Complex Analysis Graduate Texts in Mathematics Second Edition

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

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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

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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.

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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

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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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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

;

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Standard Notation and Commonly Used Symbols

A List of Symbols

Term Z Q R C b C { {R
@f

, @Nz @R jRj clR intR Xcondition .f / i./ e./ f jB U.z; r/ D Uz .r/ D H2 S1 S 2 Š C [ f1g

Meaning Integers Rationals Reals Complex numbers C [ f1g The square root of 1 The imaginary axis in C Real part of z Imaginary part of z x D
Interior of the Jordan curve  Exterior of the Jordan curve  Restriction of the function f to the subset B of its domain f 2 CI j  zj < rg U.0; 1/ fz 2 CI =z > 0g fz 2 CI jzj D 1g Riemann sphere

Chapter 1

The Fundamental Theorem in Complex Function Theory

This introductory chapter is meant to convey the need for and the intrinsic beauty found in passing from a real variable x to a complex variable z. In the first section we “solve” two natural problems using complex analysis. In the second, we state what we regard as the most important result in the theory of functions of one complex variable, which we label the fundamental theorem of complex function theory, in a form suggested by the teaching and exposition style of Lipman Bers; its proof will occupy most of this volume. The next two sections of this chapter include an outline of our plan for the proof and an outline for the text, respectively; in subsequent chapters we will define all the concepts encountered in the statement of the theorem in this chapter. The reader may not be able at this point to understand all (or any) of the statements in our fundamental theorem or to appreciate its depth and might choose initially to skim this material. All readers should periodically, throughout their journey through this book, return to this chapter, particularly to the last section, that contains an interesting account of part of the history of the subject.

1.1 Some Motivation 1.1.1 Where Do Series Converge? In the calculus of a real variable one encounters two series that converge for jxj < 1 but in no larger open interval (on the real axis): 1 D 1  x C x 2     C .1/n x n C    1Cx and 1 D 1  x 2 C x 4     C .1/n x 2n C    : 1 C x2 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 1, © Springer Science+Business Media New York 2013

1

2

1 The Fundamental Theorem in Complex Function Theory

It is natural to ask why these two series that are centered at the origin have radius 1 of convergence 1. The answer for the first one is natural: the function 1Cx has a singularity at x D 1, and so the series certainly cannot represent the function at this point, which is at distance 1 from 0. For the second series, the answer does 1 not appear readily within real analysis. However, if we view 1Cx 2 as a function of 1 the complex variable x, then we again conclude that the series representing this function should have radius of convergence 1, since that is the distance from 0 to the singularities of the function; they are at ˙{.

1.1.2 A Problem on Partitions A natural question in elementary additive number theory is the following: Is it possible to partition the positive integers Z>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
zz zCz and =z D : 2 2{

One easily verifies the following basic

3

With these operations .C; C; / is a field.

2.1 Introduction and Preliminaries

17

ιR

z=x +ιy

ιy r θ

x R

z=x−ιy

Fig. 2.1 The complex plane; rectangular and polar representations, conjugation

2.1.1 Properties of Conjugation Forz and w 2 C, (a) (b) (c) (d)

zCwD zCw zw D z w z D z if and only if z 2 R zDz

There is a simple and useful geometric interpretation of conjugation: it is represented by mirror reflection in the real axis; see Fig. 2.1. From a slightly different point of view, conjugation may be seen as a self-map of C, denoted by . W C ! C: Then its properties (a) through (d) may be restated as follows: (a) (b) (c) (d)

preserves the sum of complex numbers. preserves the product of complex numbers. fixes precisely the real numbers. is an involution of C; that is, when composed with itself, it gives the identity map on C.

It is not hard to show that any self-map of C satisfying these properties coincides with complex conjugation; see Exercise 2.19.

18

2 Foundations

Another important map, z 7! jzj or j j W C ! R0 1

1

is defined by r D jzj D .z z/ 2 D .x 2 C y 2 / 2 , where z D x C { y. Note that z z D x 2 C y 2 is always a real nonnegative number; we use the usual convention that unless otherwise specified the square root of a real nonnegative number is chosen to be nonnegative. The nonnegative real number r is called the absolute value or norm or modulus of the complex number z. The following properties follow directly from the definition.

2.1.2 Properties of Absolute Value For z and w 2 C, (a) (b) (c) (d) (e)

jzj  0, and jzj D 0 if and only if z D 0 jzwj D jzj jwj jzj D jzj j
2.1.3 Linear Representation of C As a vector space over R, we can identify C with R2 . Vector addition agrees with complex addition, and scalar multiplication by real numbers (R  C ! C) is the restriction of complex multiplication (C  C ! C). This identification provides a powerful geometric interpretation for many results on complex numbers. One example is provided by conjugation, which can be viewed as the R-linear map of R2 (with basis 1 D .1; 0/ and { D .0; 1/) that sends 1 to 1 and { to {. Another instance is provided by the next geometric interpretation of the following:

2.1.4 Additional Properties of Absolute Value From the Pythagorean equality, r D jzj is the (Euclidean) distance in the plane from z to the origin; see Fig. 2.1. Furthermore, for z and w 2 C, the following properties hold. (f) jz C wj2 C jz  wj2 D 2.jzj2 C jwj2 /. (g) jz C wj  jzj C jwj. Equality holds whenever either z or w is equal to 0. If z ¤ 0 and w ¤ 0, then equality holds if and only if w D az with a 2 R>0 . (h) jjzj  jwjj  jz  wj.

2.1 Introduction and Preliminaries

19

z +w w

z 0

z −w −w Fig. 2.2 Vector sums

Equality (f) is sometimes called the parallelogram law: the sum of the squares of the lengths of the diagonals in a parallelogram is equal to the sum of the squares of the lengths of its sides, see Fig. 2.2. This equality can be proven directly from the definition of absolute value and properties of the complex conjugation we have already stated: jz ˙ wj2 D .z ˙ w/.z ˙ w/ D jzj2 ˙ 2 <.z w/ C jwj2 :

(2.1)

Inequality (g) is called the triangle inequality: the length of a side of a triangle is at most equal to the sum of the lengths of the other two sides, with equality if and only if the triangle is degenerate (z and w lie on the same ray); see Fig. 2.2. The triangle inequality (or rather its squared version) follows from (2.1) and the previous properties of the absolute value, by observing that <.z w/  jz wj D jzj jwj and using the conditions for equality given in property (d) for the absolute value. Through the identification of R2 with C given above, we can use real or complex notation to describe geometric shapes in the plane. As we show next, sometimes the use of complex notation simplifies the description of the objects under study.

2.1.5 Lines, Circles, and Half Planes Any line in the plane R2 with orthogonal coordinates xand y is given by an equation of the form a x C b y C c D 0; (2.2) with a, b, and c real numbers, and a and b not both equal to zero.

20

2 Foundations

Similarly, any circle in the plane is given by an equation of the form .x  d /2 C .y  f /2  R2 D 0;

(2.3)

with d , f , and R real numbers and R > 0. In this case, we can read off from the equation that the center of the circle is at the point .d; f /, and its radius is R. We will now see an advantage of using complex notation: both of the above types of geometric figures may be described algebraically by a single equation, thus implying that there is a certain relation between lines and circles on the plane (this relation will be explained later: see Exercise 3.21). zz zCz and y by first in (2.2) and then in (2.3), we obtain Replacing x by 2 2{ the following two equations: B z C B z C c D 0 ; with B D

a{b ¤ 0; 2

(2.4)

and jzj2 C .d C { f / z C .d  { f / z C d 2 C f 2  R2 D 0;

(2.5)

or, equivalently, jz  Ej D R with E D d C { f:

(2.6)

We claim that both equations (2.4) and (2.5) are special cases of A jzj2 C B z C B z C C D 0 ;

(2.7)

with A and C real numbers, B complex, A  0, and jBj2 > AC . Indeed, if A D 0 then (2.7) becomes (2.4), which is equivalent to (2.2), whereas B if A > 0 then (2.7) becomes (2.5), which is equivalent to (2.3) with center E D  A q 2 jBj  AC . and radius R D A We have thus shown that any circle or line in the plane is given by Equation (2.7), depending on whether A > 0 or A D 0. Similarly, half planes in C are given by equations of the form <.B z/ > C or, equivalently, =.B z/ > 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: .
Theorem 2.14. Let S be a subset of C and let f and g be functions defined on S . If c is a limit point of S , then:

26

2 Foundations

(a) lim .f C a g/.z/ D lim f .z/ C a lim g.z/ for all a 2 C z!c

z!c

(b) lim .fg/.z/ D lim f .z/ lim g.z/ z!c ˇz!c ˇz!c ˇ ˇ (c) lim jf j .z/ D ˇˇ lim f .z/ˇˇ z!c z!c

z!c

(d) lim f .z/ D lim f .z/ z!c

z!c

Remark 2.15. The usual interpretation of the above formulae is used here and in the rest of the book: the LHS6 exists whenever the RHS exists, and then we have the stated equality. Corollary 2.16. Let S be a subset of C, let f be a function defined on S , and ˛ 2 C. Set u D
if and only if lim u.z/ D <˛ and lim v.z/ D =˛:

z!c

z!c

Definition 2.17. Let S be a subset of C, f W S ! C be a function defined on S , and c 2 S be a point in S . We say that: (a) f is continuous at c if lim f .z/ D f .c/. z!c

(b) f is continuous on S if it is continuous at each c in S . (c) f is uniformly continuous on S if for all  > 0, there is a ı > 0 such that jf .z/  f .w/j <  for all z and w in S with jz  wj < ı: Remark 2.18. A function f is (uniformly) continuous on S if and only if both
1 is defined in a neighborhood of f

c and is continuous at c. .b/ If f is continuous at c and g is continuous at f .c/, then g ı f is continuous at c. Theorem 2.20. Let K  C be a compact set and f W K ! C be a continuous function on K. Then f is uniformly continuous on K.

6

LHS (RHS) are standard abbreviations for left (right) hand side and will be used throughout this book.

2.2 Topological Preliminaries

27

Proof. A continuous mapping from a compact metric space to a metric space is uniformly continuous. u t Definition 2.21. Given a sequence of functions ffn g, all defined on the same set S in C, we say that ffn g converges uniformly to a function f on S if for all  > 0 there exists an N 2 Z>0 such that jf .z/  fn .z/j <  for all z 2 S and all n > N: Remark 2.22. ffn g converges uniformly on S (to some function f ) if and only if for all  > 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
2.3.2 The Cauchy-Riemann (CR) Equations Theorem 2.33. If f D u C { v is differentiable at c D a C { b, then u and v have partial derivatives with respect to x and y at c, and they satisfy the Cauchy– Riemann equations: ux .a; b/ D vy .a; b/; uy .a; b/ D vx .a; b/:

.CR/

2.3 Differentiability and Holomorphic Mappings

31

Furthermore, f 0 .c/ D ux .a; b/ C { vx .a; b/ D { uy .a; b/ C vy .a; b/: Proof. First take h D ˛, with ˛ real, in the limit (2.9) appearing in the definition of differentiability and compute f 0 .c/ D ux .a; b/ C { vx .a; b/: Then take h D {ˇ, with ˇ real, and compute f 0 .c/ D { uy .a; b/ C vy .a; b/: Comparing the two expressions we obtain the desired result.

t u

Remark 2.34. Let f D u C { v be a function defined in a neighborhood of c, such that the partial derivatives ux , vx , uy , and vy exist at c. Then we will use the obvious notation: fx D ux C { vx and fy D uy C { vy : In this language the CR equations (CR) for the function f are written as follows: fx .c/ D { fy .c/;

(2.10)

and Theorem 2.33 may be stated as follows. If f is differentiable at c, then f has partial derivatives with respect to x and y at c, and they satisfy the Cauchy–Riemann equation (2.10). Furthermore, f 0 .c/ D fx .c/ D { fy .c/:

(2.11)

Remark 2.35. The CR equations are not sufficient for differentiability. To see this, define ( z5 jzj4 for z ¤ 0; f .z/ D 0 for z D 0: It is easy to verify that the function f is continuous on C. Furthermore, for ˛ real f .˛/ f .{ˇ/ and nonzero we have D 1, and for ˇ real and nonzero we have D 1. ˛ {ˇ Therefore fx .0/ D 1 and fy .0/ D {, and f satisfies the Cauchy–Riemann equation (2.10) at z D 0. However, f is not differentiable at z D 0. Indeed, if it were, we would conclude from (2.11) that f 0 .0/ D 1. Now take h D .1 C {/

f .h/ D 1 so that f 0 .0/ would be equal with real and nonzero and observe that h to 1.

32

2 Foundations

Remark 2.36. We may use the CR equations to try to manufacture an entire function with a given real (or imaginary) part. Let us start with the real-valued function u.x; y/ D x 2 Cy 2 . If this were to be the real part of some entire function f D uC{ v, then the CR equations would help us to determine v. Since ux D 2x D vy and uy D 2y D vx must be satisfied, by integrating 2x D vy with respect to y we obtain that v.x; y/ D 2 x y C h.x/, for some function h of x alone; by integrating 2y D vx with respect to x we obtain that v.x; y/ D 2 y x C g.y/, for some function g of y. It is quite obvious that these two expressions for v are incompatible, and hence there is no such function f . The situation changes dramatically for u.x; y/ D x 2  y 2 , where similar calculations lead to v.x; y/ D 2 x y C h.x/ D 2 y x C g.y/, and we may choose h.x/ D g.y/ D a, for any real value of a. We have thus obtained a family of entire functions f with prescribed real part u; these are given by f .z/ D u.x; y/ C { v.x; y/ D x 2  y 2 C { .2xy C a/ D z2 C { a; with a any real number. We will determine later the class of real-valued functions u for which the construction outlined above leads us to an entire function f . Definition 2.37. For a complex-valued function f defined on a region in the complex plane, such that both fx and fy exist in this region, set fz D

1 fx  {fy 2

and 1 fx C {fy : 2 Remark 2.38. The partial derivatives just defined are computed as if z and z were independent variables. For instance, if f .z/ D z2 C 5zz3 , then it is easy to verify that fz D 2z C 5z3 and fz D 15zz2 . These partials not only simplify the notation: for example, the two Cauchy– Riemann equations (CR) are written as the single equation fz D

fz D 0;

.CR complex/

or, if f is differentiable at c, then f 0 .c/ D fz .c/; but they also allow us to produce more concise arguments (and, as we shall see later, prettier formulae), as illustrated in the proof of the lemma below. @f @f (respectively ) interchangeably with fz (respecWe use the notation @z @z tively fz ).

2.3 Differentiability and Holomorphic Mappings

33

Lemma 2.39. If f is a C1 -complex-valued function defined in a neighborhood of c 2 C, then f .z/  f .c/ D .z  c/ fz .c/ C .z  c/ fz .c/ C jz  cj ".z; c/;

(2.12)

for all z 2 C with jz  cj small, where ".z; c/ is a complex-valued function of z and c such that lim ".z; c/ D 0:

z!c

Proof. As usual we write z D x C { y, c D a C { b, and f D u C { v and abbreviate 4u D u.z/  u.c/, 4x D x  a, 4y D y  b, and 4z D z  c D 4x C { 4y. By hypothesis, the real-valued function u has continuous first partial derivatives defined in a neighborhood of c, and we can define "1 by "1 .z; c/ D

4u  ux .c/4x  uy .c/4y j4zj

for z ¤ c, and "1 .c; c/ D 0. Then it is clear that u.z/  u.c/ D .x  a/ux .a; b/ C .y  b/uy .a; b/ C jz  cj "1 .z; c/: We now show that lim "1 .z; c/ D 0:

z!c

(2.13)

If we rewrite 4u as 4u D Œu.x; y/  u.x; b/ C Œu.x; b/  u.a; b/ ; it follows from the (real) mean value theorem applied to the two summands on the RHS that 4u D uy .x; y0 /4y C ux .x0 ; b/4x; where y0 is between y and b and x0 is between x and a. Thus "1 .z; c/ D

Œuy .x; y0 /  uy .a; b/4y C Œux .x0 ; b/  ux .a; b/4x ; j4zj

for z ¤ c. Hence we see that ˇ ˇ j"1 .z; c/j  ˇuy .x; y0 /  uy .a; b/ˇ C jux .x0 ; b/  ux .a; b/j ; and the claim (2.13) follows. Similarly, v.z/  v.c/ D .x  a/vx .a; b/ C .y  b/vy .a; b/ C jz  cj "2 .z; c/;

34

2 Foundations

with lim "2 .z; c/ D 0:

z!c

(2.14)

With obvious notational conventions, we compute that 4f D 4u C { 4v

 D Œux .a; b/ C { vx .a; b/ 4x C uy .a; b/ C { vy .a; b/ 4y C j4zj ".z; c/

D

4z C 4z 4z  4z fx .c/ C { fy .c/ C j4zj ".z; c/ 2 2

D 4zfz .c/ C 4zfzN .c/ C j4zj ".z; c/; with ".z; c/ D "1 .z; c/ C { "2 .z; c/. Now equalities (2.13) and (2.14) imply that lim ".z; c/ D 0:

z!c

t u

Theorem 2.40. If the function f has continuous first partial derivatives in a neighborhood of c that satisfy the CR equations at c, then f is (complex) differentiable at c. Proof. The theorem is an immediate consequence of (2.12), since in this case t u fzN .c/ D 0 and hence f 0 .c/ D fz .c/. Corollary 2.41. If the function f has continuous first partial derivatives in an open neighborhood U of c 2 C and the CR equations hold at each point of U , then f is holomorphic at c (in fact on U ). Remark 2.42. The converse to this corollary is also true. It will take us some time to prove it. Theorem 2.43. If f is holomorphic and real-valued on a domain D, then f is constant. Proof. As usual we write f D u C { v; in this case v D 0. The CR equations say ux D vy D 0 and uy D vx D 0. Thus u is constant, since D is connected. t u Theorem 2.44. If f is holomorphic and f 0 D 0 on a domain D, then f is constant. Proof. As above f D u C { v and f 0 D ux C { vx D 0. The last equation together with the CR equations say 0 D ux D vy and ; 0 D vx D uy . Thus both u and v are constant, since D is connected. t u

Exercises 2.1. (a) Let fzn g be a sequence of complex numbers and assume jzn  zm j <

1 ; for all n and m: 1 C jn  mj

Exercises

35

Show that the sequence converges. Do you have enough information to evaluate lim zn ? n!1 What else can you say about this sequence? (b) Let fzn g be a sequence with lim zn D 0 and let fwn g be a bounded sequence. n!1 Show that lim wn zn D 0: n!1

2.2. (a) Let z and c denote two complex numbers. Show that jc z  1j2  jz  cj2 D .1  jzj2 / .1  jcj2 /: (b) Use (a) to conclude that if c is any fixed complex number with jcj < 1, then fz 2 CI jz  cj < jc z  1j g D fz 2 CI jzj < 1g ; fz 2 CI jz  cj D jc z  1j g D fz 2 CI jzj D 1g and fz 2 CI jz  cj > jc z  1j g D fz 2 CI jzj > 1g:

2.3. Let a, b, and c be three distinct points on a straight line with b between a and c. Show that ab 2 R<0 : c b 2.4. (a) Given two points z1 , z2 such that jz1 j < 1 and jz2 j < 1, show that for every point z 6D 1 in the closed triangle with vertices z1 , z2 , and 1, j1  zj  K; 1  jzj where K is a constant that depends only on z1 and z2 . (b) Determine the smallest value of K for z1 D

1C{ 1{ and z2 D . 2 2

2.5. Verify the Cauchy–Riemann equations for the function f .z/ D z3 by splitting f into its real and imaginary parts. 2.6. Suppose z D x C { y. Define f .z/ D

xy 2 .x C { y/ ; x2 C y 4

for z ¤ 0 and f .0/ D 0. Show that lim

f .z/  f .0/ D0 z

36

2 Foundations

as z ! 0 along any straight line. Show that as z ! 0 along the curve x D y 2 ; the limit of the difference quotient is 12 , thus showing that f 0 .0/ does not exist. 2.7. Let x D r cos  and y D r sin . Show that the Cauchy–Riemann equations in polar coordinates for F D U C { V , where U D U.r; / and V D V .r; /, are r

@U @V D @r @

and r

@V @U D ; @r @

or, in alternate notation, rUr D V and rVr D U : 2.8. Let f be a complex-valued function defined on a region in the complex plane, and assume that both fx and fy exist in this region. Using the definitions of fz and fz , show that for C1 -functions f , f is holomorphic if and only if fz D 0 and that in this case fz D f 0 . 2.9. Let R and ˚ be two real-valued C1 -functions of a complex variable z. Show that f D Re{˚ is holomorphic if and only if Rz C {R˚z D 0: 2.10. Show that if f and g are C1 -functions, then the (complex) chain rule is expressed as follows (here w D f .z/ and g is viewed as a function of w). .g ı f /z D gw fz C gw f z and .g ı f /z D gw fz C gw f z : 2.11. Let p be a complex-valued polynomial of two real variables: p.z/ D Write p.z/ D

X X

aij x i y j :

Pj .z/zj ;

j 0

where each Pj is of the form Pj .z/ D an entire function if and only if

P

bij zi (a polynomial in z). Prove that p is

0  P1  P2  : : : :

Exercises

37

What can you conclude in this case for the matrix Œaij ? 2.12. Deduce the analogues of the CR equations for anti-holomorphic functions, in rectangular, polar, and complex coordinates. 2.13. Let f W C ! C be a holomorphic function, and set g.z/ D f .z/ and h.z/ D f .z/, for z in C. Show that g is holomorphic and h is anti-holomorphic on C. Furthermore, h is holomorphic on C if and only if f is a constant function. 2.14. Let D be an arbitrary (nonempty) open connected set in C. Describe the class of complex-valued functions on D that are both holomorphic and anti-holomorphic. 2.15. Does there exist a holomorphic function f on C whose real part is: (a) u .x; y/ D ex ? Or (b) u .x; y/ D ex .x cos y  y sin y/? Justify your answer; that is, if yes, exhibit the holomorphic function(s) and if not, prove it. 2.16. Prove the fundamental theorem of algebra: If a0 ; : : : ; an1 are complex numbers (n  1) and p.z/ D zn C an1 zn1 C C a0 , then there exists a number z0 2 C such that p.z0 / D 0. Hint: A standard method of attack: (a) Show that there are an M > 0 and an R > 0 such that for all jzj  R, jp.z/j  M holds. (b) Show next that there is a z0 2 C such that jp.z0 /j D min f jp.z/j I z 2 Cg: (c) By the change of variable p.z C z0 / D g.z/, it suffices to show that g.0/ D 0. (d) Write g.z/ D ˛ Czm .ˇ Cc1 zC Ccnm znm / with ˇ ¤ 0. Choose such that ˛

m D  : ˇ If ˛ ¤ 0, obtain the contradiction jg. z/j < j˛j for some z. Note. We will later have several simpler proofs of this theorem using results from complex analysis, for instance, in Theorem 5.16 and Exercise 6.1. See also the April 2006 issue of The American Mathematical Monthly for yet other proofs of this fundamental result. 2.17. Conclude from the fundamental theorem of algebra that a nonconstant complex polynomial of degree n has n complex roots, counted with multiplicities. Use this result to show that a nonconstant real polynomial that cannot be factored as a product of two nonconstant real polynomials of lower degree (i.e., a real irreducible nonconstant polynomial) has degree one or two.

38

2 Foundations

2.18. Using the fundamental theorem of algebra stated in Exercise 2.16, prove the Frobenius theorem: If F is a field containing the reals whose dimension as a real vector space is finite, then either F is the reals or F is (isomorphic to) C. Hint: An outline of possible steps follows. (a) Assume dimR F D n > 1. Show that for  in F  R there exists a nonzero real polynomial p with leading coefficient 1 and such that p./ D 0. (b) Show that there exist real numbers ˇ and such that  2  2ˇ C D 0: (c) Show that there exists a positive real number ı such that .  ˇ/2 D ı 2 , and therefore D

 ˇ ı

is an element of F satisfying  2 D 1. (d) The field G D R./ D fx C y W x; y 2 Rg  F is isomorphic to C, so without loss of generality assume  D { and G D C. Conclude by showing that any element of F is the root of a complex polynomial with leading coefficient 1 and is therefore a complex number. 2.19. Prove the following statements, where automorphism is a bijection preserving sums and products. (a) Every automorphism of the real field is the identity. (b) Every automorphism of the complex field fixing the reals is either the identity or conjugation. (c) Every continuous automorphism of the complex field is either the identity or conjugation. 2.20. A domain is defined to be an open connected set. It was remarked that it could also be defined to be an open arcwise connected set. Can it be defined as an open path connected set? Justify your answer.

Chapter 3

Power Series

This chapter is devoted to an important tool for constructing holomorphic functions: convergent power series. It is the basis for the introduction of new non-algebraic holomorphic functions, the elementary transcendental functions. It turns out that power series play an even more central role in the theory of holomorphic functions, a role beyond enabling the construction of complex transcendental functions that are the extension of the real transcendental functions. A much stronger result holds. All holomorphic functions are (at least locally) convergent power series. This will be proven in the next chapter. The first section of this chapter is devoted to a discussion of elementary properties of complex power series. Some material from real analysis, not usually treated in books or courses on that subject, is studied. The concept of a convergent power series is extended from series with real coefficients to complex power series, and tests for convergence are established. In the second section, we show that convergent power series define holomorphic functions. Section 3.3 introduces important complex-valued functions of a complex variable including the exponential function, the trigonometric functions, and the logarithm. This is followed by Sects. 3.4 and 3.5, which describe an identity principle and introduce the new class of meromorphic functions; these functions are holomorphic on a domain except that they “assume the value 1” (in a controlled way) at certain isolated points, known as the poles of the function. Meromorphic functions are defined locally as ratios of functions having power series expansions. It will hence follow subsequently that these are locally ratios of holomorphic functions. After some more work we will be able to replace “locally” by “globally.” We develop the fundamental identity principle and its corollary, the principle of analytic continuation, for functions that locally have convergent power series expansions, and discuss the zeros and poles of a meromorphic function. The principle of analytic continuation is one of the most powerful results in complex function theory. Once we show that every holomorphic function is locally defined by a power series, we will see that the principle of analytic continuation says that a holomorphic function defined on an open connected set is

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 3, © Springer Science+Business Media New York 2013

39

40

3 Power Series

remarkably rigid: its behavior at a single point in the set determines its behavior at all other points of the set. Holomorphicity at a point is an extremely strong concept.

3.1 Complex Power Series Let A  C and let ffn g D ffn g1 nD0 be a sequence of functions defined on A (in the previous chapter sequences were indexed by Z>0 ; for convenience, in this chapter, they will be indexed by Z0 ). We form the new sequence (known as a series) fSN g, where SN .z/ D

N X

fn .z/; z 2 A;

nD0

and the formal infinite series

1 X

fn .z/.

nD0

The sequence of complex numbers fSN .z/gN is also known as the sequence of 1 X fn .z/ at the point z 2 A. When the partial sums associated to the infinite series nD0

indices of summation are clear from the context, we often omit them. For example, 1 X X fn ./. Other similar abbreviations and fn ./ usually means the infinite sum nD0

conventions are used. Definition 3.1. We distinguish several types of convergence for a series. We say 1 X fn : that the infinite series nD0

(i) Converges at a point c 2 A if fSN .c/g converges. In this case we write 1 X fn .c/ D lim SN .c/. nD0

N !1

(ii) Converges pointwise in A if fSN .z/g converges for every z 2 A. (iii) Converges absolutely at a point c 2 A if the infinite series

1 X nD0

jfn .c/j

converges. (iv) Converges uniformly in A if the sequence of partial sums fSN g converges uniformly in A. (v) Converges normally on a set B  A if there exists a sequence of positive constants fMn g such that: (1) jf n .z/j  Mn for all z 2 B and all X Xn. (2) Mn < 1; that is, the series Mn converges. (vi) Diverges at a point of A if it does not converge at that point.

