D K. ! D 1 on @K. 0 < ! < 1 on D K. If !1 2 CR ..D K/ [ @K/ is a competing function that satisfies the above three properties, then !1 !.

Proof. Let 0 be any nonconstant positive superharmonic function on D, whose existence is guaranteed by the hyperbolicity of D. We now adjust this function to obtain a more suitable one. 0 Let m0 be the minimum of 0 on K, and set 1 D . Then 1 is a positive, m0 nonconstant, superharmonic function on D, with 1 jK 1. There clearly exists a point z 2 K such that 1 .z/ D 1. If z is in the interior of K, then 1 D 1 on the connected component of K containing the point z, thus also on the boundary of that component. In particular, there exists a z 2 @K with . z/ D 1. There also exists a point w 2 D K with 1 .w/ < 1, since otherwise 1 1, and the fact that 1 .z/ D 1 would imply that 1 is constant. Setting D minf1; 1 g we obtain a superharmonic function on D that satisfies 0 < 1, .w/ < 1, and jK D 1. Let F consist of all functions v 2 CR ..D K/ [ @K/ satisfying the following conditions: 1. v is subharmonic on D K. 2. v j.D K/. 3. v has compact support. The family we have defined is closed under maximization and harmonization, and thus a Perron family if not empty; we proceed to establish this next. Choose a small disc in D so that its complement in D contains K. Let v0 be the solution to the Dirichlet problem on K with boundary values 1 on @K and 0 on @; then 0 v0 1 on cl. K/. Extending v0 to be 0 on D yields a subharmonic function on D K. By the maximum principle, v0 0 on D K. (This inequality holds on D since v0 j.D / D 0 and > 0, hence also on @. On @K both functions v0 and have the constant value 1; thus .v0 /j. K/ 0.) Defining !.z/ D supfv.z/g for all z 2 .D K/ [ @K; v2F

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9 Harmonic Functions

we see that • • • • • • •

v0 ! ! is harmonic on D K ! D 1 on @K ! 2 CR ..D K/ [ @K/ ! is not constant, since !.w/ 0 ! 1 on .D K/ [ @K 0 < ! < 1 on D K

.w/ < 1

Finally let ! 0 be a competing function satisfying the above conditions. Let v 2 F and consider the function ! 0 v. Assume that the compact set K 0 D contains the support of v. Since ! 0 0 and v D 0 on @K 0 , ! 0 v 0 on @.K 0 K/, hence also on .K 0 K/ [ .D K 0 / D D K and on @K. Since v 2 F is arbitrary, ! 0 !. t u Definition 9.39. For D and K as above, the function ! described by the theorem is called the harmonic measure of K.

9.9 Green’s Function and RMT Revisited In this section we introduce a new function and use it to give a second proof of the RMT. Definition 9.40. Let D C be a nonempty domain and let z 2 D. A real-valued function g on D fzg is the Green’s function for D with singularity at z if (1) g is harmonic and positive on D fzg. (2) The function w 7! g.w/ C log jw zj is (extends to be) harmonic on D (3) gQ 2 CR .D fzg/ is another function satisfying the above two conditions, then gQ g. Remark 9.41. (a) The last condition in the definition guarantees that the Green’s function is unique if it exists. (b) The entire plane C does not have a Green’s function with singularity at any point. Assume to the contrary that g is a Green’s function for C with singularity at zero (this latter assumption involves no loss of generality). It follows that z 7! g.z/ C log jzj extends to be a real-valued harmonic function u on C. Then u has a (single-valued) harmonic conjugate v, and thus e.uC{ v/ is an entire function. Let R > 0; we estimate for jzj D R ˇ u.z/{ v.z/ ˇ ˇˇ g.z/logjzj ˇˇ 1 1 ˇ D ˇe ˇe D : ˇ R jzj

9.9 Green’s Function and RMT Revisited

257

and conclude that the inequality holds for all jzj R. Since R may be arbitrarily large, we have arrived at a contradiction, the contradiction that our entire function (an exponential!) vanishes identically. (c) The Green’s function for the unit disc D with singularity at 0 is given by z 7! log jzj. For proper simply connected domains in C, we may use the RMT to construct their Green’s functions (see Exercise 9.24). We handle the reverse direction by giving a second proof of the RMT based on the existence of Green’s functions—which we now proceed to prove. The value of the Green’s function for a (hyperbolic) domain D with singularity at c 2 D at any point d 6D c in D will be denoted by gD .d; c/, and by g.d; c/ when the domain is clear from the context. Theorem 9.42. Let D be a nonempty domain in C. There exists a Green’s function on D with singularity at some point c 2 D if and only if D is hyperbolic. In the latter case D has a Green’s function with a singularity at any arbitrary point of D. Proof. Assume that there exists a Green’s function on D with singularity at some point c 2 D. It involves no loss of generality (by Exercise 9.25) to assume that D contains the unit disc and that c D 0. Let g be the Green’s function for D with singularity at 0. Then lim g.z/ D C1. z!0