3.1 Complex Power Series

41

We speak of the pointwise, uniform, absolute, or normal convergence of a series as well as the divergence of a series. Remark 3.2. The fact that the sequence fSN .c/g converges if and only if it is a Cauchy sequence allows us to rephrase conditions (i) and (ii) as if and only if statements are useful: P (i) P fn converges at a point c 2 A if and only if fSN .c/g is Cauchy. (ii) fn converges pointwise in A if and only if fSN .z/g is Cauchy for every z 2 A. Remark 3.3. Many questions on convergence of complex sequences are reduced to the real case by the trivial but important observation that absolute convergence at a point implies convergence at that point. Remark 3.4. Two other such observations, both trivial but important, are that 1 X if an infinite series fn converges at a point c, then lim fn .c/ D 0 and lim

n!1

1 X

n!1

nD0

fk .c/ D 0.

kDn

Some of the relationships between some of the different types of convergence of a series are given in the following result. Theorem 3.5 (Weierstrass M -test). Normal convergence implies uniform and absolute convergence. Proof. With notation as in the definition of normal convergence, if N1 < N are positive integers, then N X

jSN .z/  SN1 .z/j 

Mn for all z 2 B

nDN1 C1

(needed for the uniform convergence argument), and j jSN .z/j  jSN1 .z/j j 

N X

Mn for all z 2 B

nDN1 C1

(needed for the absolute P convergence argument). Since the series Mn converges, for any given  > 0, we can find a positive N X Mn < , thus concluding the integer N0 such that N > N1 > N0 implies nDN1 C1

proof.

t u

42

3 Power Series

We shall be mostly interested in series of the form

1 X

an zn with an 2 C (these

nD0

are known as power series) and the associated real-valued series

N X

jan jr n where

nD0

r D jzj. We define, for each N 2 Z0 , SN .r/ D

1 X

jan jr n for r 2 R0

nD0

and observe that SN C1 .r/  SN .r/ for all N 2 Z0 and for all r 2 R0 : We refer to SN .r/ as the real partial sum at r.  This nondecreasing sequence of real partial sums fSP N .r/g is important because if it converges at some fixed positive r, then the series an zn converges normally on the set fjzj  rg (with Mn D an r n in the definition of normal convergence). An elementary but most important example is provided by the geometric series: 1Cr Cr C D 2

1 X

r n:

(3.1)

nD0

In this case SN .r/ D

N X

r n for all nonnegative r, and therefore SN .1/ D N C 1,

nD0

1  r N C1 for 0  r < 1 and for r > 1. and SN .r/ D 1r 1 X P 1 Thus if 0  r < 1 and r n diverges if r  1. rn D 1r nD0 P It follows that the power series zn converges for all jzj < 1; the same argument 1 . Note that this power series diverges shows that in this case it converges to 1z n for all z with jzj  1, since in this case lim z D 0 is not satisfied, a necessary n!1 condition for the convergence of the series, as we saw in Remark 3.4. We now introduce two special cases of divergence of a sequence of real numbers. It will be useful to regard these sequences as convergent sequences with infinite limits. Definition 3.6. A sequence of real numbers fbn g converges to C1 if for all M > 0 there exists an N 2 Z>0 such that bn > M for all n > N . In this case we shall write lim bn D C1. A similar definition applies to real sequences converging to 1. n!1

3.1 Complex Power Series

43

With this notation either: (a) lim SN .r/ exists and is finite: that is, N !1

write

1 X

1 X

jan jr n converges. In this case we

nD0

jan jr n < C1;

nD0

or (b) lim SN .r/ D C1 : that is, N !1

case we write

1 X

1 X

jan j r n diverges (in the previous sense). In this

nD0

jan j r n D C1.

nD0

For real series, we have the comparison test. Assume 0  ak  bk for all large values of k; that is, assume there exists N 2 N such that 0  ak  bk for all k  N . Then the following results are easy to prove: P P (a) If P an D C1, then P bn D C1. (b) If bn < C1, then an < C1. As an application of the comparison test we prove the following very useful result. Lemma 3.7 (Abel’s Lemma). Assume that 0 < r < r0 . If there exists a positive number M such that jan r0n j  M for all n 2 Z>0 ; P then the series an zn converges normally for all z with jzj  r. In particular, it converges absolutely and uniformly for all z with jzj  r. Proof. For all jzj  r we have  jan zn j D jan j jzjn  jan j jrjn D jan j

r r0

n

 r0n  M

r r0

n :

P r < 1, the normal convergence of an zn follows by comparison with r0 thegeometric series or by an application of the Weierstrass M -test with Mn D n r M . t u r0

Since 0 <

If S is any nonempty set of real numbers, then the least upper bound or supremum of S is denoted by sup S , and the greatest lower bound or infimum of S is denoted by inf S . The possibilities that sup S D C1 or inf S D 1 are allowed.

44

3 Power Series

Definition 3.8. The radius of convergence  of the power series  D supfr 2 R0 I

X

P

an zn is given by

jan j r n < C1g:

Note that 0    C1. As a result of the next theorem it makes sense P to define the set fz 2 CI jzj < g as the disc of convergence of the power series an zn . P Theorem 3.9. Let an zn be a power series with radius of convergence  > 0. Then P (a) For any 0 < r < , the series an zn converges normally, absolutely, and uniformly for P jzj  r. (b) The series an zn diverges for jzj > : P Proof. (a) For any r0 satisfying r < r0 < , the series jan j r0n converges. Then lim jan j r0n D 0, and thus there exists an M > 0 with jan j r0n  M for all n in n!1 Z>0 . Now apply Abel’s lemma. (b) We claim that for jzj > , the sequence fjan j jzjn g is not even bounded. Otherwise Abel’s lemma P (with r0 D jzj) would guarantee the existence of an r with  < r < jzj and jan j r n < C1. This contradicts the definition of . u t The theorem makes no claim about the convergence or divergence of a power series on the boundary of its disc of convergence. There are power series with finite positive radii of convergence that diverge at every boundary point, as well as others that converge at all or some boundary points (see Exercise 3.1). We will explore this situation a little further in Theorem 3.22. P Corollary 3.10. Let an zn be a power P series with radius of convergence  > 0. Then the function defined by S.z/ D an zn is continuous for jzj < . Proof. It follows immediately from Theorems 2.23 and 3.9.

t u

We now turn to the obvious and important question: how do we compute ? To answer this question we introduce the concepts of limit inferior and limit superior of real sequences; these concepts appropriately belong to real analysis and are extremely useful in the current context. Definition 3.11. Let fun g be a real sequence. Here we use  to indicate equivalent names1 and define lim un  lim sup un  upper limit of fun g  limit superior of fung n

n

D lim sup fun g D inf sup fun g; p!1 np

p

np

We will also use this same notation with a different meaning in other places, such as f  0 or f  g, to emphasize that these functions are (identically) equal.

1

3.1 Complex Power Series

45

and lim un  lim inf un  lower limit of fun g  limit inferior of fun g n

n

D lim inf fun g D sup inf fun g: p!1 np

np

p

Note that every real sequence has a limit superior as well as a limit inferior, which are either real numbers or C1 or 1.

3.1.1 Properties of Limits Superior and Inferior Let fun g and fvn g be real sequences. Then we have: (a) lim inf un  lim sup un . n

n

(b) lim inf.un / D  lim sup un . n

n

(c) If r > 0, then lim inf.run / D r lim inf un ; and lim sup.run / D r lim sup un : n

n

n

n

More generally, (d) If lim vn exists, then n

lim inf.un vn / D .lim inf un /.lim vn /; and n

n

n

lim sup.un vn / D .lim sup un /.lim vn /; n

n

n

provided the right-hand sides are not indeterminates, that is, not of the form 0  .˙1/ nor .˙1/  0. (e) If un  vn for all n, then lim inf un  lim inf vn ; and lim sup un  lim sup vn : n

n

n

n

(f) lim un D L if and only if lim inf un D L D lim sup un n

n

(L is allowed to be ˙1 in this context). (g) lim inf.un C vn /  lim inf un C lim inf vn , and n

n

n

n

lim sup.un C vn /  lim sup un C lim sup vn : n

n

n

Remark and Exercise: If f is a real-valued function of a complex variable, defined on a set S in C and if c, is a limit point of S , then it is possible to define lim f .z/ and lim f .z/.

z!c

z!c

46

3 Power Series

  Example 3.12. Let un D sin n 2 ; n D 0; 1; 2; : : : ; this is the sequence: f0; 1; 0; 1; : : : g. Hence for all p 2 Z0 we have ap D sup un D 1 and thus lim sup un D lim ap D 1, and np

p!1

n

bp D inf un D 1 and thus lim inf un D lim bp D 1. np

n

p!1

3.1.2 The Radius of Convergence The ratio test will be well known to most readers (see Exercise 3.5). Suppose that vn > 0 for all nonnegative integers n. Then the following properties hold: P vnC1 (a) If lim D L < 1, then vn converges. n!1 vn P vnC1 D L > 1, then vn diverges. (b) If lim n!1 vn Perhaps less familiar (but extremely useful in the current setting) is the root test. Suppose that vn  0 for all nonnegative integers n: P 1 (a) If lim.vn / n D L < 1, then vn converges. n P 1 (b) If lim.vn / n D L > 1, then vn diverges. n

Proof. (a) Choose  > 0 such that 0 < L C  < 1; then there exists a P 2 Z>0 such that   1 sup vnn < L C  for all p  P: np

Thus vn < .L C /n for all n  P; and comparison P with the geometric series yields convergence. (b) Suppose that vn < C1. Then lim vn D 0 and thus there exists a P in Z>0 n

1

such that n  P implies that vn < 1 for all n  P . Hence .vn / n < 1 for all 1 n  P and therefore lim.vn / n  1. u t n Remark 3.13. The root test applies (with the same value of L) whenever the ratio test applies. However, the converse is not true. To see this take a sequence where the ratios are alternately 12 and 18 ; then the root test will apply with L D 14 , but the sequence of ratios is obviously divergent. Example 3.14. For general series, the two tests are not strong enough to determine convergence, as this example will show. However, we will see that we can compute the radius of convergence for any power series with the root test.

3.2 More on Power Series

Consider the series

47

X

ns , with s a positive real number. Using standard

n1

calculus techniques, one shows that 1 .n C 1/s D 1 D lim .ns / n I n!1 n!1 ns

lim

thus both the ratio and the root tests are inconclusive in this case. The integral test, not discussed in this book, easily handles this example, showing that the series converges for s > 1 and diverges (converges to C1) for 0 < s  1. We now return to the study of the complex power series

1 X

an zn and the problem

nD0

of computing its radius of convergence.

ˇ ˇ ˇ anC1 ˇ ˇ ˇD Theorem 3.15. If an z is a power series with an ¤ 0 for all n, and limˇ n an ˇ 1 L, with 0  L  C1, then its radius of convergence  is ; in other words, L P

n

1 D L;  1 D 0. C1 The hypotheses required for this result to hold are strong. As pointed out in Remark 3.13, the ratio test is not as effective as the root test. The next result provides a way of computing the radius of convergence for any power series. where

Theorem 3.16 (Cauchy–Hadamard). 1 X series an zn is given by

The radius of convergence  of the power

nD0 1 1 D lim jan j n : n  1

1

Proof. Let L D lim jan j n . Thus lim jan r n j n D rL for all r  0, and we conclude n n P by the root test that the associated series jan j r n converges for 0  r < L1 and diverges for r > L1 . Thus  D L1 . t u

3.2 More on Power Series Our next result is a technical lemma, useful in proving that the product of two convergent power series is a convergent power series.

48

3 Power Series

Lemma 3.17. If

P

un and

P

vn are absolutely convergent series and wn D

n X

up vnp ;

pD0

P

P P P wn is absolutely convergent and wn D . un / . vn /. X X Proof. Let ˛p D jun j and ˇp D jvn j. Then

then

np

np

lim ˛p D 0 D lim ˇp : p

Also

N X

jwn j 

nD0

and therefore

1 X

(3.2)

p

1 X

jun j

nD0

1 X

jvn j D ˛0 ˇ0 < C1;

nD0

jwn j < C1. Thus we have proven the absolute convergence of

nD0 P

the new series wn . To show the required equality, choose m and n with m  2n and consider ˇ m ! n !ˇ n ˇ ˇX X X ˇ ˇ wk  uk vk ˇ D L: ˇ ˇ ˇ kD0

kD0

kD0

We have to show that L ! 0 as n ! 1 (we already know each of the above series converges). We rewrite ˇ ˇ ˇX ˇ n n X k X ˇ m X ˇ L D ˇˇ ui vki  ui vj ˇˇ : ˇkD0 i D0 ˇ j D0 i D0 By looking at the diagrams in the .i; k/ plane shown in Fig. 3.1, we see that m X k X

ui vki D

kD0 i D0

m m X X

ui vki

(3.3)

i D0 kDi

D

m X i D0

ui

m X kDi

vki D

m X i D0

ui

mi X j D0

vj D

m mi X X i D0 j D0

u i vj :

3.2 More on Power Series

a

49

b

5 k

k 4

5 4

3

3

2

2

1

1

2

4

i

2

4

i

−1

−1

Fig. 3.1 The .i; k/ plane. (a) The first sum in (3.3). (b) The second sum

Thus we can estimate ˇ ˇ 2 3 ˇ ˇX n m mi X X X X ˇ ˇ n mi ˇ 4 5 u i vj  u i vj C ui vj ˇˇ LDˇ ˇ ˇ i D0 j D0 j D0 i DnC1 j D0 ˇ ˇ ˇ ˇ ˇ ˇ X ˇ ˇX mi mi ˇ ˇ m X ˇ ˇ n X ˇ ˇ ˇ u i vj ˇ C ˇ ui vj ˇˇ ˇ ˇ ˇi DnC1 j D0 ˇ ˇ i D0 j DnC1 

1 1 X X

1 1 X X ˇ ˇ ˇ ˇ jui j ˇvj ˇ C jui j ˇvj ˇ

i D0 j DnC1

i DnC1 j D0

D ˛0 ˇnC1 C ˇ0 ˛nC1 ; and the last expression approaches 0 as n goes to 1, by (3.2).

t u

It is straightforward to prove the following result (see Exercise 3.6). P P bn zn have radii of Theorem 3.18. Suppose the power series an zn and convergence that are both greater than or equal to a positive number . Then .a/

n X P P ai bni , have radii of convergence .an C bn /zn and cn zn , where cn D i D0

that are at least . .b/ For jzj < , we have X

.an C bn /zn D

X

an zn C

X

bn zn

50

3 Power Series

and

X

cn zn D

X

an zn

X

bn zn :

We already know that a power series defines a continuous function inside its disc of convergence, by Corollary 3.10. The next result shows that much more is true: it is C1 and holomorphic there. Theorem 3.19. A power series defines a holomorphic function in its disc of convergence, and all its derivatives exist and are holomorphic there. In particular, 1 X suppose the power series an zn has radius of convergence  > 0, and set nD0

S.z/ D

1 X

an zn

nD0

for jzj < . Then the power series

1 X

nan zn1 also has radius of convergence ,

nD1

and S 0 .z/ D

1 X

nan zn1 ; for all jzj < :

nD1 0

Proof. Let  be the radius of convergence of

1 X

nan zn1 . Then

nD1 1 1 1 1 1 1 D lim.n jan j/ n D lim n n  lim jan j n D 1  lim jan j n D ; n n n n 0 

where the first and last equalities follow from Hadamard’s theorem and the second one from Exercise 3.4. Thus we can define a continuous function T on jzj <  by T .z/ D

1 X

nan zn1 :

nD1

Note that if A and B 2 C and n 2 Z>1 , then An  B n D .A  B/.An1 C An2 B C    C B n1 /: Let us set for h ¤ 0 and jhj sufficiently small and for fixed z with jzj < , ˇ ˇ ˇ ˇ S.z C h/  S.z/  T .z/ˇˇ f .h/ D ˇˇ h

3.2 More on Power Series

51

ˇ1 ˇˇ ˇX .z C h/n  zn ˇ n1 ˇ  nz Dˇ an ˇ ˇ ˇ h nD2 ˇ1 ˇ ˇX ˇ

ˇ ˇ Dˇ an .z C h/n1 C .z C h/n2 z C    C zn1  nzn1 ˇ : ˇ ˇ nD2

Now we choose r so that jzj < r < , and then we choose h with jz C hj < r. Under these restrictions, ˇ ˇ ˇ.z C h/n1 C .z C h/n2 z C    C zn1  nzn1 ˇ ˇ ˇ ˇ ˇ  ˇ.z C h/n1 C z.z C h/n2 C    C zn1 ˇ C ˇnzn1 ˇ  2nr n1 ; and thus 0  f .h/ 

1 X

2 jan j nr n1 < C1:

nD2

Therefore given any  > 0 there exists a positive integer N with 1 X

2 jan j nr n1 <

nDN

 ; 2

and we can write ˇ X ˇ 1 ˇ ˇ .z C h/n  zn n1 ˇ ˇ 0  f .h/  nz ˇC 2 jan j n r n1 : jan j ˇ h nD2 nDN N 1 X

The first sum, being a finite sum, goes to 0 as h ! 0. Hence there exists a ı > 0 such that 0 < jhj < ı implies that the first term is at most 2 . Thus 0  f .h/ <  for 0 < jhj < ı, and therefore T .z/ D S 0 .z/. t u Remark 3.20. The theorem tells us that under certain circumstances we can interP change the order of computing limits: if the power series an zn has positive radius of convergence and if we let SN .z/ D

N X

an zn ;

nD0

then the theorem states that SN .z C h/  SN .z/ SN .z C h/  SN .z/ D lim lim : N !1 h!0 h!0 N !1 h h lim lim

52

3 Power Series

Corollary 3.21. If S.z/ D

1 X

an zn for jzj < , then for all n 2 Z0 and all

nD0

d nS are computed by term– jzj < , the power series for the derivatives S .n/ D dzn P term differentiation of an zn , and an D

S .n/ .0/ : nŠ

Proof. Applying the theorem and induction on n shows that 1

S .n/ .z/ D nŠ an C

X .n C 1/Š kŠ anC1 z C    D ak znk 1Š .n  k/Š kDn

t u

for all jzj < .

The results obtained so far have provided information about the behavior of a power series inside its disc of convergence. Our next result deals with a point on the boundary of this disc. P Theorem 3.22 (Abel’s Limit Theorem).P Assume that the power series an zn n has finite radiusP of convergence  > 0. If an z0 converges for some z0 with jz0 j D , then f .z/ D an zn is defined for fjzj < g [ fz0 g and we have lim f .z/ D f .z0 /;

z!z0

as long as z approaches z0 from inside the circle of convergence and jz  z0 j   jzj remains bounded. z Proof. By the change of variable w D we may assume that  D 1 D z0 (replace z0 P n an converges to f .1/. By changing a0 to a0  f .1/, we may an by an z0 ). Thus P assume that f .1/ D an D 0. Thus we are assuming that jzj < 1 (with 1  jzj j1  zj small) and that  M for some fixed M > 0. (Recall Exercise 2.4.) Let 1  jzj sn D a0 C a1 C    C an : Then lim sn D 0, n

Sn .z/ D a0 C a1 z C    C an zn D s0 C .s1  s0 /z C    C .sn  sn1 /zn

3.3 The Exponential and Related Functions

53

D s0 .1  z/ C s1 .z  z2 / C    C sn1 .zn1  zn / C sn zn D .1  z/.s0 C s1 z C    C sn1 zn1 / C sn zn ; and hence f .z/ D lim Sn .z/ D .1  z/ n!1

1 X

sn zn :

nD0

Given  > 0, choose N 2 Z>0 such that jsn j <  for n > N . Then ˇ ˇN ! 1 ˇ ˇX X ˇ n nˇ sn z ˇ C jsn j jzj jf .z/j  j1  zj ˇ ˇ ˇ nD0

nDN C1

ˇ ˇN ! ˇ ˇX jzjN C1 ˇ nˇ sn z ˇ C   j1  zj ˇ ˇ ˇ 1  jzj nD0 ˇN ˇ ˇX ˇ ˇ ˇ  j1  zj ˇ sn zn ˇ C M; ˇ ˇ nD0

and thus we conclude that lim f .z/ D 0. z!1

t u

Remark 3.23. Observe that we have not needed nor have we used polar coordinates in our formal development thus far. After completing the next section, we will be free to do so.

3.3 The Exponential Function, the Logarithm Function, and Some Complex Trigonometric Functions In this section we use power series to define and study several basic (and in some sense, familiar) transcendental functions.

3.3.1 The Exponential Function We discover the exponential function by looking for functions that are solutions of the ordinary differential equation f 0 .z/ D f .z/ subject to the initial condition f .0/ D 1:

54

3 Power Series

We try to find such a solution defined by a power series f .z/ D a0 C a1 z C    C an zn C    that converges near z D 0. Then (by Theorem 3.19) f 0 .z/ D a1 C 2a2 z C    C nan zn1 C    and thus we must have a0 D 1; a1 D a0 D 1; a2 D

1 1  a1 D ; : : : I 2 2Š

that is, anC1 D

1 1  an D (by induction): nC1 .n C 1/Š

Hence f .z/ D 1 C z C Now note that

1 n X zn z2 z CC C D : 2Š nŠ nŠ nD0

(3.4)

ˇ ˇ ˇ anC1 ˇ nŠ 1 ˇ D lim D lim ˇˇ D 0I n n .n C 1/Š  an ˇ

thus  D C1, and the power series (3.4) defines an entire transcendental function. We write ez and exp z for f .z/ and call it the exponential function. Recall that an algebraic number is a complex number that is the root of a nonconstant polynomial equation with rational (equivalently integer) coefficients, and a transcendental number is one that is not algebraic. A similar concept applies to functions. Definition 3.24. Let f be a function defined on a nonempty domain D  C. We say that f is an algebraic function if there exists a nonconstant complex polynomial F D F .z; w/ (in two variables) such that F .z; f .z// D 0 for all z in D. Otherwise, f is called transcendental.2 Under the above definition, polynomials and rational functions are algebraic in their domains of definition. For holomorphic functions f on the domain D, we will be able to conclude by the principle of analytic continuation that the property of being algebraic or transcendental continues to hold if we replace D by a subdomain or superdomain (to which it extends). Exercise 3.9 shows that the exponential function is a transcendental function.

2

It should be noted that this definition is appropriate for analysis; it is not so for algebra and number theory where it is appropriate to require that F be a polynomial with integer coefficients, that is, that it belongs to ZŒz; w rather than CŒz; w as in our case.

3.3 The Exponential and Related Functions

55

Example 3.25. At the end of the next section we will see that (well-defined, that is, single-valued) holomorphic power functions exist on appropriate subdomains of C. It will follow that there exist precisely two square root functions on any simply connected domain D not containing the origin. If f is one of these, it satisfies the polynomial equation F .z; f .z// D 0 where F .z; w/ D w2  z. Proposition 3.26. Let c 2 C. The function f .z/ D c ez is the unique power series, and also the unique entire function, satisfying f 0 .z/ D f .z/ and f .0/ D c:

(3.5)

Proof. It is trivial that z 7! cez satisfies (3.5) and is the unique power series to do so; we already know that this is an entire function. We postpone the proof that this is the unique entire function that satisfies (3.5) until after we establish the next two propositions. t u Proposition 3.27. For all z 2 C, ez ez D 1: Thus ez ¤ 0 for all z 2 C: Proof. Set h.z/ D ez ez for all z 2 C. Then h is an entire function, and the rules for differentiation tell us that h0 .z/ D ez ez  ez ez D 0 for all z. Therefore h must be constant, by Theorem 2.44; the result follows since h.0/ D 1. t u Proposition 3.28. ezCc D ez ec for all z and c in C. ezCc , with c fixed in C. The function h has a power series ec expansion that converges for all z 2 C, h0 .z/ D h.z/ for all z 2 C, and h.0/ D 1. Thus h.z/ D ez , as needed. t u

Proof. Define h.z/ D

3.3.1.1 Conclusion of Proof of Proposition 3.26 If g.z/ is any entire function satisfying (3.5), consider the function h.z/ D Then h is also entire, and the rules for differentiation tell us that h0 .z/ D

g 0 .z/ez  g.z/ez D 0 for all z 2 C: e2z

Thus h is constant, by Theorem 2.44.

g.z/ ez .

56

3 Power Series

Proposition 3.29. ez D ez for all z 2 C. Proof. This follows directly from the definition (3.4) of the exponential function. t u As an immediate consequence we have Proposition 3.30. For any z in C, write z D x C { y, with x and y in R. Then je{y j2 D e{y e{y D 1; and thus jez j D ex : The exponential function leads us immediately to our next section, the complex trigonometric functions.

3.3.2 The Complex Trigonometric Functions We define two entire functions, the cosine and the sine functions, by cos z D

z2 z4 e{z C e{z D 1 C  2 2Š 4Š

(3.6)

and e{z  e{z z3 z5 D z C   : 2{ 3Š 5Š It is then easy to verify the following familiar properties: sin z D

(3.7)

cos z C { sin z D e{z ; cos2 z C sin2 z D 1;

(3.8)

cos.z/ D cos z; sin.z/ D  sin z; and cos0 z D  sin z; sin0 z D cos z for all z in C. Remark 3.31. Observe that the equality e{z D cos z C { sin z given in (3.8), called Euler’s formula, is not (in general) the decomposition of the complex number e{z into its real and imaginary parts, since cos z and sin z are not necessarily real numbers. For the same reason, the equality cos2 z C sin2 z D 1 does not imply that the cosine and the sine functions are bounded on C, as they are on R. In fact, they are not bounded on C; see Exercise 3.13.

3.3 The Exponential and Related Functions

57

However, if z D x C { y, then we can compute the real and imaginary part of ez as ez D ex e{ y D ex cos y C { ex sin y:

(3.9)

In the next section we will formally define , and then, after some calculations, we will be able to obtain from (3.8) the beautiful Euler identity (connecting perhaps the four most interesting numbers in mathematics): e { C 1 D 0: Remark 3.32. It should be observed that the functions sine and cosine defined above agree, for real values of the independent variable z, with the familiar real-valued functions with the same names. The easiest way to conclude this is from the power series expansions of these functions at z D 0. Also note that sin z and cos z form a basis for the power series solutions to the ordinary differential equation f 00 .z/ C f .z/ D 0I that is, every solution f of this equation is of the form f .z/ D a cos z C b sin z for some complex constants a and b and for all z in C. Similarly, using either this last characterization of the sine and cosine functions, or the formula for the exponential of a sum, one establishes that for all z and c 2 C, cos.z C c/ D cos z cos c  sin z sin c and sin.z C c/ D sin z cos c C cos z sin c:

3.3.3 The Definition of  and the Logarithm Function In this section we show that the exponential function is periodic and hence not injective on C. However, since it is locally injective, it will have local inverses. In order to define a local inverse to the exponential function, our first task is then to establish the periodicity of ez . We will need to use some elementary results from calculus. For x 2 R, sin x and cos x (as defined by (3.7) and (3.6), respectively) are real numbers. From sin2 x C cos2 x D 1, we conclude that 1  cos x  1:

58

3 Power Series

Rx Rx Integrating for x  0 we have 0 cos t dt  0 dt, or sin x  x.RAlso, for x > 0, x we must have sin x < x.3 Equivalently,  sin x > x. Thus, 0 . sin t/ dt > Rx 2 x x2 0 .t/ dt. We conclude that cos x  1 >  2 or, equivalently, cos x > 1  2 . Repeating the argument, we obtain the following two inequalities for x ¤ 0: sin x > x 

x3 x2 x4 and cos x < 1  C : 6 2 24

(3.10)

Definition 3.33. Let f be a complex-valued function defined on C and let c 2 C; c ¤ 0. We say that f has period c (and call f periodic) if and only if f .zCc/ D f .z/ for all z 2 C. We show next that the exponential function is periodic (i.e., it has a period). Note that ezCc D ez for all z if and only if ec D 1. It follows from (3.9) that c D { !, with ! 2 R. Traditionally ! (and not { !) is called a period of the exponential function. We want to determine the smallest such positive !. First note that cos 0 D 1 (obvious). It follows from the inequalities (3.10) that p 9 1 3 D  < 0; 3<1 C 2 24 8 p and then continuity implies that there exists a y0 2 .0; 3/ such that cos y0 D 0. But then from cos2 y0 C sin2 y0 D 1 we obtain that sin y0 D ˙1, and thus {y0 e D cos y0 C { sin y0 D ˙{ and e4{y0 D 1. We conclude that 4y0 is a period of the exponential function. We claim that this is the smallest positive period and that any other period is an integral multiple of this one. p 2 2 Proof. If 0 < y < y0 < 3, then y 2 < 3 and 1  y6 > 12 ; thus sin y > y .1  y6 / > y 2 > 0, and we conclude that cos y is strictly decreasing on Œ0; y0 . Since cos2 y C sin2 y D 1 and sin y > 0 on .0; y0 /, we conclude that sin y is strictly increasing here. Thus 0 < sin y < sin y0 D 1. Hence e{y ¤ ˙1; ˙{, and therefore e4{y ¤ 1 for 0 < y < y0 . Thus !0 D 4y0 is the smallest positive period. If ! is an arbitrary period of exp (recall that ! 2 R), then so is j!j and there is an n 2 Z>0 such that n!0  j!j < .n C 1/!0 . If n!0 ¤ j!j, then 0 < .n C 1/!0  j!j is a positive period less than !0 . Since this is impossible by definition of !0 , we must have j!j D n !0 . t u cos

Definition 3.34 (Definition of ). We define the real number  by 4y0 D 2. Thus ez D 1 if and only if z D 2{n with n 2 Z.

The function x 7! x  sin x is certainly nondecreasing on Œ0; C1/ since its derivative is the function x 7! 1  cos x  0. The inequality sin x < x certainly holds for x > 1. If for some x0 in .0; 1 we would have sin x0 D x0 , then we would conclude from the mean value theorem that for some xQ 2 .0; x0 /, cos xQ D 1 which leads to the contradiction sin xQ D 0.

3

3.3 The Exponential and Related Functions

59

Traditionally 2 is defined as the smallest positive period of the (complex) exponential function. After some more work, it will become clear that our definition of the constant  agrees with the standard approach used in real analysis. As in real analysis, the inverse to the exponential function should be a logarithm. d We now turn to its definition. Since dx .ex / D ex > 0 for all x 2 R, ex is strictly increasing on R. Hence there exists an inverse function denoted by log (sometimes written as ln): exp W R ! .0; C1/; log W .0; C1/ ! R; and we have the well-known properties log ex D x for all x 2 R and elog x D x for all x 2 R>0 : We know that ez ¤ 0 for all z 2 C; we thus can expect to define a complex logarithm. The problem is that the exponential function exp W C ! C¤0 is not one-to-one. Let us write z 6D 0 in polar coordinates4 z D re{ (this agrees with our previous way of writing polar coordinates). Here r D jzj and  D arg z. The argument of z is defined up to addition of 2 n with n 2 Z. We also define log z D log jzj C { arg zI it is a multivalued function5 on C¤0 . The principal branch of arg z, Argz, is restricted to lie in .; . It is a function on C  f0g that has a jump discontinuity on the negative real axis. We define the principal branch of the logarithm by the formula Log z D log jzj C {Argz: It is easily seen to be a continuous function on C  .1; 0; moreover, it is C1 on this set. We note the following. Properties of the Complex Logarithm. 1. elog z D elogjzjC{ arg z D jzj e{ arg z D z for all z 2 C¤0 . 1 d 2. Log z is holomorphic on C  .1; 0, with Log z D there. dz z Proof. Write z D re{ with  <  < . Thus Log z D log r C { D u C { v:

4 5

Having defined , polar coordinates now rest on a solid foundation and can be used in proofs. Thus not a function.