Let m > 0. Then f D minfg; mg is positive and superharmonic on D f0g, and f is constant (equal to m) near 0; therefore f is positive and superharmonic on D. By choosing m sufficiently large, we can make certain that f is not constant. Thus D is hyperbolic, since it carries the nonconstant negative subharmonic function f . To establish the converse, we define F to consist of all real-valued functions v satisfying the following conditions: 1. v 0 is subharmonic on D f0g. 2. Kv D fw 2 D f0gI v.w/ 6D 0g [ f0g is compact (“roughly expressed,”v has compact support) 3. w 7! v.w/ C log jwj is (extends to be) a subharmonic function on D. To see that F is nonempty we define the function v.w/ D maxf log jwj ; 0g for w 6D 0. It is subharmonic on the plane punctured at the origin, hence certainly on D f0g, and it clearly belongs to the family F . It is quite obvious that this family is closed under maximization and harmonization; hence it is a Perron family. We show next that the family F is uniformly bounded outside every neighborhood of 0. We have not yet used the hypothesis that D is hyperbolic. It is this last assertion that requires hyperbolicity (in the form of existence of harmonic measures, see Theorem 9.38). Let 0 < r < 1 and let !r be the harmonic measure of fjzj rg. Thus !r is harmonic on D fjzj rg, 0 < !r < 1 on this set, and !r .z/ D 1 for jzj D r. Let r D maxf!r .z/I jzj D 1g; hence 0 < r < 1.

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9 Harmonic Functions

For u 2 F , let ur D maxfu.z/I jzj D rg. We claim that ur !r .z/ u.z/ 0 for all z 2 D such that jzj r:

(9.19)

Hence, in particular, we record for later use that ur r u.z/ 0 for jzj D 1:

(9.20)

To verify (9.19), note that the function ur !r u is superharmonic on D\fjzj > rg and is nonnegative on fjzj D rg. Let K D Ku be the compact support of u. Since u D 0 on @K, ur !r u 0 there. Thus (9.19) holds for K fjzj rg; this inequality certainly holds on D K, and the claim is verified. Next we use the fact that z 7! u.z/ C log jzj is continuous on fjzj 1g and subharmonic on fjzj < 1g. Thus ur C log r D maxfu.z/ W jzj D rg C log r maxfu.z/ W jzj D 1g C log 1 r u r where the last inequality follows from (9.20). We conclude that log r ; ur 1 r and since u D 0 off the compact set K, we see that maxfu.z/I z 2 D and jzj rg

log r : r 1

By Perron’s principle (Theorem 9.33) g.z/ D sup u.z/; z 2 D f0g u2F

defines a nonnegative harmonic function. This function is actually positive since it would otherwise be constant. Finally we must show that z 7! g.z/ C log jzj is harmonic on fjzj < rg. For any such z and for every u 2 F , u.z/ C log jzj ur C log r

r log r : r 1

Thus the same inequality holds when we replace u by g. To complete the proof that g is the Green’s function of D with singularity at 0, we must show that if gQ is a competing candidate, then gQ g (from this it will follow that either gQ > g or gQ D g). Now if u 2 F , then U D gQ u is superharmonic on

9.9 Green’s Function and RMT Revisited

259

D. If K is the support of u, then U 0 on D K, and by the minimum principle for superharmonic functions also on K. Since u 2 F is arbitrary, gQ g. t u Remark 9.43. The above existence proof for Green’s function generalizes to Riemann surfaces. We present next a second, simpler, argument that is valid only for plane domains and also yields additional information. Theorem 9.44. Let D C be a domain with nonempty @D that is regular for the Dirichlet problem. Then D is hyperbolic. In particular, for every c 2 D, let u 2 C.D [ @D/ be a harmonic function in D with u.z/ D log jz cj for z 2 @D. Then gD .z; c/ D u.z/ log jz cj

for all z 2 D fcg:

Proof. It suffices, of course, to prove the particular claim. Without loss of generality D D fz 2 CI jzj < 1g D and c D 0. Let G.z/ D u.z/ log jzj for z 2 D [ @D f0g. Observe that the hypothesis on D guarantees that u exists and that u.z/ log jzj D 0 for z 2 @D. For jzj D 1, u.z/ > 0 and log jzj D 0. It follows that G.z/ 0 for z 2 D D: For 0 < jzj < 1, u.z/ > 0 and log jzj < 0, hence G.z/ > 0. Hence G is a positive harmonic function on D f0g. Obviously z 7! G.z/ C log jzj defines a harmonic function on D. So G is a candidate for the Green’s function of D with singularity at 0. Let gQ be a competitor, and let d 2 D f0g and > 0. Choose a neighborhood N of @D in cl D such that d 62 N and G.z/ < for all z 2 N . Then gQ G on @.D N / D @N , and thus also on D N ; in particular, g.d Q / G.d / . Since d 2 D is arbitrary and so is > 0, gQ G and G D gD .; 0/. t u The proofs of the next two results are left as exercises for the reader. Theorem 9.45. The complex plane punctured at n 0 points is parabolic. Theorem 9.46. Let D be a hyperbolic domain in C. Then g.z; w/ D g.w; z/

for all z 6D w 2 D:

The final application of our work on subharmonic functions is a second proof of Theorem 9.47 (Riemann Mapping Theorem, Version 2). Every hyperbolic simply connected domain in C is conformally equivalent to the unit disc D. Proof. Let c 2 D. We want to use the Green’s function g.; c/ to produce a Riemann Q map f .; c/. If g had a harmonic conjugate g, Q we could define f D e.gC{ g/ . The problem is that g has a singularity at c. To get around this problem, we note that the function u.; c/ defined on D by u.z; c/ D g.z; c/ C log jz cj is harmonic and real-valued. Since D is simply connected, u.; c/ has a harmonic conjugate v.; c/ in D. It follows that F .z; c/ D e.u.z;c/C{v.z;c// defines (for fixed c) a holomorphic function of z on D. Now

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9 Harmonic Functions

u.z; c/ D g.z; c/ log jz cj D < log.F .z; c// D log jF .z; c/j or, equivalently, g.z; c/ D log j.z c/F .z; c/j : We define f .z; c/ D .z c/F .z; c/: Obviously f .; c/ is a holomorphic function on D that vanishes (has a simple zero) only at c. Since log jf .z; c/j D g.z; c/ < 0, f .z; c/ 2 D for all z 2 D. For any three points c; d; z 2 D, set '.z/ D '.c; d; z/ D

f .d; c/ f .z; c/ 1 f .d; c/f .z; c/

:

(9.21)

Now '.c; d; / is f .; c/ followed by a M¨obius transformation that leaves D invariant, and hence a holomorphic map of D into D, with '.d / D '.c; d; d / D 0. In the above argument, we fixed c and d and regarded ' as a function of z. We need to relate ' to some f ; for these purposes we may regard ' as a function of one of the three variables c, d , and z for fixed values of the other two. However, only for fixed c and d is ' a holomorphic function of z. Thus the emphasis on the variable z in the left-hand side of (9.21). We study the function './ in a neighborhood of z D d . Let n 1 be the order of vanishing of ' at d ; then we have the power series expansion '.z/ D ˛.z d /n 1 C a1 .z d / C a2 .z d /2 C ; with ˛ 6D 0, for all z with jz d j sufficiently small, and if we set 1 v.z/ D log j'.z/j; n we obtain a positive harmonic function on D punctured at the (isolated) zeros of '. We may regard v to be defined on all of D with value C1 at the zeros of '. We proceed to compare v to g.; d / D log jf .; d /j. Let F be the family of functions used to define g.; d /. Let w 2 F and let K be the compact support of w. By enlarging K if necessary, we may assume that none of the zeros of ' lie on @K. Let K0 be the set K from which we delete small discs about each of the finitely many zeros excluding d of ' in the compact set K. We claim that v.z/ w.z/ 0 for all z 2 K. Let us first establish this claim for z 2 @K0 . The boundary of K0 consists of @K and finitely many circles about zeros of '. The circles do not cause any problems since w is bounded on each of them (by a bound independent of the radius of the circle if we a priori specify that the radii of these circles be less than some fixed number) and v ! C1 as we shrink the circles further (which is clearly permissible). Now on the boundary of @K, v 0 and w D 0. Since the superharmonic function v w on K is 0 on @K0 , by the

9.9 Green’s Function and RMT Revisited

261

minimum principle for superharmonic functions, it is also nonnegative on K0 . It follows that v w on D fcg. Hence for z 2 D fcg we have 1 log j'.z/j D v.z/ sup h.z/ D g.z; d / D log jf .z; d /j n h2F or

1

j'.z/j j'.z/j n jf .z; d /j :

(9.22)