60

3 Power Series

Calculate ur D 1r , v D 1, u D 0, and vr D 0, and thus rur D v and rvr D u . Hence (as observed earlier) Log is C1 and satisfies CR on C  .1; 0 (see Exercise 2.7); thus it is holomorphic there. We can now compute formally using the chain rule: eLog z D z: Thus eLog z

d Log z D 1; dz

and we conclude that d 1 Log z D eLog z D : dz z 3. Log is injective. 4. Log z1 z2 D Log z1 C Log z2 C ı; where

8 < 2{; ı D 0; : 2{;

if  2 < Argz1 C Argz2  I if   < Argz1 C Argz2  I if  < Argz1 C Argz2  2:

t u

Definition 3.35. A continuous function f on a domain D  C not containing the origin is called a branch of the logarithm on D if for all z 2 D, we have ef .z/ D z. Later we will establish appropriate conditions on the domain D for a branch of the logarithm to exist on it. We have seen that Log is a branch of the logarithm on D D C  .1; 0 (or in any subdomain of D); other branches of the logarithm in D are given by changing the choice we made for arg. Our next result shows that this is a general property. Theorem 3.36. Let D be a domain in C with 0 … D. If f is a branch of the logarithm on D, then g is also a branch of the logarithm in D if and only if there is an n 2 Z such that g.z/ D f .z/ C 2 { n for all z in D. Proof. If g D f C 2 { n with n 2 Z, then eg.z/ D ef .z/ e2 { n D z for all z in D. For a proof of the converse, define h.z/ D

f .z/  g.z/ ; z 2 D: 2{

1 D 1. Thus, for each z 2 D, there is an n 2 Z z such that h.z/ D n. Hence h.D/  Z. Since h is continuous and D is connected, h.D/ D fng for some fixed n 2 Z. t u Then e2 { h.z/ D ef .z/ eg.z/ D z

3.3 The Exponential and Related Functions

61

Corollary 3.37. Every branch of the logarithm on a domain D (with 0 62 D) is holomorphic on D. Proof. Holomorphicity is a local property, and there exist holomorphic branches of the logarithm in every sufficiently small disc that does not contain the origin. t u Theorem 3.38. For z 2 C with jzj < 1, Log.1 C z/ D

1 X z2 z3 zn .1/n1 D z  C     : n 2 3 nD1

Proof. We firstˇ compute ˇ the radius of convergence of the given series using the ratio ˇ n ˇ 1 ˇ D 1. Thus the function defined by test: D lim ˇˇ n  n C 1ˇ f .z/ D

1 X

.1/n1

nD1

zn n

(3.11)

is holomorphic in jzj < 1. We calculate f 0 .z/ D 1  z C z2     D

1 for jzj < 1: 1Cz

Let g.z/ D ef .z/ ; then g 0 .z/ D ef .z/ f 0 .z/ D

ef .z/ ; 1Cz

and g 00 .z/ D

.1 C z/ef .z/ f 0 .z/  ef .z/ D 0: .1 C z/2

Thus g 0 .z/ D ˛, a constant, and ef .z/ D ˛.1 C z/: Now f .0/ D 0 tells us that ˛ D 1. Thus f .z/ defines a branch of log.1 C z/ in jzj < 1, and hence it differs from the principal branch of log.1 C z/ by an integral multiple of 2{. But it follows from the definition of f that for every x 2 .1; 1/, f .x/ is a real number. Thus f .z/ D Log.1 C z/. t u Definition 3.39. Let D be a domain in C with 0 … D. A branch f of the logarithm on D is the principal branch of the logarithm on this domain if f .z/ D Log z for all z 2 D \ .C  .1; 0/. This means, in particular, that if x > 0 belongs to D, then f .x/ 2 R.

62

3 Power Series

We have established in the last theorem that the principal branch of the logarithm on the disc U.1; 1/ is given by the power series: Log z D

1 X

.1/n1

nD1

.z  1/n ; z 2 U.1; 1/: n

Complex exponentials are defined by zc D ec log z for c 2 C and z 2 C¤0 and the principal branch of zc by ec Log z . Thus if L is a branch of the logarithm on a domain D, then for any c 2 C, fc W z 7! ecL.z/ is a holomorphic function on D with the usual expected properties. For example, if n 2 Z, f 1n .z/ D z for all z 2 D. n

3.4 An Identity Principle Holomorphic functions are remarkably rigid: if f is a holomorphic function defined on an open connected set D, then the knowledge of its behavior at a single point c 2 D (i.e., knowing the values of all the derivatives of f at the point c) or the values of f on a sequence in D with a limit point in D are sufficient to uniquely determine its properties (in particular, its value f .d /) at any arbitrary point d 2 D.6 We now start on the exciting journey to establish this and other beautiful results, at first for what seems to be only a subclass of the holomorphic functions. Definition 3.40. A function f defined in a neighborhood of c 2 C has a power series expansion at c if there exists an r > 0 such that f .z/ D

1 X

an .z  c/n for jz  cj < r  ;

nD0

where  > 0 is the radius of convergence of the power series w D z  c).

P

an wn (in the variable

Theorem 3.41. Let f be a function defined in a neighborhood of c 2 C that has a power series expansion at c with radius of convergence  > 0. Then .a/ f is holomorphic and C1 in a neighborhood of c. .b/ If g also has a power series expansion at c and if the product f  g is identically zero in a neighborhood of c, then either f or g is identically zero in some neighborhood of c. .c/ There exists a function h defined in a neighborhood of c that has a power series expansion at c, with the same radius of convergence , such that h0 D f . The function h is unique up to an additive constant.

6

We do not describe any algorithms for computing these values.

3.4 An Identity Principle

63

Proof. Without loss of generality we assume c D 0: (a) Already verified in Theorem 3.19. (b) For some r > 0, we have f .z/ D

1 X

an zn and g.z/ D

nD0

1 X

bn zn for all jzj < r:

nD0

Suppose that neither f nor g vanish identically in any neighborhood of c D 0, and choose the smallest nonnegative integers N and M such that aN ¤ 0 and bM ¤ 0. We know that .f  g/.z/ D

1 X

cn zn for jzj < r;

nD0

where cn D

X

ap bq :

pCqDn

Thus cn D 0 for 0  n  N C M  1 and cN CM D aN bM ¤ 0. But cN CM D

1 .N C M /Š



 d N CM .f  g/ .0/; d zN CM

and therefore f  g cannot be identically zero near 0. 1 X an nC1 z . Then the radius of convergence 0 of h satisfies (c) Define h.z/ D n C 1 nD0 ˇ ˇ1 ˇ an ˇ n 1 ˇ ˇ D lim sup jan j n1 D 1 : D lim sup ˇ 0  n C 1ˇ  n n It now follows from Theorem 3.19 that h0 .z/ D f .z/ for all jzj < . If H is another function that has a power series expansion at 0 and satisfies H 0 .z/ D f .z/ for all jzj < , then the holomorphic function H  h has derivative equal to zero for all jzj <  and must be a constant. t u We will see later that (c) can be strengthened significantly: if f is holomorphic on a simply connected open set D, then h extends to all of D. The following lemma is a useful tool with significant applications beyond the immediate one.

64

3 Power Series

ζ

z

ρ −R

R

ρ

Fig. 3.2 Radii of convergence

Lemma 3.42. If S.z/ D

1 X

an zn has radius of convergence  > 0, then for any

nD0

c 2 C with jcj < , the power series

1 X S .n/ .c/

nŠ

nD0

   jcj, and S.z/ D

1 X S .n/ .c/ nD0

nŠ

wn has radius of convergence

.z  c/n for jz  cj <   jcj :

Proof. Set R D jcj <  (see Fig. 3.2). The argument consists of two steps: (I) We show first that

1 X S .p/ .c/ pD0

pŠ

wp is absolutely convergent for jwj <   R:

We know from Corollary 3.21 that S .p/ .c/ D

1 X nDp

an

nŠ c np : .n  p/Š

If we let q D n  p and set bpCq D jan j, then it follows that 1 ˇ .p/ ˇ X .p C q/Š q ˇS .c/ˇ  R : bpCq qŠ qD0

Now choose r 2 R with R < r < . Then

3.4 An Identity Principle

65

ˇ 1 ˇ .p/ X X ˇ S .c/ ˇ .p C q/Š q p ˇ ˇ bpCq R .r  R/p : ˇ pŠ ˇ .r  R/  pŠ qŠ p; q

pD0

Returning to the original variable n D p C q, we continue to estimate the above expression D

1 X nD0

D

1 X

bn

n X

nŠ Rnp .r  R/p pŠ.n  p/Š pD0

bn .r  R C R/n

nD0

D

1 X

jan j r n < C1;

nD0

where the last series converges because the power series

1 X

an zn converges

nD0

absolutely in jzj < . (II) We show next that S.z/ D

1 X S .p/ .c/ pD0

pŠ

.z  c/p

for jz  cj <   R. We know from (I) that the series 1 1 1 X X X S .p/ .c/ an nŠ .z  c/p D c np .z  c/p pŠ pŠ.n  p/Š pD0 pD0 nDp

converges absolutely for jz  cj <   R. Hence we may rearrange the order of the terms and sums (see Exercise 3.16), and the argument proceeds exactly as in part (I). t u Example 3.43. We study the holomorphic function S.z/ D

1 ; z 2 C6D1 I 1z

it satisfies S.z/ D 1 C z C z2 C    for jzj < 1I

66

3 Power Series

that is, S has a power series expansion at the origin with radius of convergence  D 1. Thus for the function S defined on C6D1 , the power series representation with center at 0 is only valid for jzj < 1 and not for any other values of z.   Let us take c D  12 . Then S .p/ .z/ D pŠ .1  z/1p and thus S .p/  12 D 1  1Cp X  1Cp 2 . A calculation shows that the power series wp has radius of pŠ 23 3 pD0 convergence 0 D 32 , and it follows from the lemma that ˇ  1  1Cp  X ˇ 1 p 1 2 D zC for ˇˇz C S.z/ D 1z 3 2 pD0

ˇ 1 ˇˇ 3 < : 2ˇ 2

We do not yet have (as we soon will) the machinery to conclude the last expansion without calculation.Recall that the power series expansion for S at 0 is just a consequence of the convergence of the geometric series (see (3.1)). So is the expansion for S at  12 , as can be seen from 1 D 1z

3 2

2 1 1 D  :  1 2 3 1  3 z C 12  zC 2

Note that   jcj D 12 < 0 D 32 . What we see in this example is not an accident, as will soon become clear: the radius of convergence of each power series is the distance from its center to the nearest singularity of the function. P Corollary 3.44. If f .z/ D an zn for jzj < , then f has a power series expansion at each point c with jcj < . The next result is called an (perhaps, “the”) identity principle; it provides necessary and sufficient conditions for a function that has a power series expansion at each point of a connected domain of definition to vanish identically. It is usually applied in the form given by Corollary 3.46. Theorem 3.45. Let f be a function defined on a domain D in C. Assume that f has a power series expansion at each point of D, and let c 2 D. The following conditions are equivalent: (a) f .n/ .c/ D 0 for n D 0; 1; 2; : : : . (b) f  0 in a neighborhood of c. (c) There exists a sequence fzn g consisting of distinct points of D with lim zn D c and f .zn / D 0 for each n. (d) f  0 on D.

n

Proof. First note that there are obvious implications: .a/ , .b/ and .d / ) .b/ ) .c/. To complete the proof, we will now show that .c/ ) .a/ and that .a/ ) .d /.

3.5 Zeros and Poles

67

.c/ ) .a/ W We know that f .z/ D

1 X

an .z  c/n for all jz  cj < 

nD0

f .k/ .0/ for all k, and in particular a0 D f .c/ D kŠ lim f .zn / D 0. Assume by induction that with  > 0. Furthermore, ak D n

0 D f .c/ D    D f .n/ .c/ for some integer n  0. Then f .z/ D

1 X

ap .z  c/p D .z  c/nC1

pDnC1

1 X

anC1Cp .z  c/p D .z  c/nC1 g.z/:

pD0

Now, without loss of generality, we assume zn ¤ c for all n. The function g has a power series expansion at c, and obviously g.zn / D 0 if and only if f .zn / D 0. Thus g.c/ D 0 D anC1 . .a/ ) .d /: Let us define D 0 D fz 2 DI f  0 in a neighborhood of zg: The set D 0 is trivially open in D. Since we already know that (a) and (b) are equivalent, we can write D0 D

1 \

fz 2 DI f .n/ .z/ D 0gI

nD0

then D 0 is the intersection of a countable family of closed subsets of D, and hence closed in D. But D 0 is not empty because c 2 D 0 , and since D is connected we obtain D 0 D D. t u

3.5 Zeros and Poles The most important and the first practical consequence of the work of the last section is the next corollary. The results of Sect. 3.4 will also allow us to introduce an important class of functions; these are the meromorphic functions, taking values in the extended complex plane C [ f1g rather than just in C.

68

3 Power Series

Corollary 3.46 (Principle of Analytic Continuation). Let D be a domain in C, and let f and g be functions defined on D having power series expansions at each point of D. If f and g agree on a sequence of distinct points in D with a limit point in D, or if they have identical power series expansions at a single point in D, then f  g (on D). Example 3.47. The exponential function ez is the unique extension of ex in the class of functions under study; that is, it is the unique function on C that has a power series expansions at each point and agrees with ex at each point x 2 R. Remark 3.48. One often uses analytic continuation along a path. To start with the simplest case, let f be a convergent power series centered at c 2 C with radius of convergence  > 0; thus an analytic function on fjz  cj < g. If c  is a point with jc   cj < , then the power series expansion f  of f about c  has a radius of convergence     jc   cj. If  >   jc   cj, then we have extended f to fjz  cj < g [fjz  c  j <  g, and we say that f  is a direct analytic continuation to c  of the power series f at c. We have seen this phenomenon in Example 3.43. More generally, it is of interest to start with a continuous path  in C from c to d . Assume that the path can be covered by discs D0 ; D1 ; : : : ; Dn with centers c D c0 ; c1 ; : : : ; cn D d and radii r0 ; r1 ; : : : ; rn , respectively, such that Dj \ Dj C1 is not empty for 0  j  n  1. Assume further that there are convergent power series fj centered at cj with radius of convergence  rj and such that fj C1 is a direct analytic continuation to cj C1 of fj at cj ; then we say that fn is an analytic continuation along  of f0 . Example 3.49. We outline two applications of the concepts discussed for holomorphic functions. (We shall see that these are precisely the functions having convergent power series expansions.) Both can be established after the material in the next two chapters. If D is a simply connected domain and f is a holomorphic function in a neighborhood of c 2 D that can be continued analytically along any path in D starting at c, then this continuation defines an analytic function on D. In more generality, let 1 and 2 be two continuous paths in a domain D with the same end points. Let f be a holomorphic function in a neighborhood of the initial point c of the two paths that can be continued analytically along all paths in D starting at c. If 1 and 2 are homotopic in D with fixed end points,7 then the continuations of f along the two paths lead to the same holomorphic function in a neighborhood of their common end point. This result is known as the monodromy theorem. Let a, b, and c 2 C be subject to the conditions that each of the three differences c  1, a  b, and .a C b/  c is not an integer. We introduce8 the hypergeometric functions by solving the ordinary differential equation

7

These concepts are defined in the next chapter. See L.V. Ahlfors, Complex Analysis (third edition), McGraw-Hill, 1979 for more details, including the motivation for the form of the differential equation. 8

3.5 Zeros and Poles

69

z.1  z/w00 C Œc  .1 C a C b/zw0  abw D 0: This equation has a solution that is holomorphic in a neighborhood of the origin. The hypergeometric function Fa;b;c is that solution normalized to assume the value 1 at 0. While it is easy to obtain the Taylor series expansion for Fa;b;c at 0 and then compute that its radius of convergence is 1, the monodromy theorem and the theory of solutions of linear differential equations allow us to reach the same conclusion without calculations; the same facts lead us to conclude that Fa;b;c can be analytically continued to become an analytic function on any simply connected domain in C6D1 . Another immediate consequence of Theorem 3.45 is Corollary 3.50. If K is a compact subset of a domain D and f is a nonconstant function that has a power series expansion at each point of D, then f has finitely many zeros in K. Definition 3.51. Let c 2 C. Assume that f .z/ D

1 X

an .z  c/n for all jz  cj <  and for some  > 0:

nD0

If f is not identically zero, it follows from Theorem 3.45 that there exists an N 2 Z0 such that aN ¤ 0 and an D 0 for all n such that 0  n < N: Thus f .z/ D .z  c/N

1 X

aN Cp .z  c/p D .z  c/N g.z/;

pD0

with g having a power series expansion at c and g.c/ ¤ 0. We define N D c .f / D order (of the zero) of f at c: Note that N  0, and N D 0 if and only if f .c/ ¤ 0. If N D 1, then we say that f has a simple zero at c. Definition 3.52. (a) Let f be defined in a deleted neighborhood of c 2 C (see the Standard Terminology summary). We say that lim f .z/ D 1

z!c

if for all M > 0, there exists a ı > 0 such that 0 < jz  cj < ı ) jf .z/j > M:

70

3 Power Series

(b) Let ˛ 2 b C, and let f be defined in jzj > M for some M > 0 (equivalently, we say that f is defined in a deleted neighborhood of 1 in b C). We say lim f .z/ D ˛

z!1

provided lim f

z!0

  1 D ˛: z

(c) The above defines the concept of continuous maps between sets in the Riemann sphere b C. (d) A function f defined in a neighborhood of 1  is  holomorphic (has a power series expansion) at 1 if and only if g.z/ D f 1z is holomorphic (has a power series expansion) at z D 0, where we define g.0/ D f .1/. Definition 3.53. Let U  C be a neighborhood of a point c. A function f that is holomorphic in U 0 D U  fcg, a deleted neighborhood of the point c, has a removable singularity at c if there is a holomorphic function in U that agrees with f on U 0 . Otherwise c is called a singularity of f . Note that all singularities are isolated points. Let us consider two functions f and g having power series expansions at each point of a domain D in b C. Assume that neither function vanishes identically on D and fix c 2 D \ C. Let F .z/ D

g.z/ f .z/ and G.z/ D .f / c .z  c/ .z  c/ c .g/

for z 2 D. Then the functions F and G have removable singularities at c, do not vanish there, and have power series expansions at each point of D. Furthermore, we define a new function h on D by h.z/ D

.z  c/ c .f / F .z/ f .z/ D for all z 2 D g .z  c/ c .g/ G.z/

and fixed c 2 D \ C. There are exactly three distinct possibilities for the behavior of the function h at z D c, which lead to the following definitions. Definition 3.54. (I) If c .g/ > c .f /, then h.c/ D 1 (this defines h.c/, and the resulting function h is continuous at c). We say that h has a pole of order c .g/  c .f / at c. If c .g/  c .f / D 1, we say that the pole is simple. (II) If c .g/ D c .f /, then the singularity of h at c is removable, and, by F .c/ 6D 0. definition, h.c/ D G.c/

3.5 Zeros and Poles

71

(III) If c .g/ < c .f /, then the singularity is again removable and in this case h.c/ D 0. In all cases we set c .h/ D c .f /  c .g/ and call it the order or multiplicity of h at c. In cases (II) and (III) of the definition, h has a power series expansion at c as a consequence of the following result. Theorem 3.55. If a function f has a power series expansion at c and f .c/ ¤ 0, 1 also has a power series expansion at c. then f Proof. Without loss of generality we assume c D 0 and f .0/ D 1. Thus f .z/ D

1 X

an zn ; a0 D 1;

nD0

and the radius of convergence of the series is nonzero. We want to find the reciprocal power series, that is, a series g with positive radius of convergence, that we write as g.z/ D

1 X

bn zn

nD0

and satisfies

X

an zn

X

bn zn D 1:

The LHS and the RHS are both power series, where the RHS is a power series expansion whose coefficients are all equal to zero except for the first one. Equating the first two coefficients on both sides, we obtain a0 b0 D 1; from where b0 D 1; and a1 b0 C a0 b1 D 0; from where b1 D a1 b0 D a1 : Similarly, using the n-th coefficient of the power series when expanded for the LHS, for n  1, we obtain an b0 C an1 b1 C    C a0 bn D 0: Thus by induction we define bn D 

n1 X j D0

bj anj ; n  1:

72

3 Power Series

1 1 1 < C1. Since lim sup jan j n D , there exists a positive   n number k such that jan j  k n . We show by the use of induction, once again, that jbn j  2n1 k n for all n  1. For n D 1, we have b1 D a1 and hence jb1 j D ja1 j  k. Suppose the inequality holds for 1  j  n for some n  1. Then

Since  > 0, we have

jbnC1 j 

n n X X ˇ ˇˇ ˇ ˇ ˇˇ ˇ ˇbj ˇ ˇanC1j ˇ D janC1 j C ˇbj ˇ ˇanC1j ˇ j D0

 k nC1 C

j D1 n X

2j 1 k j k nC1j

j D1

D k nC1 .1 C 2n  1/: Thus there is a reciprocal series, with radius of convergence satisfying 1 1 1 D lim sup jbn j n  lim.21 n /k D 2 k n

n

and therefore nonzero.

t u

Corollary 3.56. Let D be a domain in b C and f a function defined on D. If f has 1 a power series expansion at each point of D and f .z/ ¤ 0 for all z 2 D, then f has a power series expansion at each point of D. Definition 3.57. For each domain D  b C, we define H.D/ D ff W D ! CI f has a power series expansion at each point of Dg: We will see in Chap. 5 that H.D/ is the set of holomorphic functions on D. Corollary 3.58. Assume that D is a domain in b C. The set H.D/ is an integral domain and an algebra over C. Its units are the functions that never vanish on D. b is meromorphic Definition 3.59. Let D be a domain in b C. A function f W D ! C on D if it is locally9 the ratio of two functions having power series expansions (with the denominator not identically zero). The set of meromorphic functions on D is denoted by M.D/. Recall that, by our convention, M.D/¤0 is the set of meromorphic functions with the constant function 0 omitted, where 0.z/ D 0 for all z in D.

A property P is satisfied locally on an open set D if for each point c 2 D, there exists a neighborhood U  D of c such that P is satisfied in U .

9

3.5 Zeros and Poles

73

Corollary 3.60. Let D be a domain in b C, let c be any point in D \ C, and let f 2 M.D/¤0 . There exist a connected neighborhood U of c in D, an integer n D c f , and a unit g 2 H.U / such that f .z/ D .z  c/n g.z/ for all z 2 U: Remark 3.61. If 1 2 D, there exists an appropriate version of the above Corollary for c D 1 exists; see Exercise 3.7. Corollary 3.62. If D is a domain in b C, then the set M.D/ is a field and an algebra over C. Corollary 3.63. If D be a domain and c 2 D, then c W M.D/6D0 ! Z is a homomorphism; that is, c .f  g/ D c .f / C c .g/ for all f and g in M.D/6D0 . Defining c .0/ D C1, we also have c .f C g/  minf c .f /; c .g/g for all f and g in M.D/I that is, c is a (discrete) valuation10 (of rank one) on M.D/. Remark 3.64. The converse statement also holds; it is nontrivial and not established in this book. The next corollary defines the term Laurent series, the natural generalization of power series for functions in H.D/ to functions in M.D/. Corollary 3.65. If f 2 M.D/¤0 and c 2 D \ C, then f has a Laurent series expansion at c; that is, there exists a  2 Z ( D c .f /), a sequence of complex 0 numbers fan g1 nD with a 6D 0, and a deleted neighborhood U of c such that f .z/ D

1 X

an .z  c/n

nD

for all z 2 U 0 . The corresponding power series 1 X

an .z  c/n

nDmax.0;/

converges uniformly and absolutely on compact subsets of U D U 0 [ fcg.

10

Standard, but not universal, terminology.

74

3 Power Series

Remark 3.66. If 1 2 D, then for all sufficiently large real numbers R, the Laurent series representing f in fjzj > Rg [ f1g has the form  n 1 an : f .z/ D z nD 1 X

Corollary 3.67. If f 2 M.D/, then f 0 2 M.D/. If in addition c .f / ¤ 0 for c 2 D, then c .f 0 / D c .f /  1:

Exercises 3.1. Determine the radius of convergence  of each of the following series: 1 n X z nD0

nŠ

;

1 1 1 X zn X zn X n ; ; nŠz : n2 nD1 n nD0 nD1

(1) For those cases with  < C1, determine the values jzj D  for which the series converges. (2) For the case with  D C1, can you conclude anything about convergence at infinity? 3.2. Find the radius of convergence of the power series 1 X

an zn ;

nD0

where a0 D 0, a1 D 1, and an D an1 C an2 for all n > 1. (Hint: Multiply the series by z2 C z  1.) 3.3. Prove that if jan j  M for n  0, then the power series

1 X

an zn has radius of

nD0

convergence   1.

3.4. Under the hypothesis that fan g and fbn g are sequences of positive real numbers, prove that: (a) lim an bn  lim an lim bn ; n

n

n

provided the right side is not the indeterminate form 0 1. Show by example that strict inequality may hold.

Exercises

75

(b) If lim an exists, then the equality holds in (a) provided the right side is not n indeterminate; that is, show that in this case, lim an bn D lim an lim bn : n

n

n

3.5. Give a proof of the ratio test. 3.6. Give a proof of Theorem 3.18. 3.7. Formulate the appropriate version of Corollary 3.60 for the case c D 1 2 D. 3.8. Prove using power series expansions that ez D

1 : ez

3.9. (a) Give examples of algebraic functions that are neither polynomials nor rational functions. (b) Show that the exponential function is a transcendental function. 3.10. Is it always true that Log.ez / D z? Support your answer with either a proof or a counterexample. 3.11. Find all zeros of f .z/ D 1  exp.exp z/. 3.12. (a) Show that both the sine and cosine (complex) functions are periodic with period 2. (b) Show that sin z D 0 if and only if z D  n for some n 2 Z. (c) Show that cos z D 0 if and only if z D 2 .2n C 1/ for some n 2 Z. 3.13. (a) Find <.sin z/ ; =.sin z/ ; <.cos z/ ; and =.cos z/: (b) Write z D x C {y and prove that jsin zj2 D sin2 x C sinh2 y and jcos zj2 D cos2 x C sinh2 y; where

ez C ez ez  ez and sinh z D 2 2 are the hyperbolic trigonometric functions. (c) Derive the addition formulas for cosh.z C c/ and sinh.z C c/. (d) Evaluate D sinh z; D cosh z; and cosh2 z  sinh2 z: cosh z D

3.14. Find all the roots of cos z D 2. 3.15. (a) What are all the possible values of { { ? (b) Let a and b 2 C with a 6D 0. Find necessary and sufficient conditions for ab to consist of infinitely many distinct values.

76

3 Power Series

(c) Let n be a positive integer. Find necessary and sufficient conditions for ab to consist of n distinct values. 3.16. Let fkn g be a sequence in which every positive integer appears once and only once. P P 0 0 P Let an be a series. Putting an D akn , we say that an is a rearrangement of an : P P (a) Let an 2 R and assume that an P converges but jan j does not. a 2 R. P P Let 0 0 Show that there is a rearrangement a of a such that a D a . n n n P (b) Show that an converges absolutely if and only if every rearrangement converges to the same sum. 3.17. Let fan g be a real sequence. Show that o n lim fan g D sup ˛I ˛ D lim bn n

n

with fbn g a convergent subsequence of fan g and that n o lim an D inf ˛I ˛ D lim bn n

n

with fbn g as above. In this exercise a sequence fbn g with lim bn D C1 (similarly 1) is to be considered a convergent sequence. 3.18. Let p.z/ D an zn C an1 zn1 C    C a1 z C a0 , an ¤ 0, be a polynomial of degree n  1. Consider p as a self-map of b C: (a) Let ˛ 2 b C. Show that there exists a z 2 b C such that p.z/ D ˛.11 b b (b) Let z 2 C and p.z/ D ˛ 2 C. Define appropriately mp .z/, the multiplicity of ˛ for p at z so that you can prove: for all ˛ 2 b C, X

mp .z/ D n:

(3.12)

z2b CI p.z/D˛

(c) Relate mp .z/ to z .p/ and z .p 0 /. X mp .z/ is the topological degree of the map p W b C!b C. Note: The integer z2b CI p.z/D˛

11

You may use, although other arguments are available, the fundamental theorem of algebra which will be established in Chap. 5.

Exercises

77

P 3.19. Let p D Q be a nonconstant rational map. It involves no loss of generality to assume, as we do, that P and Q do not have any common zeros. View, as in the case of polynomials, p as a self-map of b C:

(a) Show that p is surjective. (b) Define the concepts of multiplicity at a point and topological degree for the rational map p so that (3.12) holds for some positive integer n. (c) Determine n in terms of the zeros and poles of p and in terms of the degrees of the polynomials P and Q. 3.20. The unit sphere (with center at 0) S 2  R3 is defined by S 2 D f. ; ; / 2 R3 I 2 C 2 C  2 D 1g: Show that stereographic projection . ; ; / 7!

C{ 1

is a diffeomorphism from S 2  f.0; 0; 1/g onto C and that it extends to a diffeomorphism from S 2 onto b C (that sends .0; 0; 1/ to 1). 3.21. Justify the statement that stereographic projection takes circles to circles. A circle on S 2 is the intersection of a plane in R3 with S 2 . Such a circle is maximal if it is the intersection of S 2 with a plane through the point .0; 0; 1/ of R3 . Show that stereographic projection sets up a bijective correspondence between C; that the set of maximal circles on S 2 and the set of circles through 1 on b is, straight lines in C. Also show that stereographic projection sets up a bijective correspondence between the set of all circles on S 2 and the circles in b C: the union of the set of all circles in C and the set of all straight lines in C. The circles in b C will play an important role in Chap. 8. 3.22. Show that stereographic projection preserves angles. 3.23. The formula

sin z cos z defines a meromorphic function on C. Show that it has simple poles at tan z D

z D .2k C 1/

 for every integer k 2

and is holomorphic elsewhere. Show that tan maps C onto C [ f1g: (a) Show that tan z D tan  if and only if there exists an integer k such that   z D k. (b) Show that z 7! tan z is a holomorphic one to one map of

78

3 Power Series

n  o z 2 CI  <
n

z 2 CI 

 o <
(c) Show that 1 d tan z D : dz cos2 z 3.24. One of the purposes of this exercise is to establish the beautiful formula (3.13). Verify each of the following assertions and/or answer the questions: (a) The series 1

X 1 D .1/k z2k 2 1Cz kD0

defines a holomorphic function on jzj < 1. (b) Hence there is a holomorphic function f .z/ D

1 X .1/k z2kC1 kD0

2k C 1

on jzj < 1 such that f .0/ D 0 and f 0 .z/ D

1 : 1 C z2

(c) Since tan is locally injective, there exists a multivalued inverse function arctan defined on C [ f1g such that tan.arctan z/ D z for all z 2 C; hence also tan.arctan.z/ C k/ D z for all k 2 Z and all z 2 C. We can hence define the principal branch of arctan, Arctan, by requiring that 

  < <.Arctan z/  : 2 2

(d) Show that arctan z D

12

1 1 C {z 1 1 C {z log and Arctan z D Log : 2{ 1  {z 2{ 1  {z

For this and the previous onto proof you will need either some of the results of the next exercise or something like Rouch´e’s theorem, which is proven in Chap. 6.

Exercises

79

(e) Let g.z/ D f .tan z/. Show that g 0 .z/ D 1 for all z in a domain D. Describe D. (f) Conclude that f .z/ D Arctan z for jzj < 1. (g) Why does the Taylor series for Arctan at the origin not converge in a disc larger than jzj < 1? P .1/k (h) Show that Arctan 1 is given by 1 kD0 2kC1 , thus justifying D4

1 X .1/k : 2k C 1

(3.13)

kD0

3.25. (L’Hopital’s Rule) Let f and g be two functions defined by convergent power series in a neighborhood of 0. Assume that f .0/ D 0 D g.0/ and g 0 .0/ 6D 0. Show that f 0 .0/ f .z/ lim D 0 : z!0 g.z/ g .0/ 3.26. The reader might be better prepared for this problem after studying Chap. 6. 1 1 X X Suppose the series ˛j zj and ˇj zj converge for 1 < jzj < 3 and 2 < 1

1

jzj < 4, respectively, and that they have the same sum for 2 < jzj < 3. Does this imply that ˛j D ˇj for all j ? 3.27. Let p be a polynomial of degree n  0. We usually write p.z/ D

n X

bi zi ;

i D0

as a linear combination of the standard monomials zi . In this exercise we introduce a second set of monomials that will help us to evaluate the infinite series 1 X

p.k/zk :

kD0

• Show that the series converges for jzj < 1. • Define inductively (for n 2 Z0/ ) the monomials z.n/ , by setting z.0/ D 1; and for n 2 Z0/ ,

z.n/ D z.n1/ .z  n C 1/:

Show that there exists constants ai such that p.z/ D

n X i D0

ai z.i / :

80

3 Power Series

Evaluate these constants. • Show that for all n 2 Z0 and all z 2 C such that jzj < 1, S .n/ .z/ D

1 X

k.n/ zkn D

kDn

nŠ : .1  z/nC1

• Prove that for all z 2 C such that jzj < 1, 1 X

p.k/zk D

1 X n X

aj k.j / zk D

kD0 j D0

kD0

n X j D0 n X

D

n X j D0

aj zj S .j / .z/ D

aj zj

1 X

k.j / zkj

kDj

aj zj .1  z/nj j Š

j D0

.1  z/nC1

:

Chapter 4

The Cauchy Theory: A Fundamental Theorem

As with the theory of differentiation for complex-valued functions of a complex variable, the integration theory of such functions begins by mimicking and extending results from the theory for real-valued functions of a real variable, but again the resulting theory is substantially different, more robust, and more elegant. Specifically, a curve or path  in C is a continuous function from a closed interval in R to C. Thus the restriction of a complex-valued function f on C to the range of a curve has real and imaginary parts which can be viewed as real-valued functions of a real variable and thus integrated on the interval.1 Adding the integral of the real part to { times the integral of the imaginary a complex-valued integral R R part defines R of a complex-valued function (i.e., f D
When suitable conditions on
1 2

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 4, © Springer Science+Business Media New York 2013

81

82

4 The Cauchy Theory: A Fundamental Theorem

curves. Line integrals and differential forms are introduced in the next section. In the section following it, we emphasize the difference between exact and closed (locally exact) one-forms. Integration of closed forms along continuous, not necessarily rectifiable, paths is discussed next.3 This is followed by sections on the winding number of a curve about a point, curve homotopy and simple connectivity, and properties of the winding number. After that, we treat Goursat’s form of Cauchy’s theorem, which is reduced to establishing the following simple statement: the integral of a holomorphic function on a domain D over the boundary of a rectangle whose closure is contained in D is equal to zero. This result can hence be referred to as Cauchy’s theorem for a rectangle. If we were willing to assume C1 in the definition of holomorphicity, then the simple statement would follow at once from Green’s theorem. The weaker definition used complicates the proof but is needed to see the full power of holomorphicity. The chapter ends with an appendix on differential forms, and another appendix on a more classic approach to the Cauchy theory than the one we used, based on differential forms. While the proofs of Goursat’s theorem and some of the other main results become quite technical in places, the final conclusions are simple to state. This simplicity gives them a certain elegance and compactness. The two chapters that follow present the core of the proofs of the fundamental theorem. In Chap. 5 we present key consequences of the initial Cauchy theory (viz., of Cauchy’s theorem for a rectangle). This is followed by Chap. 6, where consequences related to holomorphic functions with isolated singularities are presented.

4.1 Line Integrals and Differential Forms We recall the definitions of one-sided derivatives for functions of a real variable. Definition 4.1. Let Œa; b be a closed (finite) interval on R and let g W Œa; b ! C be a function. As in calculus, for a  c < b, we define .D C g/.c/ D lim

h!0 h>0

g.c C h/  g.c/ ; h

the right-sided derivative of g at c (whenever this limit exists). Similarly, for a < c  b, we set

3

This approach has the advantage of avoiding a discussion of rectifiability, but we lose the ability to integrate some non-closed forms.

4.1 Line Integrals and Differential Forms

83

.D  g/.c/ D lim

h!0 h<0

g.c C h/  g.c/ ; h

the left-sided derivative of g at c (whenever this limit exists), and for a < c < b, we let g 0 .c/ D .Dg/.c/ D .D C g/.c/ D .D  g/.c/; the derivative of g at c, whenever the last two limits exist and are equal. We say that g is differentiable on Œa; b if g0 exists on .a; b/ and .D C g/.a/ and .D  g/.b/ exist (these define g 0 .a/ and g 0 .b/, respectively); g is called continuously differentiable on Œa; b if g 0 is continuous on Œa; b, in which case we will write g 2 C1C .Œa; b/ D C1 .Œa; b/ or, equivalently, g 0 2 C0C .Œa; b/ D C0 .Œa; b/ D CC .Œa; b/ D C.Œa; b/. The spaces C1R Œa; b/ and CR .Œa; b/ are similarly defined. Remark 4.2. The concepts we have been discussing are from real analysis; if f W Œa; b ! C is a complex-valued function, they apply to the two real-valued functions of a real variable given by u D
84

4 The Cauchy Theory: A Fundamental Theorem

Z

Z

b

!D 

Z

ŒP .x.t/; y.t// x 0 .t/ C Q.x.t/; y.t// y 0 .t/dt

a b

D

Œp1 .x.t/; y.t// x 0 .t/ C q1 .x.t/; y.t// y 0 .t/ dt

a

Z

b

C{

Œp2 .x.t/; y.t// x 0 .t/ C q2 .x.t/; y.t// y 0 .t/ dt;

a

where P D p1 C { p2 and Q D q1 C { q2 . Remark 4.5. The above definition involves again only concepts from real analysis, even though the paths and functions involve the complex numbers.

4.1.1 Reparameterization If t W Œ˛; ˇ ! Œa; b is a one-to-one, onto and differentiable function, and  W Œa; b ! C is a differentiable path, then e  D  ı t is again a differentiable path, called a reparameterization of  . For all differential forms ! defined in a neighborhood of the range of the path  and all reparameterizations e  D  ı t of  , the following equalities hold readily from the definitions: Z Z ! D ! ; if t 0 .u/  0 for all u 2 Œ˛; ˇ; e   and

Z

Z  e

! ; if t 0 .u/  0 for all u 2 Œ˛; ˇ:

!D 

Note that there are no other possibilities for the sign of the derivative of t, since t 0 cannot change signs, because t is one-to-one, and therefore either increasing or decreasing on Œ˛; ˇ. Since for any closed interval Œa; b there exists a one-to-one, onto and differentiable map t W Œ0; 1 ! Œa; b with t 0 .u/ > 0 for all u 2 Œ0; 1 (see Exercise 4.6), we can always assume, as we will when appropriate, that a given path  is parameterized by Œ0; 1.

4.1.2 Subdivision of Interval Let  W Œa; b ! C be a differentiable path and consider the partition of Œa; b defined by

4.1 Line Integrals and Differential Forms

85

a D t0 < t1 <    < tnC1 D b:

(4.1)

For a set T contained in the domain of  ,  jT , of course, denotes the restriction of  to T . If we let j D  jŒtj ;tj C1  ; for j D 0; : : : ; n;

(4.2)

then each j is a differentiable path, and Z !D 

n Z X

!:

(4.3)

j D0 j

This observation allows us to generalize the notion of integral of a form over a differentiable path to a more general class of paths, as follows.

4.1.3 The Line Integral Definition 4.6. Let  W Œa; b ! C be a continuous path. We say that  is a piecewise differentiable path (henceforth abbreviated pdp) if there exists a partition of Œa; b of the form given in (4.1) such that each of the paths defined by (4.2) is R differentiable. Then we use (4.3) to define the path integral  !. Remark 4.7. The path integral over a pdp is well defined (independent of the partition) and agrees with our earlier definition for differentiable paths. The verification of these facts is left as an exercise. Remark 4.8. There are three pictures in R2 that are naturally associated to each path  D x C { y: the picture of the range of the curve and the graphs of the functions x and y. Figure 4.1 illustrates this for a curve whose image is the boundary of the rectangle with vertices .c; e/; .d; e/; .d; f /, and .c; f /. Definition 4.9. Let  denote a pdp parameterized by Œa; b in a domain D. Then the path  traversed backward is defined by  .t/ D .t C b C a/ for all t 2 Œa; b: It follows that

Z

Z !D 

! 

for all differential forms ! defined in a neighborhood of the range of  . Definition 4.10. If 1 and 2 are pdp’s in D parameterized by Œ0; 1, with 1 .1/ D 2 .0/, then a new pdp 1  2 in D may be defined by first traversing 1 and then continuing with 2 , as follows:  1 .2t/; for 0  t  12 ; 1  2 .t/ D 2 .2t  1/; for 12  t  1:

86

4 The Cauchy Theory: A Fundamental Theorem

a

y f e

x c

d

b x

c

d

y f

c

e

t

t

Fig. 4.1 Three figures for a curve (the boundary of a rectangle). (a) Picture of the curve. (b) The graph of x. (c) The graph of y

Thus

Z

Z

Z

!D 1 2

!C 1

! 2

for all differential forms ! defined in a neighborhood of the range of 1  2 . Lemma 4.11. If D is a domain in C, then any two points in D can be joined by a pdp in D. Proof. Fix c 2 D and let E D fz 2 DI z can be joined to c by a pdp in Dg: The set E is open in D, because if z denotes any point in E, then, since D is an open set, there is a small disc U with center at z contained in D, and any point w in U may be joined to z by a radial segment in U ; the pdp consisting of this segment followed by the pdp joining z to c gives that w in E (see Definition 4.10). Similarly, D  E is also open in D. Since D is connected and c 2 E, we conclude that E D D. u t Definition 4.12. Although the concepts introduced here were used in previous chapters without explanation, this is a good place for formalizing them. Let D be a domain in C (1) We recall that a function f defined on D is of class Cp on D, with p 2 Z0 , if f has partial derivatives (with respect to x and y) up to and including order p, and these are continuous on D. We also say that f is a Cp -function on D.

4.1 Line Integrals and Differential Forms

87

Of course p D 0 just means that f is continuous on D. The vector space of complex-valued functions of class Cp on D is denoted by Cp .D/. We also say that f is of class C1 or smooth in D if it is of class Cp for all p 2 Z0 . In this case we also write f 2 C1 .D/. (2) A differential form ! D P dx C Qdy is of class Cp on D if and only if P and Q are. (3) For a given function f , we have the (real) partial derivatives fx and fy as well as the formal (complex) partial derivatives fz and fz introduced in Definition 2.37. Similarly, we consider the differentials dz D dx C { dy and dz D dx  { dy: It follows that every differential form can be written in the following two ways: 1 P dx C Qdy D Œ.P  {Q/dz C .P C {Q/dz: 2 Remark 4.13. We recommend that all concepts and definitions that are formulated in terms of x and y be reformulated by the reader in terms of z and z (and vice versa). (4) If f is a C1 -function on D, then we define df , the total differential of f , by either of the two equivalent formulae: df D fx dx C fy dy D fz dz C fz dz:

In addition to the differential operator d , we have two other important differential operators @ and @ defined by @f D fz dz and @f D fz dz; as well as the formula d D @ C @: We have defined the three differential operators on spaces of C1 -functions. They can be also defined on spaces of C1 -differential forms, and it follows from these definitions that, for example, on C2 -functions the equality d 2 D 0 holds. We shall not need these extended definitions but will outline some facts concerning the exterior differential calculus in the first appendix to this chapter. (5) A differential form ! is called exact if there exists a C1 -function F on D (called a primitive for !) such that ! D dF . A primitive (if it exists) is unique up to addition of a constant, because dF D 0 means Fx D Fy D 0, and this implies that F is constant on the connected set D.

88

4 The Cauchy Theory: A Fundamental Theorem

By abuse of language we also say that a function F is a primitive for a function f if F is a primitive for the differential form ! D f .z/dz. (6) A differential form ! on D is closed if it is locally exact; that is, if for each c 2 D there exists a neighborhood U of c in D such that !jU is exact.

4.2 The Precise Difference Between Closed and Exact Forms While the definitions of exact and closed forms are straightforward, as is the fact that every exact differential is closed, an intuitive sense of the difference between the two properties may not immediately present itself. This is because these differences arise from the topology of the domain where the differential form is defined and from the behavior of the differential form along certain paths in that domain. A closed but not exact differential is given in Example 4.30. We will see that on a disc the two properties are equivalent, but situations where they are not equivalent are especially significant. To understand this difference, we study the pairing that associates the complex number Z h; !i D ! 

to a pdp  in a domain D and a differential form ! on D (when the integral exists). Lemma 4.14. R Let ! be a differential form on a domain D. Then ! is exact on D if and only if  ! D 0 for all closed pdps  in D. Proof. Assume that ! is exact. Then there exists a C1 -function F on D with ! D Fx dx C Fy dy: If  is a pdp parameterized by Œa; b joining two points P1 to P2 in D, then Z

Z

b

!D 

a

  Z b dx dy dF C Fy dt D dt D F .P2 /  F .P1 /; Fx dt dt a dt

which equals zero if P1 D P2 , as happens for every closed curve. To prove the converse, let Z0 D .x0 ; y0 / be a fixed point in D and let Z D .x; y/ be an arbitrary point in D. Let  be a pdp in D joining Z0 to Z and define Z F .x; y/ D

!: 

To see that the function F is well defined on D, note that if 2 is another pdp in D joining Z0 to Z, then 2   is a closed pdp in D, and it follows from the hypothesis and from Definitions 4.9 and 4.10 that

4.2 Closed and Exact Forms

89

(x1,y)

L

(x,y)

δ Z0

Fig. 4.2 The integration path for F

Z

Z

0D

Z

!D 2 

! 2

!: 

We must show that F 2 C1 .D/ and dF D !. Choose  > 0 so that U D U.x;y/ ./  D; also choose x1 ¤ x such that .x1 ; y/ 2 U and the (straight) segment L from .x1 ; y/ to .x; y/ is contained in D. Let ı be any pdp in D from .x0 ; y0 / to .x1 ; y/ (see Fig. 4.2), and assume that ! D P dx C Q dy. Then Z Z x F .x; y/ D ! C P .t; y/ dt: ı

x1

It is now clear that Fx D P . Similarly, Fy D Q.

t u

For future use it is important to record the following result. Corollary 4.15 (of proof). If dF is an exact differential form on the domain D and  is a pdp in D starting at P1 and ending at P2 , then Z dF D F .P2 /  F .P1 /: 

Theorem 4.16. RLet ! be a differential form on an open disc U . Then ! is exact on U if and only if  ! D 0 for all  that are boundaries of rectangles contained in U with sides parallel to the coordinate axes. Proof. Repeat the appropriate argument in the proof of the last lemma with .x0 ; y0 / the center of U , observing that any other point in the disc may be joined to the center by either a vertical segment, a horizontal segment, or two consecutive segments, one horizontal and one vertical. t u Corollary 4.17. A differential form ! is closed on an open disc if and only if it is exact on the disc.

90

4 The Cauchy Theory: A Fundamental Theorem

Proof. Every exact form on a domain is closed, so we just need to show that the converse holds on an open disc. So assume ! is a closed form on the open disc U . By the theorem, it is enough to show that ifR R is any rectangle contained in U , with sides parallel to the coordinate axes, then @R ! D 0. Note that the rectangle R may be subdivided into smaller rectangles with sides parallel to the coordinate axes and such that each smaller rectangle is contained in an open set in U where ! is exact. It follows from the theorem that the integral of ! over the boundary of each smaller rectangle is equal to zero, and therefore the integral of ! over the boundary of R, being equal to the sum of all the integrals over the boundaries of the smaller rectangles, is also equal to zero. t u Corollary 4.18. If ! is a differential form on a domain D, then ! is closed on D if R and only if  ! D 0 for all curves  that are boundaries of rectangles contained in D with sides parallel to the coordinate axes. Definition 4.19. A region R  R2 is called .xy/-simple if it is bounded by a pdp and has the property that any horizontal or vertical line which has nonempty intersection with R intersects it an interval. Further, the set of values a 2 R for which the vertical line x D a has nonempty intersection with R is an interval, and the set of values c 2 R for which the horizontal line y D c has nonempty intersection with R is also an interval. Here an interval may consist of a single point. In particular, there exist real numbers c < d and functions h1 and h2 defined on the interval Œc; d  such that the region R may be described as follows: R D f.x; y/I c  y  d; h1 .y/  x  h2 .y/g: A similar description may be given interchanging the roles of the two variables x and y. Open discs and the interiors of rectangles and triangles are examples of .xy/simple regions. We recall4 and establish a form of a theorem that will help us to further distinguish closed from exact differentials. Theorem 4.20 (Green’s Theorem). Let R be an .xy/-simple region and let  denote its boundary oriented counterclockwise (this means that R lies to the left of the oriented curves on its boundary). Consider a C1 -form ! D P dx C Q dy on a region D R [  . Then “  R

4

From calculus courses.

@Q @P  @x @y



Z dx dy D

Z P dx C Q dy D



!: 

4.2 Closed and Exact Forms

91

Proof. Using the notation introduced in the definition of .xy/-simple regions, we have “ R

@Q dx dy D @x

Z Z

d

Z

h2 .y/

@Q dx dy @x

h1 .y/

c d

D

ŒQ.h2 .y/; y/  Q.h1 .y/; y/ dy Z

c

Z

d

D

c

Q.h2 .y/; y/ dy C Z

c

D

Q.h1 .y/; y/ dy d

Q dy: 

Similarly, “

@P dx dy D   @y R

“ R

@P dy dx D @y

Z P dx:

t u



Remark 4.21. (1) The theorem can be easily extended to any region that may be divided into a finite union of .xy/-simple regions and their boundaries (by cancelation of integrals over common boundaries oppositely oriented); for instance, the interior of a compact convex set. (2) In terms of complex derivatives, the theorem can be restated as “  R

@Q @P  @z @z



Z dz dz D

P dz C Q dz; 

where dz dz D 2{ dx dy. (3) In most real analysis courses and books (see, e.g., Theorem 5.12 of G.B. Folland Advanced Calculus, Prentice Hall, 2002), Green’s theorem is given in the following form (we are now using complex notation). Theorem 4.22 (Green’s Theorem, Version 2). Let K be a compact set in C which is the closure of its interior, with piecewise smooth positively oriented boundary @K. If f and g are C1 -functions on a neighborhood of K, then “

Z .gz  fz / dz dz D

K

f .z/ dz C g.z/ dz: @K

We can now characterize the closed C1 -forms. Theorem 4.23. Suppose that ! D P dx C Q dy is a C1 -differential form on a domain D. If ! is closed, then Px D Qy .

92

4 The Cauchy Theory: A Fundamental Theorem

Conversely, if D is an open disc, P and Q are C1 -functions on D, and Px D QY , then ! D P dx C Q dy is closed (hence exact) on the disc. Proof. If ! is closed in the domain D, then near every point in D there exists a function F such that ! D dF D Fx dx C Fy dy. But ! is C1 and thus F is C2 ; therefore Py D Fxy D Fyx D Qx . For the Rconverse on a disc D, by Theorem 4.16 and Corollary 4.17, we need only show that  ! D 0 for all paths  in D that are boundaries of rectangles R with sides parallel to the coordinate axes and such that R [   D. But “ Z   Qx  Py dx dy D 0: !D t u 

R

Corollary 4.24. If ! D P dx CQ dy is a C1 -form on a domain D, then ! is closed on D if and only if Py D Qx in D. Proof. For any point in D, consider an open disc U centered at that point and contained in D, and apply the previous theorem to ! restricted to U . t u Remark 4.25. Recall that f .z/ dz D .u C { v/.dx C { dy/ D .u dx  v dy/ C { .u dy C v dx/ D !1 C { !2 ; with !1 and !2 real differentials. Thus Z Z Z f .z/ dz D !1 C { !2 : 





Further, f .z/ dz is closed (respectively exact) if and only if both !1 and !2 are, and Fj is a primitive for !j (j D 1, 2) if and only if F1 C { F2 is a primitive for f .z/ dz. Lemma 4.26. Let f .z/ dz be of class C1 on a domain D. Then f .z/ dz is closed on D if and only if f is holomorphic in D. Proof. By the above remarks and previous Corollary, f .z/ dz is a closed form on D if and only if uy D vx and vy D ux if and only if u and v satisfy CR if and only if f is holomorphic. t u Lemma 4.27. A C1 -function F is a primitive for f .z/ dz if and only if F 0 D f . Proof. The function F is a primitive for f .z/ dz if and only if dF D Fz dzCFz dz D f .z/ dz if and only if Fz D 0 and Fz D F 0 D f . t u We have now proven the following result that gives a preliminary characterization of certain closed forms.

4.2 Closed and Exact Forms

93

Theorem 4.28. The differential form f .z/ dz is closed on a domain D if and only R if  f .z/ dz D 0, for all boundaries  of rectangles R contained in D 5 with sides parallel to the coordinate axes. If f 2 C1 .D/, then f .z/ dz is closed if and only if f is holomorphic on D. Remark 4.29. We shall see that the C1 assumption in the last part of the theorem is not needed. dz Example 4.30. Not every closed form is exact. Let D D C6D0 and ! D . z R 2 { t (a) If .t/ D e for t 2 Œ0; 1, then  ! D 2 {. Thus ! is not exact on D. 1 (b) Since f .z/ D is holomorphic and C1 on D, ! is closed on D. z Note that locally (in D) we have ! D dF , where F is a branch of the logarithm, and that we have just proved that there is no branch of the logarithm globally defined on D. We have produced two real forms on D D C6D0 , the real and imaginary parts of !: x dx C y dy y dx C x dy dz D C{ D d log jzj C { d arg z D d log z: z x2 C y 2 x2 C y 2 The first of the two real forms is exact, the second closed but not exact on D. Note that d arg z D d arctan yx (for x ¤ 0). Note also that arg z and arctan yx are multivalued functions, whose differentials agree and are single-valued. It is important to observe that if we change the domain D then ! may become dz exact; for instance, on the domain C  .1; 0 the same form ! D is exact z because ! D dLog, where Log denotes the principal branch of the logarithm in this domain. We have been working with the integral of any differential form over any pdp. We want to extend the definition to the integral over any continuous path  but only for closed differential forms. The next result will lead us in that direction. Definition 4.31. Let D be a domain in C,  W Œa; b ! D be a continuous path in D, and ! D P dx C Q dy be a closed form in D. A primitive for ! along  is a continuous function f W Œa; b ! C such that for all t0 2 Œa; b there exists a neighborhood N of .t0 / in D and a primitive F for ! in N such that F ..t// D f .t/ for all t in a neighborhood of t0 in Œa; b.

5

It should be emphasized that the rectangle R, not just its perimeter @R, is contained in D.

94

4 The Cauchy Theory: A Fundamental Theorem

4.2.1 Caution It is possible to have t1 ¤ t2 with .t1 / D .t2 / but f .t1 / ¤ f .t2 /; that is, f need not be well defined on  ; of course, it is required to be well defined on Œa; b. See Exercise 4.8.

4.2.2 Existence and Uniqueness Theorem 4.32. If  is a continuous path in a domain D and ! is a closed form on D, then there exists a primitive f of ! along  . Furthermore, f is unique up to the addition of a constant. Proof. Suppose  is parameterized by Œa; b and ! D P dx C Q dy. Uniqueness: Suppose f and g are two primitives of ! along  and let t0 2 Œa; b. Then there exist primitives F and G of ! in a connected neighborhood U of .t0 / in D such that F ..t// D f .t/ and G..t// D g.t/ for t near t0 . Hence Fx D Gx D P and Fy D Gy D Q in U ; thus F  G is constant in U , and therefore f  g is constant near t0 . We conclude that f  g is a continuous and locally constant function on the connected set Œa; b, and thus f  g is a constant function. Existence: Given t 2 Œa; b, there exists an interval I.t/  Œa; b (open in Œa; b and containing t) and an open set U..t//  D such that ! has a primitive in U..t// and .I.t//  U..t//. Then [ I.t/ t 2Œa;b

is an open cover of Œa; b and thus there exists a finite subcover I0 [ I1 [    [ In D Œa; b; with corresponding Uj . Without loss of generality we may assume I0 D Œa0 ; b0 /, Ij D .aj ; bj / for j D 1; : : : ; n  1 and In D .an ; bn , where a0 D a < a1 < b0 < a2 < b1 <    < an < bn1 < bn D b: Further, .Ij /  Uj and ! has a primitive Fj on Uj for j D 0; 1; : : : ; n. Set f .t/ D F0 ..t// for t 2 I0 . Having defined f on I0 [ I1 [    [ Ik for 0  k < n, we define f on IkC1 as follows. Let FkC1 be any primitive for ! in UkC1 and let Fk be the primitive for ! in Uk such that f .t/ D Fk ..t// for t 2 Ik . Then FkC1 and Fk are two primitives for ! in UkC1 \ Uk , and thus FkC1  Fk is constant on each connected component of UkC1 \ Uk ; in particular, let FkC1  Fk D c on the component containing .IkC1 \ Ik /. Set f .t/ D FkC1 ..t//  c for t 2 IkC1 , then f is well defined on IkC1 \ Ik . t u

4.3 Integration of Closed Forms and the Winding Number

95

4.3 Integration of Closed Forms and the Winding Number Consideration of the next example leads to the extension of the integral to more general paths. This leads in turn to the surprising result, given in Corollary 4.37, that certain integrals take on only integer values. This fact allows a precise definition corresponding to the intuitive idea of counting the number of times a curve winds around a point, the winding number of the curve with respect to a point. R Example 4.33. We use Theorem 4.32 to compute  !, where ! is a closed differential form in D and  W Œa; b ! D is a pdp in D. Subdivide Œa; b D I0 [ I1 [    [ In , where Ij D Œaj ; aj C1 , a0 D a, and anC1 D b, such that j D  jIj is a differentiable path and ! has a primitive Fj in a neighborhood of .Ij / for j D 0; : : : ; n. By the theorem, there exists a primitive f of ! along  . Then Z !D 

n Z X

dFj D f .b/  f .a/:

j D0 j

Using the last equation, we extend the concept of a line integral to continuous paths. Definition 4.34. Let ! be a closed differential form in D and  W Œa; b ! D be a continuous path in D. We define Z ! D f .b/  f .a/; 

where f is a primitive of ! along  . Remark 4.35. The integral is well defined and agrees with the earlier definition for pdp’s. Note that we have avoided any discussion of rectifiability6 of the curve  . We have however paid a price: we have not introduced the class of curves  whose length is well defined. Theorem 4.36. For every c 2 C and every continuous closed path  in C  fcg, the number Z 1 dz 2 Z: 2{  z  c Proof. We may assume c D 0. Let f be a primitive of

6

dz along the curve  . Then z

We leave it to the curious reader to consult other sources for the meaning of rectifiability.

96

4 The Cauchy Theory: A Fundamental Theorem

Z 

dz D f .b/  f .a/; z

where Œa; b parameterizes  . Since .a/ D .b/, we know from Example 4.30 that f .b/  f .a/ is just the difference between two branches of log z at the same point, hence of the form 2{ n with n in Z. t u From Example 4.30 we obtain Corollary 4.37. For  as in Theorem 4.36 with cD0, the value of

1 2

Z 

xdy  ydx x2 C y 2

is an integer. Definition 4.38. Let c 2 C and let  be a continuous closed path in C  fcg. We define the index or winding number of  with respect to c as the following integer: 1 I.; c/ D 2{

Z 

dz : zc

Example 4.39 (In polar coordinates). Let r D g./ > 0, with g 2 C1 .R/. Let n 2 Z>0 and define ./ D g./e{ , where  2 Œ0; 2 n. Assume that g.0/ D g.2 n/. Observe that the conditions on g imply that the curve  winds around the origin n times in the counterclockwise direction and, as expected, I.; 0/ D

1 2{

Z Z



1 dz D z 2{

Z

2 n 0

d.g./e{ / g./e{

g 0 ./e{ C {g./e{ d g./e{ 0  Z 2 n  0 g ./ 1 C { d D n: D 2{ 0 g./

D

1 2{

2 n

In general, let  W Œa; b ! C  fcg be a continuous closed path and let f be a dz primitive of on  . Then f .t/ agrees with a branch of log..t/  c/; that is, zc ef .t / D .t/  c for all t 2 Œa; b: Hence I.; c/ D

f .b/  f .a/ : 2{

We see that the C1 assumption on g is unnecessary as we will also be able to conclude using homotopy of curves discussed in the next section.