Setting z D c (since by continuity the last set of inequalities are also valid at c), we conclude that jf .d; c/j jf .c; d /j, and since c and d are arbitrary, we may interchange them to obtain jf .c; d /j D jf .d; c/j ; an equality that also follows from Theorem 9.46 (which we did not prove and hence are not using). We consider another holomorphic function from D into the closed unit disc. We define '.z/ h.z/ D ; z 2 D: f .z; d / From (9.22), we conclude that jh.z/j 1; since ˇ ˇ ˇ ˇ ˇ ˇ ˇ '.c/ ˇ ˇ f .d; c/ f .c; c/ 1 ˇˇ ˇˇ f .d; c/ ˇˇ ˇDˇ D D 1; jh.c/j D ˇˇ f .c; d / ˇ ˇ 1 f .d; c/f .c; c/ f .c; d / ˇ ˇ f .c; d / ˇ h is constant, and there exists a 2 C with j j D 1 and such that ' D f .; d /, or, equivalently, f .d; c/ f .z; c/ f .z; d / D for all z 2 D: 1 f .d; c/f .z; d / We conclude that f .d; c/ D f .z; c/ iff f .z; d / D 0 iff z D d I that is, f .; c/ is an injective holomorphic map of D into D that takes c to 0. The proof concludes by showing that f .; c/ is surjective, in a manner quite similar to that used in the earlier proof of the RMT. Assume that we do not have surjectivity, and let D f .; c/.D/ D. The domain is simply connected and contains 0. Let t 2 be any point in D . We construct a function h W ! D by the formula q zt 2 {t 1Nt 2 z q h.z/ D ; z 2 : zt 2 1 C { tN 1N 2 t z The square root is defined (and holomorphic) since p is simply connected and zt 2 D 6 0 for z 2 ; we choose the square root so that t 2 D {t. Straightforward 2 1Nt z

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9 Harmonic Functions

calculations now show that ˇ ˇ 1 C jtj2 > 1; jh.z/j < 1; h.0/ D 0; ˇh0 .0/ˇ D 2 jtj

for all z 2 :

We now define W D h ı f .; c/ and observe that W is a holomorphic function from D into D, with a simple zero at c and no other zeros. Hence log jW j is a competing function for the Green’s function g.; c/. Thus log jW j g.; c/ D log jf .; c/j or jW j jf .; c/j : We thus see that ˇ ˇ ˇ h.z/ ˇ ˇ ˇ ˇ z ˇ1

ˇ ˇ for small positive jzj and ˇh0 .0/ˇ 1I t u

a contradiction.

Remark 9.48. Let D C be a nonempty simply connected domain and let c 2 D. We have introduced two numerical invariants for the pair .D; c/. The first is jf 0 .c/j, where f is any Riemann map from D onto D with f .c/ D 0. The second is the value at c of gD .; c/ C log j. c/j. The two invariants agree.

Exercises 9.1. Prove that the equivalent forms for the Laplacian given in equations (9.1) and (9.2) are correct. 9.2. Show that a function z 7! f .z/ on a symmetric region is harmonic (analytic) if and only if the function z 7! f .Nz/ is. 9.3. Show that if ! D f dz C g dz is a continuous differential form on a domain D, then ! D { .f dz g dz/: 9.4. We have shown that if u1 and u2 are real-valued harmonic functions on D, then u1 du2 u2 du1 is a closed form on D and have asserted that it also holds for complex-valued harmonic functions. Prove this assertion. 9.5. Prove the maximum and minimum principles for real-valued harmonic functions: 1. As a general result for real-valued functions that satisfy the MVP.

Exercises

263

2. As a consequence of Harnack’s inequalities for positive harmonic functions. 9.6. Let K be a compact subset of a domain D C, and let u be a positive harmonic function on D. Show that there exists a constant c 1 that depends only on K and D, but not on u, such that 1 u.z1 / c; c u.z2 / for all z1 and z2 2 K. 9.7. Complete the proof of Theorem 9.14 and show that both possibilities for its conclusion do occur. 9.8. Let u be a continuous real-valued function on a domain D. @2 u @2 u Suppose that the partial derivatives and exist and satisfy Laplace’s @x 2 @y 2 equation u D 0 in D. Show that u is harmonic on D. Hint: Use the notation in the proof of Theorem 9.20. Let c D a C { b in D. Show first that for all > 0, the function F .z/ D u.z/ v.z/ C .x a/2 satisfies the maximum principle in fjz j r0 g. 9.9. Does the area MVP imply harmonicity for continuous functions? 9.10. If u is real-valued and harmonic { on jzj < 1, continuous on jzj 1, and { u.e / D cos 2 C sin 2, find u . 2 9.11. Suppose that u.0/ D 1, where u is harmonic and positive in a neighborhood of fz 2 CI jzj 1g. Prove that 17 u 34 7. 9.12. Let ˛ be a real number. For D e{ with 2 R, let './ D cos C { ˛ sin : Which of the following assertions are true for all ˛ in R? Which are true for some values of ˛? Z './ 1 d is holomorphic for jzj < 1. (a) The function f .z/ D 2{ jjD1 z (b) There exists a function f holomorphic for jzj < 1, continuous for jzj 1 and satisfying f ./ D './ for jj D 1. (c) There exists a function f holomorphic for jzj < 1 such that

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