4.4 Homotopy and Simple Connectivity

97

4.4 Homotopy and Simple Connectivity In order to give the integration results the clearest formulation (see for instance Corollary 4.52), we introduce the topological concepts of homotopic curves and simply connected domains. Definition 4.40. Let 0 and 1 be two continuous paths in a domain D, parameterized by I D Œ0; 1 with the same end points; that is, 0 .0/ D 1 .0/ and 0 .1/ D 1 .1/. We say that 0 and 1 are homotopic on D (with fixed end points) if there exists a continuous function ı W I I ! D such that (1) (2) (3) (4)

ı.t; 0/ D 0 .t/ for all t 2 I ı.t; 1/ D 1 .t/ for all t 2 I ı.0; u/ D 0 .0/ D 1 .0/ for all u 2 I ı.1; u/ D 0 .1/ D 1 .1/ for all u 2 I

We call ı a homotopy with fixed end points between 0 and 1 ; see Fig. 4.3, with u .t/ D ı.t; u/, for fixed u in I . Let 0 and 1 be two continuous closed paths in a domain D parameterized by I D Œ0; 1; that is, 0 .0/ D 0 .1/ and 1 .0/ D 1 .1/. We say that 0 and 1 are homotopic as closed paths on D if there exists a continuous function ı W I I ! D such that (1) ı.t; 0/ D 0 .t/ for all t 2 I (2) ı.t; 1/ D 1 .t/ for all t 2 I (3) ı.0; u/ D ı.1; u/ for all u 2 I The map ı is called a homotopy of closed curves or paths; see Fig. 4.4, with u .t/ D ı.t; u/ for fixed u in I . A continuous closed path is homotopic to a point if it is homotopic to a constant path (as a closed path). γ0(t)

γ0(0)

γu(t) γ1(t)

γ0(1)

Fig. 4.3 Homotopy with fixed end points

98

4 The Cauchy Theory: A Fundamental Theorem γ1(t) γu(t)

γ0(t)

Fig. 4.4 Homotopy of closed paths

Example 4.41. Continuing with Example 4.39, it is easy to see that for continuous g, the path  is homotopic as a closed path in C  f0g to the circle S 1 traversed n times in the positive direction. Thus both have the same winding number about the origin. Remark 4.42. Note that the notion of being homotopic (in all its versions) depends on the domain D. For instance, the closed path .t/ D exp.2{t/ for 0  t  1 is homotopic to a point in C (set ı.t; u/ D u.t/), but it is not homotopic to a point in C  f0g, as we will soon see (Remark 4.55). Definition 4.43. Let I = Œ0; 1, let ı W I I ! D  C be a continuous map, and let ! be a closed form on D. A function f W I I ! C is said to be a primitive for ! along ı provided for every .t0 ; u0 / 2 I I , there exists a neighborhood V of ı.t0 ; u0 / in D and a primitive F for ! on V such that f .t; u/ D F .ı.t; u// for all .t; u/ in some neighborhood of .t0 ; u0 / in I I . Remark 4.44. (1) Such a function f is automatically continuous on I I . (2) For fixed u 2 I , f .; u/ is a primitive for ! along the path t 7! ı.t; u/ (see Definition 4.31). Theorem 4.45. If ! is a closed form on D and ı W Œ0; 1 Œ0; 1 ! D is a continuous map, then a primitive f for ! along ı exists and is unique up to an additive constant. Proof. We leave the proof as an exercise for the reader.

t u

We now observe that all integrals of a closed form along homotopic paths coincide.

4.4 Homotopy and Simple Connectivity

99

Theorem 4.46. Let 0 and 1 be continuous paths in a domain D and let ! be a closed form on D. If 0 is homotopic to 1 with fixed end points, then Z

Z !D 0

!: 1

Proof. We assume that both paths are parameterized by the interval I D Œ0; 1. Let ı W I I ! D be a homotopy between our two paths and let f be a primitive of ! along ı. Thus u 7! f .0; u/ is a primitive of ! along the constant curve u 7! ı.0; u/ D 0 .0/, and hence f .0; u/ is a constant ˛ independent of u. Similarly f .1; u/ D ˇ 2 C. But then Z ! D f .1; 0/  f .0; 0/ D ˇ  ˛ 0

and

Z ! D f .1; 1/  f .0; 1/ D ˇ  ˛: 1

t u Remark 4.47. A similar result holds for two curves that are homotopic as closed paths (see Exercise 4.10). Corollary 4.48. If  is homotopic to a point in D and ! is a closed form in D, then Z ! D 0: 

This corollary motivates the following. Definition 4.49. A region D  C is called simply connected if every continuous closed path in D is homotopic to a point in D. Example 4.50. (1) The complex plane C is simply connected. More generally, (2) Discs are simply connected: let c 2 C, and for R 2 .0; C1/ set D D Uc .R/ D fz 2 CI jz  cj < Rg. Without loss of generality we assume c D 0 and R D 1. Let  be a continuous closed path in D parameterized by Œ0; 1, and define ı.t; u/ D u.t/. Corollary 4.51.R If D is a simply connected domain and  is a continuous closed path in D, then  ! D 0 for all closed forms ! on D. We obtain the simplest formulation of the main result: Corollary 4.52. In a simply connected domain a differential form is closed if and only if it is exact. Remark 4.53. The property appearing in the last corollary actually characterizes simply connected domains. We will not prove nor use this fact.

100

4 The Cauchy Theory: A Fundamental Theorem

An immediate corollary gives the existence of branches of the logarithm: Corollary 4.54. In every simply connected domain not containing the point 0, there exists a branch of log z. dz is closed and thus exact in the given domain. z Hence there exists a holomorphic function F (on the same domain) such that dF D !. This function F is a branch of the logarithm. t u

Proof. The differential form ! D

Remark 4.55. Annuli and punctured discs are not simply connected. To see this, let R1 , R, and R2 be positive real numbers such that R1 < R < R2 . For any complex number z0 , let A be the annulus A D fzI R1 < jz  z0 j < R2 g, and let D denote the punctured disc D D fzI 0 < jz  z0 j < R2 g. The range of the continuous closed path .t/ D R exp.2 { t/, for 0  t  1, is contained in A and in D, and R dz D 2{ ¤ 0.  z  z0 To give more examples of simple connectivity, we introduce Definition 4.56. A region D is convex if every pair of points in D can be joined by a segment in D. Note that convex implies simply connected, but the converse is not true.

4.5 More on the Winding Number In Sect. 4.3 we defined the winding number I.; c/ of a curve  with respect to a point c not on its range. In this section we will see that arguments involving the winding number allow us to draw strong conclusions about the behavior of a function defined in a disc (Theorem 4.57). We begin with some properties of I.; c/ for  a closed path and c … range  . 1. If 0 and 1 are homotopic as closed paths in D D C  fcg, then I.0 ; c/ D I.1 ; c/. t u Z Z dz Proof. The differential ! D !D !. is closed on D. Thus zc 0 1 2. If  W Œa; b ! C is a closed path, then z 7! I.; z/ is a locally constant function on the open set E D C  range  ; hence it is constant on each connected component of E. Proof. Let c 2 E; we need to show that for all h 2 C with jhj sufficiently small, I.; c C h/ D I.; c/. Without loss of generality, we assume Œa; b D Œ0; 1. Choose ı0 to be any positive number less than the distance of c to range  , then U..t/; ı0 /  C  fcg

4.5 More on the Winding Number

101

1 2

0

3

Fig. 4.5 Winding numbers

for all t 2 Œ0; 1. If we fix any h 2 C with jhj < ı0 , then Z 

dz D z  .c C h/

Z 

dz D .z  h/  c

Z 0

dz0 ; c

z0

where z0 D z  h and  0 D   h; that is,  0 .t/ D .t/  h for all t 2 Œ0; 1. Now  0 is a closed path in C  fcg, and  0 is homotopic to  there (as closed paths), via the homotopy defined on Œ0; 1 Œ0; 1 by ı.t; u/ D .t/  uh; for t; u 2 Œ0; 1: Thus I.; c C h/ D I. 0 ; c/ D I.; c/.

t u

3. If range   D  C  fcg with D simply connected, then I.; c/ D 0. Proof. The differential

dz is closed in D, therefore exact. zc

t u

4. I.; c/ D 0 for all c in the unbounded component of the complement of the range of  . 5. Let  W t 7! Re{t , with R > 0 and t 2 Œ0; 2. Then ı I.; 0/ D 1, by Example 4.39 ı I.; z/ D 1 for jzj < R, by (2) ı I.; z/ D 0 for jzj > R, by (3) In Fig. 4.5 we depict the range of a closed path  . The numbers indicate the value of the winding number I.; c/ for a point c in the respective component of the complement of the range of  . We now show that if the image of the boundary of a disc under a continuous function f winds nontrivially around a point c, then f assumes the value c somewhere inside the disc. More precisely,

102

4 The Cauchy Theory: A Fundamental Theorem

Theorem 4.57. Let f W fz 2 C W jzj  Rg ! C be a continuous map (with R > 0) and let ./ D f .Re2{ / for  2 Œ0; 1. If c … range  and I.; c/ ¤ 0, then there exists a z such that jzj < R and f .z/ D c. Proof. Assume f .z/ ¤ c for all jzj < R. Then f .z/ ¤ c for all jzj  R, because c … range  . Define ı.; / D f .Re2{ / on Œ0; 1 Œ0; 1. Then ı is continuous, ı.1; / D ./, ı.0; / D f .0/, ı.; 0/ D ı.; 1/, and ı.; / 2 C  fcg for all ;  2 Œ0; 1. Thus  is homotopic to a point in C  fcg, and hence I.; c/ D 0; we have arrived at the needed contradiction. t u Definition 4.58. Let 1 and 2 be continuous paths parameterized by Œ0; 1. We define two new continuous paths, also parameterized by Œ0; 1: 1 2 W t 7! 1 .t/2 .t/; 1 C 2 W t 7! 1 .t/ C 2 .t/: Note that the above definition of a product of two paths differs from the one given earlier in Definition 4.10, where the paths are traversed in succession but at twice the speed. Theorem 4.59. If 1 and 2 are continuous closed paths not passing through 0, then I.1 2 ; 0/ D I.1 ; 0/ C I.2 ; 0/: dz and j W Œ0; 1 ! C  f0g. Choose continuous functions fj W z fj .t / D j .t/ for all t 2 I . Then ef1 .t /Cf2 .t / D 1 2 .t/. Thus Œ0; 1 ! C so that e f D f1 C f2 is a primitive of ! along 1 2 , and Proof. Let ! D

I.1 2 ; 0/ D

f1 .1/  f1 .0/ f2 .1/  f2 .0/ f .1/  f .0/ D C : 2{ 2{ 2{ t u

Theorem 4.60. Let 1 and  be continuous closed paths in C parameterized by Œ0; 1. Assume that 0 < j1 .t/j < j.t/j for all t 2 Œ0; 1: Then I.1 C ; 0/ D I.; 0/: Proof. Note that   1 .t/ D .t/ˇ.t/ .t/ C 1 .t/ D .t/ 1 C .t/

4.6 Cauchy Theory: Initial Version

with ˇ.t/ D 1 C

103

1 .t/ for all t 2 Œ0; 1, and thus .t/ I.1 C ; 0/ D I.; 0/ C I.ˇ; 0/:

Now observe that jˇ.t/  1j < 1; and thus ˇ is a closed path in the simply connected domain U.1; 1/. Therefore I.ˇ; 0/ D 0. t u

4.6 Cauchy Theory: Initial Version The most important technical result of this chapter is the following. Theorem 4.61 (Goursat’s Theorem). If f is a holomorphic function on a domain D, then f .z/ dz is a closed differential form on D. Equivalently, if  denotes the boundary of R oriented counterclockwise, where R is any rectangle in D with sides R parallel to the coordinate axes, then  ! D 0. This theorem has many significant consequences. To prove it we need some preliminaries. Recall that the only issue that needs to be addressed involves the smoothness of the function f ; we are not assuming that the function has continuous partial derivatives. For C1 -functions, we already have the result (see Theorem 4.28). We follow a beautiful classic line of reasoning for the proof. Definition 4.62. Let  be a pdp parameterized by the unit interval Œ0; 1 in a domain D and let f be a continuous function on D. For such restricted paths, we define (1) The integral of f on  with respect to arc length Z

Z

1

f .z/ jdzj D 

ˇ ˇ f ..t// ˇ 0 .t/ˇ dt

0

(2) The length of the curve  Z L. / D length of  D

jdzj 

The following results follow straightforwardly from these definitions. Proposition 4.63. Let  be a pdp parameterized by the unit interval Œ0; 1 in a domain D, and let f be a continuous function on D. Then

104

4 The Cauchy Theory: A Fundamental Theorem

R1

R2 γ1

γ2

R3

R4 γ3

γ4

Fig. 4.6 The integrals along the common side of R1 and R2 are in opposite directions

Z

Z f .z/ jdzj D 

f .z/ jdzj ; 

where  denotes the pdp  traversed backward (recall Definition 4.9), and ˇ Z ˇZ ˇ ˇ ˇ f .z/ dzˇ  jf .z/j jdzj  .supfjf .z/j W z 2 range  g/  L. /: ˇ ˇ 



Proof of Theorem 4.61 (Due to E. Goursat). Let R be any rectangle in D with sides parallel to the coordinate axes, and let R  denotes R the boundary of R oriented counterclockwise. We need to show that  ! D  f .z/dz D 0. (Even though the proof of this last statement is due to Goursat, the Rresult is known as Cauchy’s theorem for a rectangle.) Assume to the contrary that  ! D ˛ ¤ 0, and divide R into four congruent rectangles R1 ; : : : ; R4 , with boundaries  1 ; : : : ;  4 . Then Z !D

˛D 

4 Z X i D1

!D i

4 X

˛ .i / :

i D1

The second equality follows from the fact that certain paths on the boundaries of the subrectangles have opposite directions, giving cancelations in the integrals (see Fig. 4.6). It is clear that we must have ˇ .i / ˇ j˛j ˇ˛ ˇ  4 for at least one index i . Call the corresponding rectangle R1 , its boundary 1 , and the corresponding integral ˛1 . By repeating this procedure, we obtain a sequence of closed rectangles R R1    Rk    R with boundaries k D @Rk so that for ˛k D k !, we have

4.6 Cauchy Theory: Initial Version

105

j˛k j 

j˛j : 4k

Each rectangle Rk is a closed subset of D and also of C; furthermore, lim Area .Rk / D lim

k!1

k!1

Area .R/ D 0: 4k

The Bolzano–Weierstrass theorem states that there exists a unique c 2 f is holomorphic at c,

T

k Rk . Since

f .z/ D f .c/ C f 0 .c/.z  c/ C ".z/ jz  cj ; where lim ".z/ D 0:

z!c

 Let k denote the length of a diagonal of Rk , then k D k , where  is the length 2 of a diagonal of R. Now Z

Z

˛k D

0

f .z/ dz D f .c/ k

Z

Z

dz C f .c/ k

.z  c/ dz C k

jz  cj ".z/ dz: k

The first two integrals following the second equal sign are equal to zero since the 1 integrands are exact forms (dz and d.z  c/2 , respectively). Thus we have 2 Z ˛k D jz  cj ".z/ dz: k

Now on Rk we have jz  cj  k D 2k : Given  > 0, there exists a K 2 Z>0 such that j".z/j <  for all z in RK and thus also for all z in Rk and all k  K. Hence we have Z j˛k j  jz  cj j".z/j jdzj  2k   L.k / D  4k  L. / k

for k  K. We conclude that j˛j  4k j˛k j    L. /: Since  is arbitrary, we must have that ˛ D 0.

t u

106

4 The Cauchy Theory: A Fundamental Theorem

Corollary 4.64 (Cauchy’s Theorem). If f is holomorphic on a domain R D and  is a continuous closed path in D that is homotopic to a point in D, then  f .z/ dzD0. Corollary 4.65. If f is holomorphic in a domain D, then locally f .z/ dz has a primitive in D. Remark 4.66. We have previously shown that (1) implies (4) in the fundamental theorem. We have just established another path for obtaining the same conclusion.

4.7 Appendix I: The Exterior Differential Calculus To place the discussion of differential forms in its broader context, we outline some basic concepts of the exterior differential calculus for domains D  C using only complex coordinates z D x C {y (w D u C {v).7 The reader is invited to translate all concepts under discussion to the real (rectangular) coordinates .x; y/.8 For every nonnegative integer n, let n .D/ denote the space of n-forms on D. Since D is two dimensional over the reals, we need to consider only n D 0; 1; 2, because under any reasonable set of definitions n .D/ D f0g for n > 2. The set 0 .D/ consists of smooth (C1 -) functions on D. The space 1 .D/ can be viewed as a module over 0 .D/ generated by the 1-forms d z and dz; that is, expressions of the form f .z/dz C g.z/dz with f and g smooth, subject however to the appropriate transformation rule under complex coordinate changes (w is an injective holomorphic function of z): F .w/dw C G.w/dw D f .z/dz C g.z/dz; where

f .z/ D F .w.z//w0 .z/ and g.z/ D G.w.z//w0 .z/;

and the general rule about exterior multiplication ^ of forms is given as follows: dz ^ dz D 0 D dz ^ dz; dz ^ dz D dz ^ dz D 2{dx ^ dy: The set 2 .D/ consists of elements of the form f .z/dz ^ dz D F .w/dw ^ dw; where f and F are C1 -functions with ˇ ˇ2 f .z/ D F .w.z// ˇw0 .z/ˇ ; where w is an injective holomorphic function of z.

7 8

Usually w is an analytic function of z under appropriate circumstances. Allowing arbitrary C1 coordinate changes.

4.8 Alternative Approach

107

The space of one-forms 1 .D/ decomposes as a direct sum 1 .D/ D .1;0/ .D/ ˚ .0;1/ .D/; where .1;0/ .D/ D f! 2 1 .D/I ! D f .z/dzg; and .0;1/ .D/ D f! 2 1 .D/I ! D g.z/dzg: Three differential operators, @, @, and d D @C@, act on forms; each of these follows the product rule; for example, d.!1 ^ !2 / D d.!1 / ^ !2 C !1 ^ d.!2 /: Thus these operators are completely determined by their actions on functions (of z) and the distinguished one forms dz and dz, for example, @.f .z// D fz dz; @.dz/ D 0 D @.z/: One easily check that 2

0 D @2 D @@ C @@ D @ and d 2 D @@ C @@: The differential operators lead to three set of maps: @

@

@

@

@

@

f0g ! 0 .D/ ! .1;0/ .D/ ! 2 .D/ ! f0g; f0g ! 0 .D/ ! .0;1/ .D/ ! 2 .D/ ! f0g; and d

d

d

f0g ! 0 .D/ ! 1 .D/ ! 2 .D/ ! f0g:

4.8 Appendix II: An Alternative Approach to the Cauchy Theory There is an alternative approach to the Cauchy theory that does not begin with differential forms but rather ends with it. In the main body of the text we based the theory on integrating differential forms. The alternative development starts with integration of functions, mimicking as much as possible ideas from a classical treatment of undergraduate calculus. It is a matter of taste which development one uses or prefers. Both treatments yield the same or equivalent theorems. In this appendix we outline the alternative development leaving many details to the

108

4 The Cauchy Theory: A Fundamental Theorem

reader, essentially by listing the order of definitions and results we have already encountered and providing full statements for results, usually theorems and their corollaries, that appear different than in the main body of this book. Proofs are included as needed, particularly when they involve new ideas. This outline follows the approach that Bers used in many of his courses and is similar to the approach of Ahlfors in his text. We end this outline after we prove a strengthened version of Goursat’s theorem (another version is in the next chapter). Since this approach does not rely on power series, we include a self-contained proof that the derivative of an analytic function is also analytic. For the rest of this appendix f D
4.8.1 Integration of Functions We start by considering four types of integrals, essentially mimicking and expanding slightly ideas from elementary calculus. • For a complex-valued continuous function w defined on a real interval Œa; b with w.t/ D u.t/ C { v.t/ (as usual, u D <w and v D =w), we set Z

Z

b

a

Z

b

w.t/ dt D

u.t/ dt C {

a

b

v.t/ dt: a

• Let  W Œa; b ! C be a pdp and f a continuous function on the range of  . The complex line integral of f over  is Z

Z

b

f .z/ dz D 

f ..t//  0 .t/ dt:

a

Hence the integral is reduced to the previous case. • Under the same conditions for f and  , we have the integral with respect to arc length Z Z b ˇ ˇ f .z/ jdzj D f ..t// ˇ 0 .t/ˇ dt: 

a

This integral and the next concept were encountered in our proof of Goursat’s theorem. R Rb • The length of the pdp  is L. / D  jdzj D a j 0 .t/j dt: There are a number of properties that these integrals satisfy, including, for example, invariance under orientation preserving change of parameter. We do not

4.8 Alternative Approach

109

discuss these here since they should be familiar to the reader who studied this chapter. We have previously used, and find useful here, the calculation Z

Z f .z/ dz D 

.u C {v/.dx C { dy/ Z



D

.udx  vdy/ C {.vdx C udy/ Z



D

p dx C q dy: 

It is convenient to introduce, for appropriate c and d 2 C, the abbreviation Z

d

f .z/ dz c

to denote the integral along the line segment L from c to d . Using the mean value integral theorem (from calculus), we easily compute that Z

d

f .z/ dz D .d  c/.
c

where z1 and z2 2 L.

4.8.2 The Key Theorem We proceed in an informal way to the key results. We fix a domain D, z0 2 D, and R > 0 such that U.z0 ; R/  D. (1) Goursat’s Theorem for Triangles: If f 0 .z/ exists forRall z in D, and T is any triangle contained in D (including its interior), then @T f .z/dz D 0. Proof. The proof is identical to the one of Goursat’s theorem in this chapter. Since we are dealing with triangles, we join the midpoints of its sides to divide a triangle into four smaller triangles, each with perimeter equal to half the perimeter of the original triangle. t u (2) Strengthened Goursat’s Theorem for Triangles: If f is continuous on D, f 0 .z/ exists for all z in D except for a discrete subset, and T is a triangle contained R in D (including its interior), then @T f .z/dz D 0. ˇR ˇ Proof. It suffices to show that ˇ @T f .z/dzˇ   for all  > 0. Let M > 0 be an upper bound for f on T . Since T is compact, there are only finitely many, say N , bad points in this set where f 0 does not exists; if N D 0 we are done,

110

4 The Cauchy Theory: A Fundamental Theorem

by Goursat’s theorem for triangles, so assume N > 0. Divide T into smaller triangles T1 ; : : : ; Tk (several times as in the above proof) so that the perimeter  of each triangle is of length  6MN . Now, as usual, Z f .z/dz D @T

k Z X

f .z/dz:

j D1 @Tj

R

f .z/dz D 0 by Goursat’s theˇR ˇ ˇ ˇ orem for triangles. For a triangle Tj that contains a bad point, ˇ @Tj f .z/dzˇ   M 6MN . Since each bad point can be in at most six triangles and there are N bad points, the required estimate follows. t u If Tj does not contain any bad points, then

@Tj

(3) If f is continuous on D, f 0 .z/ exists for all z in D except for a discrete subset, and D 0 is any convex subdomain of D, then there is F W D 0 ! C with F 0 .z/ D f .z/. Rz Proof. Fix z0 2 D 0 , define F .z/ D z0 f .t/dt for z in D 0 , and use the strengthened form of Goursat’s theorem for triangles to conclude that F 0 .z/ D f .z/. Specifically, by the strengthened form of Goursat’s theorem for triangles, for all small jhj > 0 (such that the triangle with vertices z0 , z, and z C h, including its interior, is contained in D 0 ), we obtain Z F .z C h/ D

zCh

Z f .t/dt D

z0

z

Z f .t/dt C

z0

zCh

f .t/dt: z

Thus Z

zCh

F .z C h/ D F .z/ C

f .t/dt z

D F .z/ C h.
F 0 .z/ D lim

t u

(4) Fundamental Theorem of Calculus (FTC) (complex version): If the function F W D ! C is complex differentiable and F 0 is continuous in D, then Z

F 0 .z/dz D F ..b//  F ..a// 

for all pdp’s  W Œa; b ! D.

4.8 Alternative Approach

111

Proof. Use the definition of the complex line integral, CR and FTC (real version)). t u 0 0 (5) If for every convex R subdomain D contained in D there 0is an F W D ! C with 0 F D f , then  f .z/dz D 0 for any closed pdp  in D . R R Proof.  f .z/dz D  F 0 .z/dz D 0 by FTC. t u

(6) If  is a closed pdp in C and z0 is not in its image, then I.; z0 / D

1 2{

Z 

dz 2 Z: z  z0

We have established this fact for any closed curve, not just a pdp, because of work with closed differential forms and logarithms. We now show that these ingredients are unnecessary if we restrict to pdp. Proof. For  W Œa; b ! C, set Z

t

h.t/ D a

 0 .s/ ds .s/  z0

for a  t  b. Then h is continuous in Œa; b, Z h.a/ D 0; h.b/ D 

dz  0 .t/ ; and h0 .t/ D z  z0 .t/  z0

at the points t where  0 .t/ is continuous. Let g.t/ D eh.t / ..t/  z0 /: Then g is continuous in Œa; b, g 0 .t/ D 0 except at a finite number of points t, and hence g is constant: g.t/ D g.a/ D .a/  z0 for all t in Œa; b. Therefore eh.t / D But then eh.b/ D

.t/  z0 : .a/  z0

.b/  z0 D1 .a/  z0

since  is closed, and therefore there exists k in Z such that

112

4 The Cauchy Theory: A Fundamental Theorem

Z

dz D h.b/ D 2{k: z  z0



t u

The reader may notice that part of our argument here is very similar to one used in the main body of our book under different circumstances (see the proof of Theorem 3.38). Many properties of the index can be developed. Among them: For the curve .t/ D z0 C Re2{t , 0  t  1, and jz  z0 j ¤ R, Z 

d

D

z



2{; if jz  z0 j < R; 0; if jz  z0 j > R:

(7) If f 0 exists in D and fjz  z0 j  Rg  D, then f .z/ D

1 2{

Z j z0 jDR

f . / d

z

for all z such that jz  z0 j < R. In the last formula, z0 C Re2{t , 0  t  1.

R j z0 jDR

represents integration over the path .t/ D

Proof. Fix z such that jz  z0 j < R and define the function g W D ! C as follows: 8 < f . /  f .z/ ; for ¤ z; g. / D

z : f 0 .z/; for D z: Then g is continuous in D, differentiable except perhaps at z, and it follows from (4.8.2) above that there exists a complex differentiable G W fjz  z0 j  Rg ! C with G 0 D g. Thus by (4.8.2) that Z

Z g. /d D

0D j z0 jDR

j z0 jDR

f . /  f .z/ d ;

z

and the desired integral formula follows.

t u

(8) It remains (under this approach) to be shown that the derivative of a holomorphic function is again holomorphic. This follows from the next lemma, whose similarity to Exercise 4.11 should be noted. Lemma 4.67. If  is a pdp, g a continuous function on the range of  , and n 2 Z>0 , then the function defined by Z Fn .z/ D 

g. / d ; z 2 C  range ; .  z/n

(4.4)

4.8 Alternative Approach

113

is analytic in each of the regions in the complement of the range of  and satisfies Fn0 D nFnC1 there. Proof. We first prove that F1 is continuous at z0 in , a component of the complement of the range of  . Choose ı > 0 such that U.z0 ; ı/  . For z such that jz  z0 j < ı2 , we have j  zj > 2ı for all in range  ; hence ˇ ˇ Z ˇ ˇ g. / d

ˇ ˇ jF1 .z/  F1 .z0 /j D ˇ.z  z0 / ˇ  .  z/.  z0 / Z jg. /j jd j  jz  z0 j ı ı  2 Z 2 D jz  z0 j 2 jg. /j jd j ; ı  and the last term goes to zero when z ! z0 . We also obtain, for all small positive values of jz  z0 j, F1 .z/  F1 .z0 / D z  z0

Z 

g. / d

: .  z/.  z0 /

Thus F10 .z0 /

Z D lim

z!z0

Z D 



g. / d

.  z/.  z0 /

g. / d

D F2 .z0 /: .  z0 /2

The general case is proved by induction: assume that n is at least equal to two, that Fn1 is analytic for any function g continuous on the range of  , and 0 that Fn1 .z/ D .n  1/Fn .z/ for all z in the complement of the range of  . g. / Note that the function e g . / D is continuous on the range of  , and

 z0 therefore the function Z e g . / d

Fn1 .z/ D .

 z/n1 

e

is analytic in the complement of range  and satisfies

e

fn .z/ Fn1 0 .z/ D .n  1/F there.

114

4 The Cauchy Theory: A Fundamental Theorem

We need to show that Fn , defined in (4.4), is analytic and satisfies Fn0 .z/ D nFnC1 .z/ in the complement of the range of  . To do this, first observe that 1 1 .  z0 /n  .  z/n  D n n .  z/ .  z0 / .  z/n .  z0 /n D

.  z0 /n1 .  z C z  z0 /  .  z/n .  z/n .  z0 /n

D

1 1 z  z0  ; C .  z/n1 .  z0 / .  z0 /n .  z/n .  z0 /

and then conclude that  Z Fn .z/  Fn .z0 / D g. / Z





1 1  .  z/n .  z0 /n



1 z  z0 D g. /  C n1 .  z / .  z /n n .  z / .

 z/ .

 z/ 0 0 0  Z g. / d

: D Fn1 .z/  Fn1 .z0 / C .z  z0 / .

 z/n .  z0 /  1

e



e

Thus lim Fn .z/ D Fn .z0 /, and Fn is continuous at z0 . In addition we obtain, as z!z0

before,

e

e

Fn1 .z/  Fn1 .z0 / Fn .z/  Fn .z0 / D C z  z0 z  z0

Z 

g. / d

; .  z/n .  z0 /

and hence

e

Z

g. / d

.

 z0 /nC1  Z g. / d

fn .z0 / C D .n  1/F nC1  .  z0 / Z Z g. / d

g. / d

D .n  1/ C D nFnC1 .z0 /: nC1 nC1  .  z0 /  .  z0 /

Fn0 .z0 / D Fn1 0 .z0 / C

t u

(9) If f is continuous on D, and for all R > 0 such that fjz  z0 j  Rg  D, the equality Z f . / 1 d

f .z/ D 2{ j z0 jDR  z

Exercises

115

holds for all z such that jz  z0 j < R, then f .n/ .z/ exists for all z in D and for all n 2 Z0 . In particular, if f 0 .z/ exists for all z in D, then the conclusion holds. Proof. Fix z0 in D and choose R > 0 such that f .z/ D

Z

1 2{

j z0 jDR

f . / d

z

for all z such that jz  z0 j < R. It follows from Lemma 4.67 that then f 0 .z/ D

1 2{

Z j z0 jDR

f . / d ; .  z/2

for all z such that jz  z0 j < R and, more generally, that for each n  0 and z such that jz  z0 j < R: f

.n/

nŠ .z/ D 2{

Z j z0 jDR

f . / d : .  z/nC1 t u

This material suffices to produce the Cauchy estimates and Morera’s and Liouville’s theorems, as done in the next chapter.

Exercises 4.1. Evaluate the line or contour integral

R C

jzj dz directly from the definition if

(1) C is a straight line segment from { to {: (2) C is the left half of the unit circle traversed from { to {. (3) C is the right half of the unit circle traversed from { to {. R 4.2. Evaluate the line or contour integral C xdz directly from the definition when C is the line segment from 0 to i C 1. R 4.3. Let z0 2 C. Evaluate the line or contour integral C .z  z0 /m dz directly from the definition, where C is the circle centered at z0 with radius r > 0 and (1) m is an integer, m  0. (2) m is an integer, m < 0. 4.4. Evaluate the line or contour integral the path .t/; 0  t  1, where (1) .t/ D 1 C i t. (2) .t/ D e{t .

R 

z3 dz directly from the definition over

116

4 The Cauchy Theory: A Fundamental Theorem

(3) .t/ D e{t . (4) .t/ D 1 C {t C t 2 . R 1 dz over the same paths.  z 4.5. Let D1 and D2 be simply connected plane domains whose intersection is nonempty and connected. Prove that their intersection and their union are both simply connected. Evaluate the integrals

R



zN dz and

4.6. Show that for any closed interval Œa; b there exists a one-to-one, onto, and differentiable map t W Œ0; 1 ! Œa; b with t 0 .u/ > 0 for all u. 4.7. Establish Remark 4.7. 4.8. Verify that f .t/ D 2{ t for 0  t  1 is a primitive of the closed form !D

d along the path .t/ D exp.2{t/; 0  t  1; z

on the domain D D C6D0 . Observe that f .0/ ¤ f .1/ and .0/ D .1/; see the warning after Definition 4.31. 4.9. Let P and Q be smooth functions on a domain D  C. Find necessary and sufficient conditions for the form P dz C Qdz to be closed. R 4.10. Let D be a domain in C. We have studied the pairing  !, where  is a closed path in D and ! is a closed differential form in D. Show that R (1)  ! depends only on the homotopy class of the closed path  ; that is, if we replace  by a closed path  0 homotopic to  (as closed paths), then the integral is unchanged. R (2) (Only for those who know some algebraic topology)  ! depends only on the homology class of the closed path  ; that is, if we replace  by a closed path  0 homologous to  , then the integral is unchanged. R (3)  ! depends only on the cohomology class of the closed form !; that is, if we replace ! by ! C df with f 2 C2 .D/, then the integral is unchanged. 4.11. In this exercise we consider “differentiation under the integral sign.” (1) Let  W Œa; b ! C denote a pdp, and let ' W range  ! C be a continuous function. Define g W D D C  range  ! C by Z g.z/ D 

'.u/ du: uz

Show that g has derivatives of all orders n  0 and that

Exercises

117

Z g

.n/

.z/ D nŠ 

'.u/ du .u  z/nC1

for all n in Z0 . Thus, in particular, g is holomorphic on D. (2) Let z0 2 C, R > 0 and .t/ D z0 C R exp.2{t/ for 0  t  1. Use the first part of this exercise to show that 1 2{

Z 

dz D zw



1; if jz  wj < R; 0; if jz  wj > R:

Chapter 5

The Cauchy Theory: Key Consequences

This chapter is devoted to some immediate consequences of the fundamental result for the Cauchy theory, Theorem 4.61, of the last chapter. Although the chapter is very short, it includes proofs of many of the implications of the fundamental theorem in complex function theory (Theorem 1.1). We point out that these relatively compact proofs of a host of major theorems result from the work put into Chap. 4 and earlier chapters. The appendix to this chapter contains a version of Cauchy’s integral formula for smooth (not necessarily holomorphic) functions.

5.1 Consequences of the Cauchy Theory We begin with a technical strengthening of Theorem 4.61 allowing functions that are holomorphic on a domain except on a line segment. It will lead to Cauchy’s integral formula, once described as the most beautiful theorem in complex variables. Theorem 5.1 (Goursat’s Theorem, Strengthened Version). If f is continuous in a domain D and holomorphic except possibly on a line segment in D, then f .z/ dz is closed in D. Proof. Without loss of generality, D is the unit disc, and the line segment is all or part of the real axis in D. R We must show that the integral  f .z/ dz vanishes whenever  is the (positively oriented) boundary of an open rectangle R whose closure is contained in D and whose sides are parallel to the coordinate axes. There are three possibilities for such rectangles: (1) The closure of R does not intersect the real axis. (2) The closure of R has one side on R. (3) (The interior of) R intersects R. 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 5, © Springer Science+Business Media New York 2013

119

120

5 The Cauchy Theory: Key Consequences

Rδ R δ

a

b

Fig. 5.1 The rectangles R and Rı

In case (1), there is nothing to do by the results of the last chapter. In case (3), we reduce to the case of two rectangles as in case (2). Thus it suffices to consider a rectangle of type (2). Assume that the rectangle R lies in the upper semi-disc with one side on R from a to b > a. (The possibility of R in the lower half disc is handled similarly.) Let Rı be the rectangle R with the portion below height ı chopped off, for ı > 0 sufficiently small (see Fig. 5.1), and to be chosen later. Then the difference of the integrals over the boundary of R and the boundary of Rı is an integral over an appropriate rectangle: Z

Z

Z

f .z/ dz  @R

b

f .z/ dz D @Rı

a

Z f .x; 0/ dx C { a b

b

D

Z

0

f .x; ı/ dx C {

C Z

f .b; y/ dy 0

Z

ı

a

f .a; y/ dy ı

.f .x; 0/  f .x; ı// dx Z

ı

C{

.f .b; y/  f .a; y// dy: 0

Now given  > 0, there exists a ı with 0 < ı <  such that jz  cj < ı implies that jf .z/  f .c/j <  for all z and c in R (by the uniform continuity of f on R). Also there exists an M > 0 such that jf .z/j  M for all z 2 R. Thus ˇZ ˇ ˇ ˇ

Z f .z/ dz  @R

@Rı

ˇ ˇ f .z/ dzˇˇ  .b  a/ C 2M:

5.1 Consequences of the Cauchy Theory

121

Since  is arbitrary, this tells us that Z Z f .z/ dz  But we know that

@R

R @Rı

f .z/ dz D 0: @Rı

f .z/ dz D 0; hence also

R @R

f .z/ dz D 0.

t u

We apply this strengthened theorem to obtain Theorem 5.2 (Cauchy’s Integral Formula). If f is holomorphic on a domain D and  is a continuous closed path homotopic to a point in D, then for all c 2 C  range  , we have Z 1 f .z/ dz D I.; c/  f .c/: (5.1) 2{  z  c Proof. Define, for z 2 D, 8 < f .z/  f .c/ ; if z ¤ c; g.z/ D zc : f 0 .c/; if z D c: Then g is continuous on D and holomorphic except (possibly) at c and thus, by Theorem 5.1, g.z/ dz is closed in D. It follows that Z

Z

0D

g.z/ dz D 

Thus

Z 



f .z/  f .c/ dz: zc

f .z/ dz D f .c/ zc

Z 

dz : zc t u

Example 5.3. Let D be a domain in C, f a holomorphic function defined on D, and c 2 D. Choose R > 0 such that clU.c; R/  D and let ./ D c C Re2{ , for 0    1. Then I.; w/ D 1 for jw  cj < R and I.; w/ D 0 for jw  cj > R, by property (4) of the winding numbers given in Sect. 4.5. Thus (1) 1 2{

Z 

(2) 1 2{

f .z/ dz D f .w/ for jw  cj < R and zw Z 

f .z/ dz D 0 for jw  cj > R: zw

Remark 5.4. Equation (5.1) gives the amazing result that the value of a holomorphic function at a point interior to a circle (or eventually to any simple closed curve)

122

5 The Cauchy Theory: Key Consequences

is completely determined by the values of the function on the boundary circle. The function must, of course, be holomorphic in a neighborhood of the closed disc bounded by the circle. The requirement that the function be analytic in a neighborhood of the disc can be relaxed considerably, but this topic is not discussed in the book. It follows from (5.1) and Exercise 4.11 or from the next theorem that a holomorphic function has derivatives of all orders. We prove a more general result next. Theorem 5.5 (Taylor (Power) Series Expansions for Holomorphic Functions). If f is holomorphic in the open disc fjzj < Rg, with R 2 .0; C1, then f has a power series expansion at each point in this disc. In particular, there exists a power 1 X series ak zk with radius of convergence   R such that kD0

f .z/ D

1 X

ak zk for jzj < R:

kD0

Proof. It suffices to establish the particular claim. Choose 0 < r0 < R and define ./ D r0 e2{ for 0    1. Then jzj D r < r0 implies I.; z/ D 1. We start for such z with Z 1 f .t/ f .z/ D dt; 2{  t  z by Theorem 5.2. Our plan is to write the integrand as a power series in z and then interchange the order of the operations (integration and summation). Now  z nC1

 z nC1 n 1  1 1 t 1 1 X  z k t  : D D z t z t 1 z t t t 1 kD0 t t Hence

 z nC1 n 1 X  z k 1 t 1 D C t z t t t 1 z kD0 t

and 2

 z nC1 3 Z Z n X 1 6 f .t/ f .t/ t 7 f .z/ D zk dt C 4 z dt 5 kC1 2{ t t   1 kD0 t D

n X kD0

ak zk C Rn ;

5.1 Consequences of the Cauchy Theory

123

where

ak D

1 2{

Z 

f .t/ 1 dt ; and Rn D t kC1 2{

Z 

 z nC1 f .t/ t z dt: t 1 t

On the range of  we have t D r0 e2{ , jtj D r0 , and jdtj D 2 r0 d. Let ˇ ˇ M.r0 / D supfˇf .r0 e2{ /ˇ W 0    1g ˇ z ˇˇ r ˇ and observe that ˇ1  ˇ  1  . Hence t r0  Z

1

r r0

nC1

M.r0 / r r0 2 d r0 0 1 r0  nC1 r r D M.r0 / 0 r ! 0 as n ! 1: 1 r0

jRn j 

We conclude that f .z/ D

1 2

1 X

ak zk for jzj < R; further,

kD0

1 ak D 2{

Z 

f .t/ dt t kC1

is independent of r0 for 0 < r0 < R. We have also obtained the estimates jak j 

M.r0 / r0k

(5.2)

and therefore 1

1

jak j k 

M.r0 / k : r0

Thus (we have obtained a second proof that) the radius of convergence  of satisfies   r0 for all r0 < R; in particular,   R.

P

ak zk t u

We will make further use of (5.2) shortly. Note that the theorem (once again) allows us to interchange the orders of operations: Z Z 1 1 X X zk f .t/ k dt D z dt: 2{f .z/ D f .t/ kC1 kC1 t   t kD0

kD0

124

5 The Cauchy Theory: Key Consequences

Corollary 5.6. A function f is holomorphic in an open set D if and only if f has a power series expansion at each point of D. For a holomorphic function f on D, the Taylor series expansion of f at c 2 D f .z/ D

1 X

ak .z  c/k

kD0

has radius of convergence   supfr > 0I U.c; r/  Dg: Remark 5.7. For c D 0, the above Taylor series is also called a Maclaurin series. Corollary 5.8. If f is holomorphic on a domain D, then f is C1 in D, and for each n 2 Z0 , f .n/ is holomorphic on D. Corollary 5.9 (Cauchy’s Generalized Integral Formula). Let f be holomorphic on a domain D containing clU.c; R/ for some c 2 D and R > 0. If ./ D c C R e{ for 0    2, then for n D 0; 1; 2; : : : , f .n/ .c/ D

nŠ 2{

Z 

f .t/ dt: .t  c/nC1

Proof. Recall that for n D 0; 1; 2; : : : , 1 an D 2{

Z 

f .n/ .c/ f .t/ : dt D .t  c/nC1 nŠ t u

Theorem 5.10 (Morera’s Theorem). then f is holomorphic on D.

If f 2 C .D/ and f .z/ dz is closed on D, 0

Proof. Since the differential form ! D f .z/ dz is locally exact, for each point c 2 D there is a neighborhood U of c in D and a primitive F of ! in U . That is, there is a C1 -function F on U with Fz D f and Fz D 0; thus F is holomorphic on U and so is its derivative f , by Corollary 5.8. Since being holomorphic is a local property, f is holomorphic on D. t u An immediate consequence of Morera’s theorem together with Theorem 5.1 is Corollary 5.11. If f is continuous in D and holomorphic except possibly on a line segment in D, then f is holomorphic in D. We have by now established the following important Theorem 5.12. Let f be a complex-valued function defined on an open set D in C. Then the following conditions are equivalent:

5.1 Consequences of the Cauchy Theory

(a) (b) (c) (d) (e)

f f f f f

125

is holomorphic on D. is C1 and satisfies CR on D. is C0 and f .z/ dz is closed on D. is C0 on D and holomorphic except possibly on a line segment in D. has a power series expansion at each point in D.

Remark 5.13. As a consequence of the theorem, the space H.D/ defined in Chap. 3 (see Definition 3.57) consists precisely of the holomorphic functions on D, and a meromorphic function (an element of M.D/, see Definition 3.59) is locally the ratio of two holomorphic functions. Recall that we have established the estimates (5.2) from the Cauchy integral formula. An immediate consequence is the following result. Corollary 5.14 (Cauchy’s Inequalities). Let c 2 C and assume that f .z/ D

1 X

an .z  c/n

nD0

has radius of convergence  > 0. Then an D and

f .n/ .c/ nŠ

ˇ .n/ ˇ ˇ f .c/ ˇ M.r/ ˇ ; jan j D ˇˇ nŠ ˇ rn

(5.3)

for all 0 < r < , where M.r/ D supfjf .z/j I jz  cj D rg: Theorem 5.15 (Liouville’s Theorem). A bounded entire function is constant. Proof. Observe that the Taylor series expansion

1 X

an zn of the entire function f at

nD0

the origin has radius of convergence equal to 1 and that the estimates (5.3) hold for all r > 0. Since there exists M > 0 such that jf .z/j  M for all z 2 C, we obtain jan j 

M rn

for all positive r and all n 2 Zn0 ; taking limit as r goes to 1 shows that an D 0 for all n  1. u t Theorem 5.16 (Fundamental Theorem of Algebra). If P is a polynomial of degree n  1, then there exist a1 ; : : : ; an 2 C and b 2 C6D0 such that

126

5 The Cauchy Theory: Key Consequences

P .z/ D b

n Y

.z  aj / for all z 2 C:

j D1

1 Proof. It suffices to show that P has a root. If not, is an entire function. It is also P 1 bounded since lim D 0 and thus must be constant. t u z!1 P .z/

5.2 Cycles and Homology In order to prove the general version of Cauchy’s integral formula, we will need a more general form of Cauchy’s theorem that deals with integrals over cycles that are homologous to zero. However, except for the next section and Sect. 9.3.3, we develop the subsequent chapters without reference to the material in this section. Definition 5.17. A cycle  is a finite sequence of continuous closed paths in the complex plane. If  is a cycle, we refer to the continuous closed paths 1 ; 2 ; : : : ; n that make up  as its component curves, and we write  D .1 ; 2 ; : : : ; n /. Note that the component curves of a cycle need not be distinct; we also mention that the order of the component curves will not be relevant in our considerations. We consider the range of  to be the union of the ranges of its components. We extend the notion of the integral of a function over a single closed path to the integral over a cycle as follows. Definition 5.18. If  D .1 ; 2 ; : : : ; n / is a cycle, then for any holomorphic function f defined on a domain D such that range   D, we set Z

Z

Z

f .z/ dz D 

f .z/ dz C    C 1

f .z/ dz:

(5.4)

n

We can extend the notion of the index of a point with respect to a path to the index of a point with respect to a cycle as follows. Definition 5.19. The index of a cycle  D .1 ; 2 ; : : : ; n / with respect to a point c 2 C  range  is denoted by I.; c/ and defined by I.; c/ D I.1 ; c/ C    C I.n ; c/:

(5.5)

Definition 5.20. A cycle  with range contained in a domain D  C is said to be homologous to zero in D if I.; c/ D 0 for every c 2 C  D.

5.2 Cycles and Homology

127

Observe that if a continuous closed path  is homotopic to a point in D, then the cycle . / with the single component  is homologous to zero in D. However, the two notions are different, see Exercise 5.3. With these definitions and some work,1 we can obtain the most general forms of Cauchy’s theorem and integral formula. Theorem 5.21 (Cauchy’s Theorem and Integral Formula: General Form). If f is analytic in a domain D  C and  is a cycle homologous to zero in D, then R (a)  f .z/ dz D 0. (b) For all c 2 D  range  , we have (5.1). Proof. If E D fz 2 C  range  I I.; z/ D 0g; then the set E is open in C and contains the unbounded component of the complement of the range of  in C, because it contains the unbounded component of the complement of the range of each component curve of  , as we saw in Sect. 4.5. Moreover E  .C  D/, since  is homologous to zero in D. Define g W D D ! C by 8 f .z/  f .w/ ˆ ˆ for z 6D w; < zw g.w; z/ D ˆ ˆ : 0 f .w/ for z D w: The function g is continuous in D D, and for fixed z 2 D, g.; z/ is holomorphic on D. Furthermore, for all c 2 D  range  , we have Z Z f .z/  f .c/ dz g.c; z/ dz D zc   Z Z f .z/ dz D dz  f .c/ z  c z   c Z f .z/ dz  f .c/ 2 { I.; c/: (5.6) D z  c We define next h.w/ D

8R g.w; z/ dz for w 2 D; ˆ ˆ < 

R f .z/ dz ˆ ˆ : for w 2 E:  zw Noting that D [ E D C, we see from (5.6) that for w 2 D \ E, 1 We are following a course outlined by J. D. Dixon, A brief proof of Cauchy’s integral formula, Proc. Amer. Math. Soc. 29 (1971), 625–626.

128

5 The Cauchy Theory: Key Consequences

Z

Z g.w; z/ dz D





f .z/ dz; zw

because I.; w/ D 0, and thus h is a well-defined function on the plane. The set E contains the complement of a large disc, and the function h is clearly bounded there. By Exercise 4.11, h is complex differentiable; thus h is a bounded analytic function in C and hence constant, by Liouville’s theorem. Since lim h.w/ D 0, h is the zero function. In particular,

z!1

Z g.w; z/ dz D 0 

for all w 2 D  range  , and (b) follows from (5.6). We now fix a point c 2 D  range  and apply part (b) to the analytic function defined on D by z 7! .z  c/f .z/ and the cycle  , to obtain 1 I.; w/.w  c/f .w/ D 2{

Z 

.z  c/f .z/ dz zw

for all w 2 D  range  . We obtain part (a) by evaluating the last equation at w D c. t u Remark 5.22. 1. A topologist would develop the concept of homology in much more detail using chains and cycles. However, for our purposes, the above definitions suffice. 2. To help the reader with some of the problems of the last chapter, we review a standard definition from algebraic topology: two cycles  D .1 ; : : : ; n / and ı D .ı1 ; : : : ; ım / with ranges contained in a domain D are homologous in D if the cycle with components .1 ; : : : ; n ; ı1 ; : : : ; ım / is homologous to zero in D, where ıi is the curve ıi traversed backward (see Definition 4.9). 3. It is also standard to define the next relation between curves, that we have not had any reason to use. Two non-closed paths 1 and 2 (with ranges contained in D) are homologous in D if they have the same initial point and the same end point and the cycle .1 2 / is homologous to zero in D; note that the only component of this cycle is the closed path 1 2 . 4. The notions of a cycle  D .1 ; 2 ; : : : ; n / and the sum 1 C 2 C    C n of its components, as in Definition 4.58, are different and should not be confused. 5. A domain D in C is simply connected if and only if I.; c/ D 0 for all cycles  in D and all c 2 C  D.

5.3 Jordan Curves

129

5.3 Jordan Curves We recall that the continuous closed path  W Œ0; 1 ! C is a simple closed path or a Jordan curve whenever .t1 / D .t2 / with 0  t1 < t2  1 implies t1 D 0 and t2 D 1. In this case, the range of  is a homeomorphic image of the unit circle S 1 . To see this, we define h.e2{t / D .t/ and note that h maps S 1 onto the range of  . Observe that h is well defined, continuous, and injective. Since the circle is compact, h is a homeomorphism. Theorem 5.23 (Jordan Curve Theorem2 ). If  is a simple closed path in C, then (a) C  range  has exactly two connected components, one of which is bounded. (b) Range  is the boundary of each of these components, and (c) I.; c/ D 0 for all c in the unbounded component of the complement of the range of  . I.; c/ D ˙1 for all c in the bounded component of the complement of the range of  . The choice of sign depends only on the choice of direction for traversal on  . Definition 5.24. For a simple closed path  in C we define the interior of  , i. /, to be the bounded component of C  range  and the exterior of  , e. /, to be the unbounded component of C  range  . If I.; c/ D C1 (respectively 1) for c in i. / then we say that  is a Jordan curve with positive (respectively negative) orientation. We shall not prove the above theorem. It is a deep result. In all of our applications, it will be obvious that our Jordan curves have the above properties. Remark 5.25. Another important (and nontrivial to prove) property of Jordan curves is the fact that the interior of a Jordan curve is always a simply connected domain in C. If we view the Jordan curve as lying on the Riemann sphere b C, then each component of the complement of its range is simply connected. This property allows us to prove the following result. Theorem 5.26 (Cauchy’s Theorem (Extended Version)). Let 0 ; : : : ; n be nC1 positively oriented Jordan curves. Assume that range j  e.k / \ i.0 /

2 For a proof see the appendix to Ch. IX of J. Dieudonn´e, Foundations of Modern Analysis, Pure and Applied Mathematics, vol. X, Academic Press, 1960 or Chap. 10 of J. R. Munkres, Topology (Second Edition), Dover, 2000.

130

5 The Cauchy Theory: Key Consequences γ0

δ2

γ1 γ2 D

δ1

Fig. 5.2 Jordan curves and the domain they define

for all 1  j ¤ k  n, see Fig. 5.2. If f is a holomorphic function on a neighborhood N of the closure of the domain D D i.0 / \ e.1 / \    \ e.n /; then

Z f .z/ dz D 0

n Z X

f .z/ dz:

kD1 k

Proof. Adjoin nonintersecting curves ıj in D from 0 to j for j D 1; : : : ; n, as in Fig. 5.2. Then the cycle ı D .0 ; ı1 1 ı1 ; : : : ; ın n ın / is homologous to zero in N . Thus Theorem 5.21 implies that Z C 0

n Z X

C ık

kD1

Z !

Z

f .z/ dz D 0;

C k

ık

and the result follows by noting that the integral over each ık is canceled by the corresponding integral over ık . u t An immediate consequence is Theorem 5.27 (Cauchy’s Integral Formula (Extended Version)). With the hypotheses as in the extended version of Cauchy’s Theorem 5.26, we have Z 0

for all c 2 D.

X f .z/ dz  zc n

2{f .c/ D

Z

kD1 k

f .z/ dz zc

5.4 The Mean Value Property

131

Proof. We can apply Theorem 5.2 to the function f , using the neighborhood N of Theorem 5.26 and the cycle ı constructed in its proof, since ı is homologous to zero in N and I.ı; c/ D C1: As before, the integral over each ık is canceled by the corresponding integral over ık . t u

5.4 The Mean Value Property The next concept applies in a broader context than that of holomorphic functions, as we will see in Chap. 9. Definition 5.28. Let f be a function defined on a domain D in C. We say that f has the mean value property (MVP) if for each c 2 D there exists r0 > 0 with U.c; r0 /  D and f .c/ D

1 2

Z

2

f .c C r e{ / d for all 0  r < r0 :

(5.7)

0

Remark 5.29. A holomorphic function f on a domain D has the MVP (with r0 D the distance of c 2 D to @D). Hence so do its real and imaginary parts. Theorem 5.30 (Maximum Modulus Principle). Suppose f is a continuous complex-valued function defined on a domain D in C that has the MVP. If jf j has a relative maximum at a point c 2 D, then f is constant in a neighborhood of c. Proof. The result is clear if f .c/ D 0. If f .c/ 6D 0, replacing f by e{ f for some  2 R, we may assume that f .c/ > 0. Write f D u C { v and choose r0 > 0 such that (1) clU.c; r0 /  D. (2) Equation (5.7) holds. (3) jf .z/j  f .c/ for z 2 cl U.c; r0 /. If we define M.r/ D supfjf .z/j I jz  cj D rg for 0  r  r0 ; then it follows from (3) that M.r/  f .c/ for 0  r  r0 : Since the MVP implies that 1 f .c/ D 2

Z

2

f .c C r ei / d for 0  r < r0 ; 0

132

5 The Cauchy Theory: Key Consequences

we also have f .c/  M.r/, and we conclude that f .c/ D M.r/ for 0  r  r0 . Now observe that 1 2

Z

2

M.r/ d D M.r/ D f .c/ D

0

1 2

Z

2

u.c C r e{ / d;

(5.8)

0

where the last equality holds because f .c/ is real. Also note that M.r/  u.c C r e{ /  0;

(5.9)

from the definition of M.r/. But from (5.8) we obtain that Z

2



 M.r/  u.c C re{ / d D 0;

0

and hence we must have equality in (5.9) for all . Finally, 1

1

M.r/  .u2 .c C r e{ / C v2 .c C r e{ // 2 D .M.r/2 C v2 .c C r e{ // 2 which implies that v.c C r e{ / D 0 for 0  r  r0 and 0    2. Therefore f .z/ D u.z/ D M.jzj/ D f .c/ for all jzj  r0 . t u It is now easy to deduce that if f is a nonconstant holomorphic function on a bounded domain that extends to a continuous function on the closure of the domain, then jf j assumes its maximum on the boundary of that domain. More is true, as seen in Corollary 5.31. Suppose D is a bounded domain and f 2 C0 .clD/ satisfies the MVP in D. If M D supfjf .z/j I z 2 @Dg; then (a) jf .z/j  M for all z 2 D. (b) If jf .c/j D M for some c 2 D, then f is constant in D. Proof. If M 0 D supfjf .z/j I z 2 clDg; then M  M 0 < C1: We know that there exists a c in clD such that jf .c/j D M 0 . If c 2 D, then f is constant in a neighborhood of c by the MMP. Let D 0 D fz 2 DI jf .z/j D M 0 g:

5.4 The Mean Value Property

133

The set D 0 is closed and open in D; hence, if nonempty, it is all of D. In this latter case f is constant on D, since it is locally constant and D is connected, and thus also constant on clD. Then M D M 0 , and (a) and (b) are trivially true. On the other hand, if D 0 D ;, then M D M 0 ; (a) follows and (b) is trivially true. t u In particular, since a function that is holomorphic in a domain satisfies the MVP, we have Corollary 5.32 (The Maximum Modulus Principle for Analytic Functions). If f is a nonconstant holomorphic function on a domain D, then jf j has no relative maximum in D. Further, if D is bounded and f is continuous on the boundary of D, then jf j assumes its maximum on the boundary of D. Remark 5.33. By studying the proof of Theorem 5.30, one can prove the maximum principle, an interesting result that may be stated as follows. Suppose f is a continuous real-valued function defined on a domain D in C that satisfies the MVP. If f has a relative maximum at a point c 2 D, then f is constant in a neighborhood of c. Similarly, the minimum principle asserts that a continuous real-valued function defined on a domain D in C that satisfies the MVP on D and has a relative minimum at a point c 2 D must be constant in a neighborhood of c (apply the maximum principle to the negative of the function). An important consequence of Corollary 5.31 is Theorem 5.34 (Schwarz’s Lemma). If f is a holomorphic function defined on U.0; 1/ satisfying jf .z/j < 1 for jzj < 1 and f .0/ D 0, then jf .z/j  jzj for jzj < 1 and jf 0 .0/j  1. Furthermore, if jf .c/j D jcj for some c with 0 < jcj < 1 or if jf 0 .0/j D 1, then there exists a  2 C with jj D 1 such that f .z/ D z for all jzj < 1; that is, f is a rotation around zero, with angle Arg./. Proof. Using the Taylor series expansion for f at 0, we can write f .z/ D

1 X nD1

an zn ;

this power series has radius of convergence   1, by Theorem 5.5. Then the function defined by 8 f .z/ ˆ ˆ ; for 0 < jzj < 1; < z g.z/ D ˆ ˆ : a1 D f 0 .0/; for z D 0; satisfies g.z/ D

1 X nD1

an zn1 for jzj < 1

134

5 The Cauchy Theory: Key Consequences

and is holomorphic on U.0; 1/. Now for any r with 0 < r < 1 and any z with jzj D r, we have ˇ ˇ ˇ f .z/ ˇ 1 ˇ I jg.z/j D ˇˇ z ˇ r by the MMP, the same inequality holds for all jzj < r. Hence jg.z/j  1 for all jzj < 1, or, equivalently, ˇ ˇ jf .z/j  jzj for all jzj < 1 and ˇf 0 .0/ˇ  1: If jg.c/j D 1 for some c with jcj < 1, then g is constant, again by the MMP.

t u

Remark 5.35. Schwarz’s lemma implies that a holomorphic function from the unit disc U.0; 1/ to itself fixing zero either decreases the distance to the origin for all points in the disc or maintains it for all points. We will return to a natural geometric generalization of this fact in Sect. 8.4.4. Remark 5.36. In Schwarz’s lemma, the strong hypothesis jf .z/j < 1 for all jzj < 1 can be replaced by the weaker hypothesis that jf .z/j  1 for all jzj < 1. Under this weaker hypothesis we conclude the stronger one, for otherwise there exists a c in U.0; 1/ with jf .c/j D 1, and by the MMP, the function f must then be constant— obviously impossible.

5.5 Appendix: Cauchy’s Integral Formula for Smooth Functions In Sect. 4.2, we proved a simple version of Green’s theorem (Theorem 4.20) and stated another version as Theorem 4.22. As a consequence of the second version, we now prove Theorem 5.37 (Cauchy’s Integral Formula for Smooth Functions). Let K be a compact set in C that is the closure of its interior, with piecewise smooth positively oriented boundary @K. If f is a C1 -function on a neighborhood of K and c is a point in the interior of K, then f .c/ D

1 2{

Z @K

f .z/ dz C zc

“ K

fz .z/ dzdz : zc

(5.10)

Proof. Choose  > 0 such that the closure of the ball U.c; / is contained in the interior of K, and let K D K  U.c; /. We apply Green’s theorem to the smooth f .z/ dz on K and obtain differential form zc “ Z f .z/ fz .z/ dz D  dzdz: (5.11) z  c z @K K  c

Exercises

135

But @K consists of two parts: @K and the clockwise oriented circle with center at c and radius . Hence Z @K

f .z/ dz D zc

Z @K

f .z/ dz  { zc

Z

2

f .c C e{ / d:

0

Letting  ! 0 in (5.11) yields (5.10).

t u

Remark 5.38. • For holomorphic functions (5.10) reduces to (5.1). • The last result shows, once again, that the Cauchy theory is a consequence of Green’s theorem for C1 -functions that satisfy CR.

Exercises 5.1. Let D be a domain in C. Prove that the following conditions are equivalent. (1) D is simply connected. (2) b C  D is connected. (3) For each holomorphic function f on D such that f .z/ 6D 0 for all z 2 D, there exists a g 2 H.D/ such that f D eg (g is a logarithm of f ). (4) For each holomorphic function f on D such that f .z/ 6D 0 for all z 2 D and for each positive integer n there exists an h 2 H.D/ such that f D hn . How unique are the functions g and h? 5.2. Let f be analytic in a simply connected domain D and let  be a closed pdp in D. Set ˇ D f ı  . Show that I.ˇ; c/ D 0 for all c 2 C; c … f .D/. 5.3. (a) Show that if a continuous closed curve  is homotopic to a point in a domain D, then the cycle . / is homologous to zero in D. (b) Is the converse true? Hint: Consider the curve  of Fig. 5.3, with D D C  fp; qg, and verify that I.; p/ D 0 D I.; q/. 5.4. Let f be a holomorphic function on jzj < 1 with jf .z/j < 1 for all jzj < 1. (1) Prove the invariant form of Schwarz’s lemma (also known as the Schwarz–Pick lemma): 1 jf 0 .z/j  for all jzj < 1: 2 1  jf .z/j 1  jzj2 Hint: Use properties of M¨obius transformation (developed subsequently) to reduce to the standard Schwarz lemma. (2) Find necessary and sufficient conditions forˇequality 

ˇ in the last equation. (3) If f 12 D 13 , find a sharp upper bound for ˇf 0 12 ˇ.

136

5 The Cauchy Theory: Key Consequences

Fig. 5.3 Homologous to zero but not homotopic to a point?

5.5. Let f be a holomorphic function on U.0; R/, R > 0. Assume there exist an M 2 R>0 such that jf .z/j  M for all z 2 U.0; R/ and an n 2 Z0 such that 0 D f .0/ D f 0 .0/ D    D f .n/ .0/: (1) Prove that

 jf .z/j  M

jzj R

nC1 for all z 2 U.0; R/;

ˇ .nC1/ ˇ ˇf .0/ ˇˇ M ˇ ˇ .n C 1/Š ˇ  RnC1 : ˇ ˇ M (2) Assume that either ˇf .nC1/ .0/ˇ D .n C 1/Š nC1 or R  nC1 jcj for some c with 0 < jcj < R. Prove that there exists an jf .c/j D M R  z nC1 for all z 2 U.0; R/. ˛ 2 C, j˛j D 1 such that f .z/ D ˛M R 5.6. Let f be a holomorphic function on the punctured plane C¤0 . Assume that there exist a positive constant C and a real constant M such that and

jf .z/j  C jzjM for 0 < jzj <

1 : 2

Show that z D 0 is either a pole or a removable singularity for f , and find sharp bounds for 0 .f / the order of f at 0. 5.7. Prove by use of Schwarz’s lemma that every one-to-one holomorphic mapping of a disc onto another disc is given by a fractional linear transformation. Here the term “disc”is also meant to include half planes (with 1 adjoined to the boundary), az C b , where and a fractional linear transformation is a map of the form T .z/ D cz C d a; b; c; d are complex numbers with ad  bc 6D 0.

Exercises

137

Formulate and prove a corresponding theorem for one-to-one surjective holomorphic self mappings of 1 < jzj < C1. 5.8. Let f be an entire function with jf .z/j  a jzjb C c for all z, where a, b, and c are positive constants. Prove that f is a polynomial of degree at most b. 5.9. Let f be an entire function such that f .0/ D 0 and that <.f .z// ! 0 as jzj ! 1. Show that f is identically 0. 5.10. Let D be a bounded domain in C. Let f W clD ! C be a nonconstant continuous function, which is analytic in D and satisfies jf .z/j D 1 for all z 2 @D. Show that f .z0 / D 0 for some z0 2 D. 5.11. Prove the maximum and the minimum principles stated in Remark 5.33. Furthermore, if D is a bounded domain in C and f is a continuous real-valued function on the closure of D and satisfies the MVP in D, with m  f  M on @D for some real constants m and M , show that then m  f  M on D. 5.12. Let f be a nonconstant holomorphic function on the disc U.0; R/, 0 < R  C1. For 0  r < R, let M.r/ D supfjf .z/j I jzj D rg: Show that M is a strictly increasing function on the open interval .0; R/.

Chapter 6

Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions

In this chapter we use the Cauchy theory to study functions that are holomorphic on an annulus, and analytic functions with isolated singularities. We describe a classification for isolated singularities. Functions that are holomorphic on an annulus have Laurent series expansions, an analogue of power series expansions for holomorphic functions on discs. Holomorphic functions with a finite number of isolated singularities in a domain can be integrated using the residue theorem, an analogue of the Cauchy integral formula. We discuss local properties of these functions. The study of zeros and poles of meromorphic functions leads to a theorem of Rouch´e that connects the number of zeros and poles to an integral. The theorem is not only aesthetically pleasing in its own right but also allows us to give alternate proofs of many important results. In the penultimate section of this chapter we illustrate the use of complex function theory in the evaluation of real definite integrals. The appendix discusses Cauchy principal values, a way to integrate functions with certain singularities.

6.1 Functions Holomorphic on An Annulus Theorem 6.1 (Laurent Series Expansion). Let c 2 C and let f be holomorphic in the annulus A D fz 2 CI 0  R1 < jz  cj < R2  C1g:1 Then f .z/ D

1 X

an .z  c/n for all z 2 A;

(6.1)

nD1

1

We are including here the cases of degenerate annuli: those with R1 D 0 and/or R2 D C1.

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 6, © Springer Science+Business Media New York 2013

139

140

6 Cauchy Theory: Local Behavior and Singularities

where the series converges uniformly and absolutely on compact subsets of A, and an D

1 2{

Z r

f .t/ dt .t  c/nC1

for R1 < r < R2 ;

with r ./ D c C re{ ; 0    2: This series is called a Laurent series for f . It is uniquely determined by f and A. Proof. Without loss of generality we assume c D 0. Consider two concentric circles rj D fzI jzj D rj g (j D 1; 2), bounding a smaller annulus R1 < r1 < jzj < r2 < R2 : If for j 2 f1; 2g, we let fj .z/ D

1 2{

Z rj

f .t/ dt; t z

then it follows from the extended version of Cauchy’s integral formula (Theorem 5.26) that 1 f .z/ D 2{

Z r2

1 f .t/ dt  t z 2{

Z r1

f .t/ dt D f2 .z/  f1 .z/: t z

Since f2 can be extended to be a holomorphic function in the disc fzI jzj < r2 g, by Exercise 4.11, we obtain Z 1 X f .t/ 1 an zn ; with an D dtI f2 .z/ D nC1 2{ r2 t nD0 furthermore, this power series converges for jzj < R2 , and an is independent of r2 , since any two circles centered at zero and contained in A are homotopic in A. As for f1 , note that 1   1 1 1 1X t k 1 D D D  .t  z/ .z  t/ z .1  tz / z z kD0

for jtj D r1 and jzj > r1 . Thus 1

f1 .z/ D

1 1X 2{ z kD0

Z

1

r1

X f .t/ k 1 t dt D zk1 k z 2{ kD0

Z f .t/t k dt: r1

6.1 Functions Holomorphic on An Annulus

141

Letting k  1 D n we obtain f1 .z/ D

X

an zn ; with an D

n1

1 2{

Z r1

f .t/ dt; t nC1

and an is independent of r1 , since, as before, any two circles contained in A and centered at zero are homotopic in A. Observe that f1 can be extended to be a holomorphic function in jzj > R1 , including the point 1, and f1 .1/ D 0. t u Corollary 6.2. If f is holomorphic in A, then f D f2 f1 where f2 is holomorphic in jz  cj < R2 and f1 is holomorphic in jz  cj > R1 (including the point at 1). The functions fj are unique if we require that f1 .1/ D 0. Proof. Once again, without loss of generality, we assume c D 0. Existence: Already done. Uniqueness: Suppose f D f2  f1 D g2  g1 with appropriate fi and gj . Then 0 D .f2  g2 /  .f1  g1 / in A, the function .f2  g2 / is holomorphic in jzj < R2 , and the function .f1  g1 / is holomorphic in jzj > R1 and vanishes at 1. Define  h.z/ D

.f1  g1 /.z/; if jzj > R1 ; .f2  g2 /.z/; if jzj < R2 :

The function h is well defined and holomorphic on C [ f1g, and vanishes at 1. Hence it is identically zero. t u Example 6.3. The rational function f .z/ D

1 z.1  z//.2  z/

is holomorphic on three annuli centered at zero. On each of these it has a different C1 X Laurent series expansion of the form an zn ; the uniqueness of expansion is nD1

valid only if we specify both the function and annulus. (1) On A1 D f0 < jzj < 1g: we do not need to evaluate the coefficients an by the integration process described in the proof of the last theorem. On this annulus, we have 1 D 1 C z C z2 C    C zn C    .valid for jzj < 1/; 1z and

  1 1 z z2 zn 1   D z D 2 1C 2 C 22 C    C 2n C    .valid for jzj < 2/: 2z 2 1 2

142

6 Cauchy Theory: Local Behavior and Singularities

Thus f .z/ D

   z2 z zn 1  1 C z C z2 C    C zn C    1 C C 2 C    C n C    2z 2 2 2

  1 1 3 7 1 D z C C z C    C 1  nC2 zn C    for 0 < jzj < 1: 2 4 8 2 There is another, perhaps simpler, method for computing the series. We use the partial fraction decomposition of the function f .z/ D

A B C C C I z 1z 2z

the constants are computed to be AD

1 1 ; B D 1; C D  ; 2 2

and we use the respective geometric series for the three fractions to obtain once again:   1 f .z/ D z1 C 1 C z C z2 C    C zn C    2   z z2 1 zn 1C C 2 CC n C  4 2 2 2  C1 X  1 1  nC2 zn for 0 < jzj < 1: D 2 nD1 (2) On A2 D f1 < jzj < 2g: we use again the partial fraction decomposition of f . We need to know in this case that   1 1 1 1 1 1  D D  1 C C 2 C    C n C    for jzj > 1 1 1z z z z z z 1 z and   1 z z2 1 zn 1  D  1 C C C    C C    for jzj < 2: D z 2z 2 2 22 2n 2 1 2 Therefore   1 f .z/ D z1  z1 C z2 C    C zn C    2

6.2 Isolated Singularities

143

  z z2 1 zn 1C C 2 CC n C  4 2 2 2 2 X

1

1 X 1 n  z  z for 1 < jzj < 2: D 2z nD0 2nC2 nD1 n

(3) On A3 D f2 < jzjg: in this case we use   1 1 1 1 1 1  D 1 C C 2 C    C n C    for jzj > 1 D  1 1z z z z z z 1 z and   1 1 2 22 1 2n  D 1 C C 2 C    C n C    for jzj > 2: D  2 2z z z z z z 1 z Then   1 f .z/ D z1  z1 C z2 C    C zn C    2  1  1 z C 2z2 C 22 z3 C    C 2n zn1 C    C 2 D

3 X 

 1 C 2n2 zn for jzj > 2:

nD1

6.2 Isolated Singularities In this section we study the behavior of a function holomorphic in a punctured disc. Definition 6.4. We consider the special case of functions holomorphic on a degenerate annulus with R1 D 0 and R2 2 .0; C1I that is, we fix a point c 2 C and a holomorphic function f on the punctured disc A D fz 2 CI 0 < jz  cj < R2 g. In this case c is called an isolated singularity of f . We know that then f has a Laurent series expansion (6.1). There are three possibilities for the coefficients fan gn2Z<0 ; we now analyze each possibility. 1. If an D 0 for all n in Z<0 , then f has a removable singularity at z D c, and f can be extended to be a holomorphic function in the disc jz  cj < R2 by defining f .c/ D a0 .

144

6 Cauchy Theory: Local Behavior and Singularities

Shortly (in Theorem 6.7), we will establish a useful criterion for proving that such an isolated singularity is removable. 2. Only finitely many, at least one, nonzero coefficients with negative indices appear in the Laurent series; that is, there exists N in Z>0 such that an D 0 for all n > N and aN ¤ 0. We can hence write 1 X

f .z/ D

an .z  c/n C

nDN

for 0 < jz  cj < R2 . In this case,

1 X

an .z  c/n

nD0 1 X

an .z  c/n is called the principal or

nDN

singular part of f at c. Observe that lim .z  c/N f .z/ D aN ¤ 0;

z!c

and N is characterized by this property (that the limit exists and is different from zero). Note that in this case the function z 7! .z  c/N f .z/ has a removable singularity at c, and hence can be extended to be holomorphic in the disc jz  cj < R2 . Therefore f is meromorphic in the disc jz  cj < R2 , and f has a pole of order N at z D c. 3. Infinitely many nonzero coefficients with negative indices appear in the Laurent series. Then c is called an essential singularity of f . We extend the definition of isolated singularity to include the point at 1. Definition 6.5. A function f holomorphic in a deleted neighborhood of 1 (see   1 has an isolated Definition 3.52) has an isolated singularity at 1 if g.z/ D f z singularity at z D 0. If f has a pole of order N at 1, the principal part of f at 1 is the polynomial N X an zn , where the Laurent series expansion for g near zero is given by nD1

  X N 1 X 1 n g.z/ D f D an z C an zn z nD1 nD0 for jzj > 0 small. Example 6.6. We have   X 1 n z 1 D exp z nŠ nD0

6.2 Isolated Singularities

145

for jzj > 0. Here R1 D 0 and R2 D C1; 0 is an essential singularity of the function and 1 is a removable singularity (f .1/ D 1). Theorem 6.7. Let c 2 C and f be a holomorphic function on A D f0 < jz  cj < R2 g. Assume that (6.1) is the Laurent series of f on the punctured disc A. If there exist an M > 0 and 0 < r0 < R2 such that jf .z/j  M for 0 < jz  cj < r0 ; then f has a removable singularity at z D c. Z f .t/ 1 Proof. We know that an D dt for all n 2 Z, where r ./ D 2{ r .t  c/nC1 M c C r e{ , for  2 Œ0; 2 and 0 < r < r0 . We estimate jan j  n . For n < 0, we r t u let r ! 0 and conclude that an D 0. Theorem 6.8 (Casorati–Weierstrass). If f is holomorphic on f0 < jz  cj < R2 g and has an essential singularity at z D c, then for all w 2 C the function g.z/ D

1 f .z/  w

is unbounded in any punctured neighborhood of z D c. Therefore the range of f restricted to any such neighborhood is dense in C. Proof. Assume that, for some w 2 C, the function g is unbounded in some punctured neighborhood N of z D c. Then there is " > 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

jzj<1

The denominator of the integrand in the last integral is a quadratic polynomial in z with precisely one root inside the unit circle (the product of the roots is C1). We thus have one residue to evaluate and conclude that 1

I D 2.a2  b 2 / 2 : 3. The next type of integral to be discussed here is Z

1

I D

Q.x/e{x dx; 1

where Q is a rational function with (at least) a simple zero at infinity and, in general, with no singularities on R. We illustrate with a more complicated example, where Q.z/ D 1z has a simple pole at the origin. Here the ordinary integral is replaced by its principal value (pr. v.) defined below: Z R2 Z ı ! { x Z 1 {x e e dx D lim dx: C pr. v. x ı!0C ı 1 x R1 R1 !C1 R2 !C1

We must choose a nice contour for integration; start with large X1 ; X2 , and Y; and small ı, all four positive. Our closed path  has several segments: 1 2 3 4 5 6

W W W W W W

from X1 to ı on R the semicircle in the lower half plane of radius ı and center 0 from ı to X2 on R at x D X2 go up to height Y at height Y travel from X2 back to X1 at x D X1 go down from height Y to the real axis.

6.6 Evaluation of Definite Integrals

Z

We start with



where f .z/ D

159

e{ z dz D 2{ Res.f .z/ d z 0/; z

{z

e 1 D C g.z/, with g entire. Thus z z Z {z e dz D 2 {:  z

We now estimate the integral over 4 : ˇZ ˇ ˇ ˇ

Y 0

ˇ Z Y Z Y ˇ 1 e{ .X2 C{ y/ 1 1 y ˇ { dy ˇ  dy  e ey dy < : X2 C { y C { yj X X jX2 2 0 2 0

Next we estimate the integral over 5 : ˇZ ˇ ˇ ˇ

ˇ Z X2 ˇ e{ .xC{ Y / eY dx ˇˇ  dx xC{Y X1 jx C { Y j Z X2 eY 1 Y dx D ŒX2 C X1 : e Y X1 Y

X1 X2

Also the integral over 6 : ˇZ ˇ ˇ ˇ

0 Y

ˇ ˇ 1 e{ .X1 C{ y/ { dy ˇˇ < : X1 C { y X1

We conclude that Z 

e{ z dz D z

Z lim

ı!0C 1 [2 [3 X1 !C1 X2 !C1

e{ z dz: z

Finally, Z

X2

lim

ı!0C

ı

Z

ı

!

C X1

But

e{ x dx D lim x ı!0C

Z lim

ı!0C

2

1

.z

Z 1 [2 [3

e{ z dz C z

Z C g.z// dz D lim

ı!0C

2

Z 2 

 e{ z dz : z

z1 dz

because g is bounded on a neighborhood of 0 and the length of the path of integration goes to zero (Fig. 6.2). Now

160

6 Cauchy Theory: Local Behavior and Singularities γ5

−X1+ıY

X2+ıY

γ6 γ4

−X1 γ1

γ3

γ2

X2

Fig. 6.2 The path of integration for third example

Z lim

ı!0C

2

Z

z1 dz D lim

ı!0C

and we conclude that

Z

1

pr: v: 1

0



1 ıe{  { d D {; ıe{ 

e{ x dx D {: x

Using the fact that e{ x D cos x C { sin x, we see that we have evaluated two real integrals: Z

1

pr: v: 1

cos x dx D 0 and x

Z

1 0

 sin x dx D : x 2

4. We end with an example that illustrates how multi-valued functions can be used to evaluate definite integrals. We evaluate Z

1 0

x˛ dx 1 C x2

with 0 < ˛ < 1. The method illustrated here will work for integrating on the positive real axis functions of the form x ˛ R.x/; where 0 < ˛ < 1 and R is a rational function without singularities on R0 and vanishing at infinity to sufficiently high order. In our example we will use the multi-valued function F .z/ D

z˛ 1 C z2

to integrate over an appropriate contour where we can use a single-valued branch. We choose the curve  as follows: start with positive constants  (small), 0 (small), and R (large). These constants will be restricted in the arguments that

6.6 Evaluation of Definite Integrals

161

γ2 γ1

ρeıθ0 γ4

0

γ3

1

Re−ıθ0

Fig. 6.3 The path of integration for fourth example

follow. The oriented curve  D 1  2  3  4 consists of four components traversed successively and will involve two circles (jzj D  and jzj D R) and two straight line segments (arg z D ˙0 ) (See Fig. 6.3). • The curve 1 consists of the straight line from e{0 to Re{0 . It hence makes sense to restrict 0 <  < 1 < R. • The curve 2 consists of the following portion of the positively oriented circle with center at 0 and radius R: fRe{ I 0    .2  0 g; since we want ˙{ to be in the interior of the  range of  , we require 0 < 0 < . 2 • The curve 3 consists of the straight line segment from Re{0 to e{0 . • Finally, the curve 4 consists of the following portion of the negatively oriented circle with center at 0 and radius : fe{.2 / I 0    2  0 g. We begin evaluating and estimating various integrals. We use the branch of z˛ whose arguments lie in .0; 2 ˛/; thus a holomorphic function on C  R0 , with a jump discontinuity at each point on R>0 . 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 R<0 . Write f .z/ D

 C g.z/ for jzj < ı; z

with M0 a bound for jgj in U0;ı . Hence Z ˛f ı;k

Z

   { C g.ıe / d.ıe{ / ıe{  Z ˛ı;k Z ˛ı;k d C {ı e{ g.ıe{ /d: D{

!D

˛k





The first of the integrals in the last line is easily evaluated to yield ˛ı;k ; the absolute value of the second is bounded by M0 j˛ı;k j < 2M0 . From this the lemma follows. t u Theorem 6.34 (Residue Theorem, Version 2). Let  be a closed positively oriented Jordan curve in a domain D C with i. / D. Let f be a holomorphic function in the domain D except for isolated singularities at z1 ; : : : ; zn in i. / and simple poles w1 ; : : : ; wm on range . /. Then

Exercises

165

Z f .z/ dz D 2{

pr: v: 

n X

Res.f .z/ dz; zj / C {

j D1

m X

˛k Res.f .z/ dz; wk /;

kD1

where the ˛k are defined by Remark 6.32. Proof. As in previous arguments, ı is a small positive number, but in this case at most the minimum of the distances from any of the wk to the zj . We will use much of the notation previously introduced in this section. By introducing new line segments and thus integrating over a finite number of paths, rather than just a single path, it suffices to assume that m D 1 and n D 0. Thus we may take  to be a path in the closure of i. / from aı;1 to w1 (where f .z/ dz has a simple pole) followed by a second one from w1 to bı;1 and a third from bı;1 back to aı;1 . As before by translating and rotating the picture we may assume that w1 D 0 and aı;1 2 R<0 . Note that Z f .z/ dz D 0: ı

From the lemma (equation (6.4)) Z f .z/ dz D {˛1 Res.f .z/ dz; w1 /;

pr: v: 

and this suffices to establish the theorem.

t u

Exercises 6.1. Use Rouch´e’s theorem to prove the fundamental theorem of algebra. 6.2. Let g be a holomorphic function on jzj < R, R > 1, with jg.z/j  1 for all jzj < R. 1. Show that for all t 2 C with jtj < 1, the equation z D tg.z/ has a unique solution z D s.t/ in the disc jzj < 1. 2. Show that t 7! s.t/ is a holomorphic function on the disc jtj < 1 with s.0/ D 0. 3. When is s a constant function? 4. When is s injective? 6.3. Verify (6.3) (in the proof of the argument principle) using Laurent series expansions for f and F . 6.4. If f is a holomorphic function on 0 < jzj < 1 and f does not assume any value w with jw  1j < 2, what can you conclude?

166

6 Cauchy Theory: Local Behavior and Singularities

6.5. Evaluate the following integrals: Z  d (a) p 5 C cos  0 Z z6 dz (b) 7 jzjD1 7z  1 Z (c) z cot zdz jz100jD 199 2 

Z (d)

jz 2 jD3:15

z tan z dz

6.6. Compute: Z 1 dx (a) 1 C x6 1 Z 1 .1 C x/ (b) dx 4 1 1 C x Z 1 x dx (c) 4 C x4 1 Z 1 dx (d) 1 C x3 0 Z 1 ˛ x (e) dx, where 0 < ˛ < 1 1 C x3 0 6.7. Let f be an entire function such that jf .z/j D 1 for jzj D 1. What are the possible values for f .0/ and for f .17/? 6.8. Find all functions f that are meromorphic in a neighborhood of fjzj  1g and satisfy jf .z/j D 1 for jzj D 1, f has a double pole at z D 12 , a triple zero at z D  13 , and no other zeros or poles in fjzj < 1g.   6.9. Suppose f is an entire function satisfying f .n/ D n5 and f  n2 D n7 for all n 2 Z>0 . 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 //

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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 <

  C D : 2 2 t u

Corollary 7.12. The topology of the metric space .C.D/; / is independent of the choice of exhaustion fKn gn2Z>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 / <

  ı , we have Ki ; 2

ı : 2

Hence .g; f / D d.g  f / < Mi .g  f / C 2i < ıI that is, g 2 Nf .ı/.

t u

Remark 7.18. The theorem proves that fn ! f in the -metric if and only if for all compact K  D and all  > 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

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 <

 I 2N0 r0n

this last finite set of inequalities can be satisfied because lim jan;k  an;m j D 0 for each n;

k;m!1

(

since for each n

.n/

an;k

f .c/ D k nŠ

) k

is a Cauchy sequence of complex numbers, because we are assuming that .n/ lim fk .c/ exists for each n. t u k!1

Theorem 7.30 (Compactness Theorem). Let D be a domain in C. Then every closed subset A of H.D/ that is bounded in the strong sense is compact. Proof. Cover D by countably many open discs fU.zi ; ri /gi 2Z>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

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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 /:

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 <

1 : j

Now the rational function R1;j C Pi;j has poles only in S . We claim that f D lim .R1;j C Pi;j /: j !1

To verify this claim, let  > 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 <

1  C < : 2 j

Hence f 2 B.K; S /. • Thus it is sufficient to prove the lemma for S  C; for this we will rely on Lemma 7.39. • Let U D C  K and let V D fw 2 C  KI z 7! .z  w/1 2 B.K; S /g: Then S  V  U . We want to show that U D V . We show first that if a 2 V and jb  aj < d.a; K/; then b 2 V:

(7.8)

Assume a 2 V and jb  aj < d.a; K/. Then there is a real number r, 0 < r < 1, such that jb  aj < r jz  aj for all z 2 K. Note that 1

.z  b/

D .z  a/

1

  b  a 1 I 1 za

(7.9)

jb  aj < r < 1 for all z 2 K, we use the Weierstrass M -test to conclude jz  aj that the series (in the variable z)

since

   1  b  a 1 X b  a n 1 D za za nD0

(7.10)

converges uniformly on K. Lemma 7.38 and equation (7.9) imply that (7.8) holds; thus V is an open subset of C. We show next that @V \ U D ;. Indeed, if b 2 @V , let fan g be a sequence of elements of V converging to b. Since b … V , it follows that jb  an j  d.an ; K/, and letting n ! 1 we obtain 0 D d.b; K/; that is, b 2 K, and therefore b … U . We now apply Lemma 7.39. If H is any connected component of U D CK, then by the definition of S there exists s 2 H \ S . But then s 2 H \ V ¤ ; and the lemma implies H  V . Therefore every connected component of U lies in V ; consequently U  V , and thus U D V . t u As pointed out earlier, we have completed the proof of Runge’s theorem. An important special case is the following

196

7 Sequences and Series of Holomorphic Functions

Corollary 7.43. If D is any simply connected domain in the plane and f is a holomorphic function in D, then f can be approximated uniformly on compact subsets of D by polynomials.

Exercises 7.1. (a) Show that Theorem 7.2 has no analogue for real variables in that the absolute value function on R, which has no derivative at 0, can be uniformly approximated by differentiable functions. (b) Use (a) to construct a sequence of continuous functions on a domain D  C, converging at every point of D and such that the limit function is not continuous. 7.2. Construct an example of a sequence of real differentiable functions converging uniformly to a real differentiable function on a closed interval such that the sequence given by the derivatives does not converge uniformly there. (Hint: The sequence fn .x/ D x n does not converge uniformly on Œ0; 1.) 7.3. Show that both possibilities in Theorem 7.5 do occur. 7.4. Show that the series

7.5. Prove that

1 X nD1

1 X 1 2 . D 2 n 6 nD1

1 does not converge. zn

7.6. Prove Lemma 7.27. 7.7. This exercise requires some familiarity with standard topics in functional analysis. (1) Prove Lemma 7.38; that is, that B.K; S / is an algebra closed under uniform limits on K. (2) Introduce a norm on B.K; S / so that it becomes a Banach algebra. (3) There are several other (more function theoretic) proofs of Runge’s theorem that rely on tools not presented in this book (for example, the Hahn–Banach theorem and/or the Riesz representation theorem). After consulting the literature, outline an alternate proof.

Exercises

197

7.8. Show that the family S D ff 2 H.D/I f is injective; f .0/ D 0; f 0 .0/ D 1g is closed in H.D/. The members of S are usually called “schlicht functions.”It is true that S is compact, but proving it is strongly bounded requires Koebe’s distortion theorem, beyond the scope of this book.

Chapter 8

Conformal Equivalence and Hyperbolic Geometry

In this chapter, we study conformal maps between domains in the extended complex plane b C; these are one-to-one meromorphic functions. Our goal here is to characterize all simply connected domains in the extended complex plane. The first two sections of this chapter study the action of a quotient of the group of two-by-two nonsingular complex matrices on the extended complex plane, namely, the group PSL.2; C/ and the projective special linear group. This group is also known as the M¨obius group. In the third section we characterize simply connected proper domains in the complex plane by establishing the Riemann mapping theorem (RMT). This extraordinary theorem tells us that there are conformal maps between any two such domains. The study of the M¨obius group is intimately connected with hyperbolic geometry, a subject that has increasingly become an essential part of complex variable theory. In the next to last section of this chapter we study this geometry: we define the non-Euclidean metric, also known as the hyperbolic or Poincar´e metric, study the disc and half-plane models for the hyperbolic plane, find their geodesics, and show that their sense-preserving isometries are subgroups of the M¨obius group. We end this section by using Schwarz’s lemma to establish the deep connection between complex variables and geometry given in Theorem 8.41, which states that a holomorphic self-map of a proper simply connected domain in the plane is either an isometry or a contraction in the hyperbolic metric. As a further application based on M¨obius transformations, the last section is devoted to a study of certain bounded analytic functions on the unit disc known as finite Blaschke products. Definition 8.1. An injective meromorphic function is called a conformal map.1 A map f is anti-conformal if its conjugate fN is conformal.

In geometry, C1 -maps are called conformal if they preserve angles. We have seen in Proposition 6.27 that in the orientation-preserving case these are precisely the holomorphic functions with nowhere vanishing derivatives. Thus the two definitions agree locally for sense-preserving transformations. In our definition we also require global injectivity. 1

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 8, © Springer Science+Business Media New York 2013

199

200

8 Conformal Equivalence and Hyperbolic Geometry

Our definition of conformality is the correct notion of isomorphism in the category of meromorphic mappings, since the inverse of a conformal map is also conformal. Thus the concept introduces a natural equivalence relation on the family of domains on the sphere, called conformal equivalence. Definition 8.2. Let D be a domain in b C. Aut D is defined as the group (under composition) of conformal automorphisms (or conformal bijections) of D; that is, it consists of the conformal maps from D onto itself. There are two naturally related problems: Problem I. Describe Aut D for a given D. Problem II. Given two domains D and D 0 , determine when they are conformally equivalent. We solve Problem Ifor D D b C, D D C, and D D D (the unit disc fz 2 CI jzj < C. 1g in b C), and Problem IIfor D and D 0 any pair of simply connected domains in b

8.1 Fractional Linear (M¨obius) Transformations We describe the (orientation preserving) M¨obius group, and show that for the domains D D b C, C, a disc or a half plane, the group Aut D is a subgroup of this group. Definition 8.3. A fractional linear transformation (or M¨obius transformation) is a meromorphic function A W b C!b C of the form z 7! A.z/ D

az C b ; cz C d

(8.1)

where a; b; c; and d are complex numbers such that ad  bc ¤ 0. Specifically, 8 az C b ˆ ˆ ˆ ˆ cz C d ˆ ˆ ˆ ˆ ˆ ˆ a ˆ ˆ < c A.z/ D ˆ ˆ1 ˆ ˆ ˆ ˆa b ˆ ˆ zC ˆ ˆ ˆ d d ˆ : 1

d if c ¤ 0; z ¤ 1 and z ¤  ; c if c ¤ 0 and z D 1; d if c ¤ 0 and z D  ; c if c D 0 and z 6D 1;

(8.2)

if c D 0 and z D 1:

From now on, the abbreviated notation (8.1) will be interpreted as the expanded version (8.2). Without loss of generality we assume subsequently that ad  bc D 1

8.1 Fractional Linear (M¨obius) Transformations

201

(the reader should prove that there is really no loss of generality in this assumption; that is, establish Exercise 8.1). Also, whenever convenient we will multiply each of the four constants a, b, c, and d by 1, since this does not alter the M¨obius transformation’s action on b C nor the condition ad  bc D 1. Remark 8.4. A M¨obius transformation is an element of the group Aut.b C/, and the set of all M¨obius transformations is a group under composition, the M¨obius group. We will soon see that these two groups coincide. Remark 8.5. Other related groups are the matrix group  SL.2; C/ D

  ab I a; b; c; d 2 C; ad  bc D 1 ; cd

the corresponding quotient group PSL.2; C/ D SL.2; C/=f˙I g; 

 10 where I D is the identity matrix, and the extended M¨obius group of 01 orientation preserving and reversing transformations, consisting of the maps z 7!

az C b az C b and z 7! ; with ad  bc D 1: cz C d cz C d

Here orientation reversing means that angles are preserved in magnitude but reversed in sense (as the map z ! zN does). It is clear that 1 ! f˙I g ! SL.2; C/ ! Aut.b C/

(8.3)

is an exact sequence,  where the first two arrows denote inclusion, and by the last ab arrow, a matrix in SL.2; C/ is sent to the element of Aut.b C/ given by (8.1). cd An exact sequence is, of course, one where for any pair of consecutive maps in the sequence, the kernel of the second map coincides with the image of the first one. It is also clear that the image of the last arrow in the sequence (8.3) is precisely the M¨obius group, and, therefore, that it is isomorphic to PSL.2; C/, the quotient of SL.2; C/ by ˙I as defined above. It is natural to ask whether the last arrow is surjective; that is, whether the M¨obius group coincides with Aut.b C/. We will see that this is the case in Theorem 8.17. Let A be an element of PSL.2; C/. The square of the trace of a preimage of A in SL.2; C/ is the same for both of the two preimages of A. Thus even though the trace of an element in the M¨obius group is not well defined, the trace squared of an element in PSL.2; C/ is. Thus it makes sense to have the following:

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8 Conformal Equivalence and Hyperbolic Geometry

Definition 8.6. For A in the M¨obius group, given by (8.1) with ad  bc D 1, we define tr2 A D .a C d /2 .

8.1.1 Fixed Points of M¨obius Transformations Let A be any element of the M¨obius group different from the identity map. We are interested in the fixed points of A in b C; that is, those z 2 b C with A.z/ D z. az C b with ad  bc D 1, then for a fixed point z of A we have either If A.z/ D cz C d z D 1, or z 2 C and cz2 C .d  a/z  b D 0. We consider two cases: Case 1: c D 0. In this case 1 is a fixed point of A and ad D 1. If d D a then b A.z/ D z C with ab ¤ 0 (b 6D 0 because A is not the identity map), and A has no a a b other fixed point. If d ¤ a, then A.z/ D z C , and A has one more fixed point, d d b at in C. d a We note that in this case A has precisely one fixed point if and only if tr2 A D 4. Case 2: c ¤ 0. In this case 1 is not fixed by A, and the fixed points of A are given by p p .a  d / ˙ tr2 A  4 a  d ˙ .a  d /2 C 4bc D  2c 2c We have thus proved. Proposition 8.7. If A is a M¨obius transformation different from the identity map, then A has either one or two fixed points in b C. It has exactly one if and only if tr2 A D 4.

8.1.2 Cross Ratios Proposition 8.8. Given three distinct points z2 ; z3 ; z4 in b C, there exists a unique M¨obius transformation S with S.z2 / D 1, S.z3 / D 0, and S.z4 / D 1. Proof. The proof has two parts. Uniqueness: If S1 and S2 are M¨obius transformations that solve our problem, then S1 ı S21 is a M¨obius transformation that fixes 1, 0 and 1 and hence, by Proposition 8.7, it is the identity map. Existence: If the zi are complex numbers, then S.z/ D

z  z3 z2  z4 z  z4 z2  z3

8.1 Fractional Linear (M¨obius) Transformations

203

is the required map. If one of the zi equals 1, use a limiting procedure to obtain 8 z  z3 ˆ ; if z2 D 1; ˆ ˆ z  z4 ˆ ˆ ˆ ˆ ˆ < z2  z4 ; if z3 D 1; S.z/ D ˆ z  z4 ˆ ˆ ˆ ˆ ˆ ˆ z  z3 ˆ : ; if z4 D 1; z2  z3 t u

respectively.

Corollary 8.9. If fzi g and fwi g (i D 2; 3; 4) are two triples of distinct points in b C, then there exists a unique M¨obius transformation S with S.zi / D wi ; thus the M¨obius group is uniquely triply transitive on b C. Definition 8.10. The cross ratio .z1 ; z2 ; z3 ; z4 / of four distinct points in b C is the image of z1 under the M¨obius transformation taking z2 to 1, z3 to 0, and z4 to 1; that is, z1  z3 z2  z4 .z1 ; z2 ; z3 ; z4 / D z1  z4 z2  z3 if the four points are finite, with the corresponding limiting values if one of the zi equals 1. As we will see in the next proposition, it is useful to view the cross ratio as a M¨obius transformation (a function of z1 ) S D Sz2 ;z3 ;z4 that takes the four distinct ordered points z1 ; z2 ; z3 ; z4 to the four distinct ordered points w1 D S.z1 / D .z1 ; z2 ; z3 ; z4 /; w2 D 1; w3 D 0; and w4 D 1. It hence makes sense to allow one repetition among the four points zj and hence have S defined on b C and conclude that .z2 ; z2 ; z3 ; z4 / D 1, for example. This point of view will be used from now on when needed. Proposition 8.11. If z1 ; z2 ; z3 ; z4 are four distinct points in b C, and T is any M¨obius transformation, then .T .z1 /; T .z2 /; T .z3 /; T .z4 // D .z1 ; z2 ; z3 ; z4 /: Proof. If we define S.z/ D .z; z2 ; z3 ; z4 / for z 2 Cf0; 1g, then S ı T 1 is a M¨obius transformation taking T .z2 / to 1, T .z3 / to 0 and T .z4 / to 1. Therefore .T .z1 /; T .z2 /; T .z3 /; T .z4 // D S ı T 1 .T .z1 // D S.z1 / D .z1 ; z2 ; z3 ; z4 /: t u b is either an Euclidean (ordinary) circle in C, or Definition 8.12. A circle in C a straight line in C together with 1 (this is a circle passing through 1). See Exercise 3.21 for a justification for the name.

204

8 Conformal Equivalence and Hyperbolic Geometry

z1

a

b z2

z2 z1 z3

z4 z3

z4

Fig. 8.1 The cross ratio arguments. (a) On a circle. (b) Not on a circle

Proposition 8.13. The cross ratio of four distinct points in b C is a real number if and only if the four points lie on a circle in b C. Proof. This is an elementary geometric argument that goes as follows. It is clear that arg.z1 ; z2 ; z3 ; z4 / D arg

z1  z3 z2  z3  arg : z1  z4 z2  z4

It is also clear from the geometry of the situation (see Fig. 8.1 and Exercise 8.3) that the two quantities on the right-hand side differ by  n, with n 2 Z, if and only if the four points lie on a circle in b C. t u Theorem 8.14. A M¨obius transformation maps circles in b C to circles in b C. Proof. This follows immediately from Propositions 8.11 and 8.13.

t u

We use the following standard notation in the rest of this chapter: D denotes the unit disc fz 2 CI jzj < 1g and H2 the upper half plane fz 2 CI =z > 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

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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 <.z/ D <.w/, take C let L be the perpendicular bisector of the Euclidean line segment connecting z and e to be the circle with w. If c is the point where L intersects the real line, we take C e in H2 gives the center c passing through z and w. See Fig. 8.2. The portion C of C sought geodesic. t u Definition 8.28. Let z and w be two distinct points in H2 . The arc of the unique geodesic determined by z and w between them is the geodesic segment or geodesic path joining z and w. The next two lemmas compute the hyperbolic length of the geodesic segment between two points in H2 . Lemma 8.29. Let P and Q be two points in H2 lying on an Euclidean circle C centered on the real axis, and let  be the arc of C in H2 between P and Q. Assume further that the radii from the center of C to P and Q make respective angles ˛ and ˇ with the positive real axis. Then ˇ ˇ ˇ csc.ˇ/  cot.ˇ/ ˇˇ lH2 . / D ˇˇlog : csc.˛/  cot.˛/ ˇ Proof. Assume the circle C has radius r and is centered at c (see Fig. 8.3). Let z D .x; y/ be an arbitrary point on  and let t be the angle that the radius from z to the center of C makes with the positive real axis; then x D c C r cos t and

216

8 Conformal Equivalence and Hyperbolic Geometry

Fig. 8.3 Two points on a circle centered on R

ˇR ˇ ˇ ˇ ˇ y D r sin t. Therefore dx D r sin t and dy D r cos t. Thus lH2 . / D ˇ ˛ csc t dt ˇ and the result follows. t u Similarly one establishes Lemma 8.30. Let P D xP C { yP and Q D xP C { yQ be two points in H2 lying on a straight line C perpendicular to the real axis, and let  be the segment of C in H2 between P and Q. Then ˇ ˇ ˇ yP ˇˇ ˇ : lH2 . / D ˇlog y ˇ Q

The next definition and the following two lemmas allow us to prove that the hyperbolic length of a geodesic segment minimizes the hyperbolic lengths of all pdp’s joining two distinct points in H2 ; they will also provide an explicit formula for the hyperbolic distance in H2 . Definition 8.31. We have shown in Lemma 8.27 that any two distinct points z and e centered on the real axis, and on a unique geodesic w in H2 lie on a unique circle C e centered on the real C . If =w we let z D 1 and w D <.w/. See Fig. 8.2. Lemma 8.32. Let z and w distinct points in H2 . There exists a unique T 2 Aut.H2 / such that T .z / D 0, T .z/ D {, T .w/ D { y with y > 1, and T .w / D 1, where z and w are as in Definition 8.31. e centered on the real axis and passing through z Proof. Consider the unique circle C and w. Since the M¨obius group is triply transitive, there exists a unique M¨obius transformation T that maps z ; z; w to 0; {; 1, respectively. Since M¨obius e onto the imaginary axis union transformations map circles to circles, T maps C f1g, and hence T .w/ D { y for some real y. Since M¨obius transformations preserve orthogonality of curves, T maps R [ f1g onto itself. Since T maps z 2 H2 to { 2 H2 , T is in Aut.H2 /, and because it is orientation preserving, y > 1. t u

8.4 Hyperbolic Geometry

217

Lemma 8.33. If z and w are two distinct points in H2 , then the hyperbolic length of the geodesic segment e  joining them is shorter than the hyperbolic length of any other pdp  in H2 joining them. Proof. Write z D xz C {yz and w D xw C {yw . First consider the case xz D xw . Assume the curve  is parameterized by the closed interval Œa; b  R and .t/ D x.t/ C {y.t/. Then x and y are differentiable functions except at finitely many points, and ˇ ˇZ p ˇ ˇ ˇ b x 0 .t/2 C y 0 .t/2 ˇ Z b jy 0 .t/j ˇ yw ˇ ˇ ˇ dt ˇ  dt  ˇˇlog ˇˇ D lH2 .e  /; lH2 . / D ˇ ˇ ˇ a y.t/ y.t/ yz a where the last equality follows from Lemma 8.30. Furthermore, equality of lengths is attained if and only if x is constant and y 0 does not change sign where it exists; that is, if and only if  is a reparametrization of e . For the case xz 6D xw , by Lemma 8.32 we can find a T in Aut.H2 / such that T .z / D 0 and T .w / D 1. Furthermore, the image under T of the geodesic segment between z and w is the segment on the imaginary axis between T .z/ and T .w/, and both segments have the same hyperbolic length. Similarly, T ı  is a pdp in H2 joining T .z/ and T .w/, with the same hyperbolic length as  , and we are reduced to the previous case. t u We have established Theorem 8.34. For any two distinct points z and w in H2 , the geodesic segment joining z to w is the unique curve that achieves the infimum defined by (8.9). Using cross ratios to simplify notation, a routine computation establishes Proposition 8.35. For any two distinct points z and w in H2 , the hyperbolic distance between z and w is equal to the length of the geodesic segment  joining z and w, and is given by H2 .z; w/ D lH2 . / D log j.z ; w ; w; z/j D log j.0; 1; { y; {/j D log y D log

jz  wj C jz  wj ; jz  wj  jz  wj

(8.10) (8.11)

where y is the real number (greater than one) given in Lemma 8.32. Proof. A fractional linear transformation preserves the cross ratio of any four points, and fractional linear transformations mapping H2 to itself are isometries for the hyperbolic metric. This yields all the equalities except for the last one, which is a calculation (see Exercise 8.15). t u Remark 8.36. It is useful to record the following simplified form for the last identity for { y and { v in { R>0 ˇ y ˇˇ ˇ H2 .{ y; { v/ D ˇlog ˇ : v

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8 Conformal Equivalence and Hyperbolic Geometry

As mentioned at the end of the previous subsection, the set PSL.2; R/ of conformal automorphisms of H2 acts as a group of hyperbolic isometries of H2 . This also follows from two facts: fractional linear transformations preserve the cross ratios and map circles to circles. We now proceed to establish the converse. Proposition 8.37. An orientation preserving isometry f of .H2 ; H2 / that fixes the imaginary axis pointwise is the identity map. Proof. Let z D x C { y and f D u C { v. For all positive real numbers t we have H2 .z; {t/ D H2 .f .z/; f .{t// D H2 .u.z/ C {v.z/; {t/: We calculate using (8.10) or Exercise 8.15 that this is equivalent to Œx 2 C .y  t/2 v.z/ D Œu.z/2 C .v.z/  t/2 yI hence also equivalent to .x 2 C y 2 /v.z/  .u2 .z/ C v2 .z//y D .y  v.z//t 2 for all positive t. Since the LHS of the last equation is independent of t, the RHS must vanish identically. Hence v.z/ D y, u.z/2 D x 2 , and f .z/ D ˙ x C { y. Because f is continuous, the same sign holds for all z; thus either f .z/ D z or f .z/ D z for all z; since f is orientation preserving, we conclude that f is the identity map. t u Theorem 8.38. The set of orientation-preserving isometries of H2 with respect to the hyperbolic metric is precisely the set of fractional linear transformations mapping H2 to itself; that is, PSL.2; R/. Proof. If g is such an isometry of H2 , it preserves geodesics. Thus there is a fractional linear transformation f that preserves H2 and such that f ı g leaves invariant the imaginary axis. Following this map by an isometry of the form z 7! kz 1 with k 2 R>0 and then (if necessary) by z 7! , we may assume that f ı g fixes { z and leaves invariant theˇ intervals  ˇ .{; 1/ and .0; {/ on the imaginary axis. Using the fact that H2 .{y; {v/ D ˇlog wy ˇ for y; v 2 R>0 , we see that f ı g is the identity on the imaginary axis and hence also on H2 , by the previous proposition. We conclude that g is a M¨obius transformation. t u

8.4.3 Unit Disc Model Statements about the hyperbolic metric on the upper half plane can be translated 2 jdzj to the unit disc model, where the length differential is ds D D .z/jdzj D . 1  jzj2

8.4 Hyperbolic Geometry

219

Most of these translations are routine, some are left as exercises, and other, do not require translation. We emphasize the following two results. Theorem 8.39. The set of orientation-preserving isometries of D consists of the fractional linear transformations mapping D to itself, that is, Aut.D/. Proposition 8.40. For all z and w 2 D, D .w; z/ D log

j1  wzj C jw  zj : j1  wzj  jw  zj

(8.12)

1 C jzj : 1  jzj

(8.13)

In particular, D .0; z/ D log

t u

Proof. See Exercise 8.16.

8.4.4 Contractions and the Schwarz’s Lemma A deep connection between function theory and geometry is established through Schwarz’s lemma. Recall that not every holomorphic self-map of D is a M¨obius transformation (for instance, z 7! z2 ), only conformal automorphisms are, and, as we have seen, these are isometries in the hyperbolic metric. However, the following result holds. Theorem 8.41. Holomorphic self-maps of the unit disc do not increase distances with respect to the hyperbolic metric; that is, for all holomorphic self-maps F of D and all z and w 2 D, D .F .z/; F .w//  D .z; w/ and D .F .z//jF 0 .z/j  D .z/: Furthermore, if for some distinct z and w 2 D D .F .z/; F .w// D D .z; w/; or for some z 2 D ˇ 0 ˇ ˇF .z/ˇ D

D .z/ ; D .F .z//

then F is a conformal self-map of D. Proof. The two inequalities certainly hold for constant maps F . So assume that F W D ! D is holomorphic and nonconstant. Assume first that F .0/ D 0. By Schwarz’s lemma (Theorem 5.34), ˇ ˇ jF .w/j  jwj for jwj < 1 and ˇF 0 .0/ˇ  1:

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8 Conformal Equivalence and Hyperbolic Geometry

These are the Euclidean analogues of the two inequalities in the theorem for the points z D 0 and arbitrary w 2 D. We intend to use equation (8.13) to translate these to the non-Euclidean setting. We start with 1 C jF .w/j D .1 C jF .w/j/.1 C jF .w/j C jF .w/j2 C    / 1  jF .w/j  .1 C jwj/.1 C jwj C jwj2 C    / D

1 C jwj 1  jwj

and then apply (8.13) to obtain D .0; F .w//  D .0; w/: Let c and d denote two different points in D; choose conformal automorphisms A and B of the unit disc such that B.0/ D c and A.F .c// D 0. Then A ı F ı B is a holomorphic self-map of the unit disc that fixes 0. Hence, by what was already established, D .A.F .B.0///; A.F .B.z////  D .0; z/ for all z in D. Since B and A are isometries, D .0; z/ D D .B.0/; B.z// and D .A.F .B.0///; A.F .B.z//// D D .F .B.0//; F .B.z/// for all z in D, and we conclude that D .F .B.0//; F .B.z///  D .B.0/; B.z// for all z in D. Taking z D B 1 .d /, we obtain the first required inequality D .F .c/; F .d //  D .c; d /:

(8.14)

If we multiply each side of the last equation by ˇ ˇ ˇ ˇ ˇ 1 ˇ ˇ F .c/  F .d / ˇˇ 1 ˇDˇ ˇ  ˇ ˇ c  d ˇ ˇ F .c/  F .d / cd and take the limit as c approaches d , we get the infinitesimal form of our required formula (the second inequality) ˇ ˇ D .F .d // ˇF 0 .d /ˇ  D .d /:

(8.15)

8.5 Finite Blaschke Products

221

We leave it to the reader to verify that equality in either (8.14) or (8.15) implies that F is conformal (thus also an isometry). t u Definition 8.42. Let M be a metric space with distance d . A self-map f of M is a contraction if d.f .x/; f .y// < d.x; y/ for all x and y in M with x 6D y. Theorem 8.41 can be restated as Theorem 8.43. A holomorphic self-map of the unit disc is either an isometry or a contraction with respect to the hyperbolic metric.

8.5 Finite Blaschke Products For a 2 D, set Ba .z/ D 

jaj z  a ; z 2 C; a 1  az

jaj D 1 if a D 0. a Of course, Ba is the Riemann map f for D normalized (as before) by sending a to 0 but with (the change that) arg f 0 .a/ D   arg a; it is an automorphism of D. Let A D fa0 ; a1 ; : : :g be a nonempty finite or countable sequence of complex numbers lying in the unit disc D. Define with the understanding that

BA D B D

Y

Bai :

i

Definition 8.44. The function BA is the (finite or infinite) Blaschke product associated to A. In the infinite case there are, of course, convergence issues (see Sect. 10.6). For the rest of this section we will study finite Blaschke products. It follows immediately from the definition that we have Proposition 8.45. If A D fa0 ; a1 ; : : : ; an g is a nonempty finite sequence of points in D, then 5 (a) B D BA is a meromorphic function on C [ f1g,  zeros precisely at the  with 1 , and n C 1 points fai g and poles at the n C 1 points ai

5

If ai appears times in our list A, then

ai .B/ D and 1 .B/ D  : ai

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8 Conformal Equivalence and Hyperbolic Geometry

(b) B is a self-map of the closed unit disc, that maps the open unit Q disc holomorphically onto itself, and the unit circle onto itself, with B.0/ D i jai j. Blaschke products transform beautifully under automorphisms of D, as shown next. Proposition 8.46. Let A D fa0 ; a1 ; : : : ; an g be a nonempty finite sequence of points in D, and let T be any element of Aut.D/. Then BA ı T D  BT 1 .A/ ; where  is a constant of absolute value 1, and T 1 .A/ D fT 1 .a0 /; T 1 .a1 /; : : :g: Proof. Since T belongs to Aut.D/, there exist complex numbers a and b, with jaj2  azCb for all z in D. It suffices to compute the action jbj2 D 1, such that T .z/ D bzCa zc , where c 2 D. A calculation shows that of T on the function z 7! 1  cz ac  b z T .z/  c bc Ca !: D 1  c T .z/ a  bc ac  b z 1 bc C a a  bc ˇ ˇ ˇ a  bc ˇ ˇ ˇ D 1, and recalling that The proof is completed by observing that ˇ ˇ a  bc aw  b T 1 .w/ D . t u bw C a Theorem 8.47. Let f be a holomorphic self-map of the open unit disc D, A D fa0 ; a1 ; : : : ; an g a nonempty finite collection of points in D and B D BA . Assume that f .ai / D 0 for each ai in A, with multiplicities; that is, ai .f /  ai .B/ for all i . Then (a) jf .z/j  jB.z/j for all z 2 D, and jf 0 .ai /j  jB 0 .ai /j for all i in f0; 1; : : : ; ng (b) If jf .a/j D jB.a/j for some a 2 D with a 6D ai for all i , then there is  2 C with jj D 1 such that f .z/ D  B.z/ for all z 2 D (c) If ai appears times in the sequence A and if 0 D f .ai / D f 0 .ai / D    D f . 1/ .ai /

(8.16)

8.5 Finite Blaschke Products

and

223

ˇ ˇ ˇ ˇ . / ˇf .ai /ˇ D ˇB . / .ai /ˇ ;

then there is  2 C with jj D 1 such that (8.16) holds. Proof. To prove (a), it suffices to consider the special case with a0 D 0. To verify this claim, let T be an automorphism of D that sends 0 to a0 ; note that if we have the first inequality in (a) for f ı T and BT 1 .A/ , then, as a result of the last proposition, for all z 2 D, ˇ ˇ jf .T .z//j  ˇBT 1 .A/ .z/ˇ D jB.T .z//j ; and since T .z/ is an arbitrary point in D, we thus also have the inequality for f and B. We now assume that a0 D 0 (thus f .0/ D 0) and let A1 denote the sequence obtained from A after removing all occurrences of 0, and let B 1 be the Blaschke product associated to A1 , if this sequence is nonempty. If A1 is empty, set B 1  1. f .z/ is certainly holomorphic on D and F .0/ D The function F .z/ D .B/1 1 z0 B .z/ 0. We claim that jF .z/j  1

ˇ ˇ for all z 2 D. Fix such a point z. The restrictions of ˇB 1 ˇ to circles of radius r, with 0  r  1, yield a family of functions that uniformly approach the constant ˇfunction ˇ 1 as r approaches 1. Hence, for all  > 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 <

1 2

for all n 2 Z>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.

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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
(3) (4) (5) (6)

Proof. Assume that f is holomorphic, let w D f .z/, and use the chain rule (see Exercise 2.10) to conclude:

9.1 Harmonic Functions and the Laplacian

231

.g ı f /z D gw fz C gw f z D gw fz and .g ı f /zz D gww fz fz C gww f z fz C gw fzz D 0: The argument in the anti-holomorphic case is similar.

t u

(7) If f 2 C2 .D/ and f is locally on D the real part of an analytic function, then f is harmonic on D. Example 9.3. log jzj is harmonic on D D C¤0 , since it belongs to C2 .D/, and it is locally the real part of log z, a multi-valued but holomorphic function in D. Proposition 9.4. If g is real-valued and harmonic in D, then it is locally the real part of an analytic function. The analytic function is unique up to an additive constant. Proof. Let D 0  D be a simply connected region. Since g is harmonic in D 0 , it follows from (2) above that 2gz dz is closed on D 0 , and hence an exact form on D 0 . Choose a holomorphic function f on D 0 with df D 2gz dz. Then df D 2gz dz and hence 1 d.f C f / D dgI 2 that is, g D
on D .

t u

Corollary 9.5. A real-valued harmonic function on a simply connected domain is the real part of a holomorphic function on the same domain. Corollary 9.6. A harmonic function is C1 . Corollary 9.7. Harmonic functions have the MVP, and hence they satisfy the maximum modulus principle. Real-valued harmonic functions also satisfy the maximum and minimum principles. Proof. See Definition 5.28 and the properties that follow thereof.

t u

Remark 9.8. The maximum (minimum) principle asserts that if f is real-valued and harmonic on a domain D and if f has a relative maximum (minimum) at a point c 2 D, then f is constant in a neighborhood of c. Furthermore, if D is bounded and if f is also continuous on the closure of D, with m  f  M on @D for some real constants m and M , then m  f  M on D.

232

9 Harmonic Functions

9.2 Integral Representation of Harmonic Functions We apply the Cauchy theory towards our present main goal, which is to solve a boundary value problem. Given a harmonic function defined in a disc, we derive an integral formula for it, known as the Poisson formula; a major tool in the solution of our boundary value problem. Proposition 9.9 (The Poisson Formula). If g is a harmonic function on the domain jzj <  for some  > 0, then, for each 0 < r < , g.z/ D

1 2

Z

2 0

r 2  jzj2 g.re{  /  ˇ ˇ d for jzj < r: ˇre{   zˇ2

(9.3)

Proof. It suffices to assume that g is real-valued. To establish this formula we can thus apply Proposition 9.4 and choose the holomorphic function f on this domain with
1 X

an zn ; with a0 D f .0/ D g.0/ 2 R:

(9.4)

nD0

Then   1 1 X n f .z/ C f .z/ D an z C an zn 2 2 nD0 1

g.z/ D

1

D a0 C

 1 X n  { n r an e C an e{ n : 2 nD1

(9.5)

Integration of (9.5) along the curve ./ D r e{  , for 0    2, yields a0 D

1 2

Z

2

g.r e{  / d: 0

Multiplying (9.5) by e{ n  for n 2 Z>0 and integrating along the same curve, we obtain Z 2 1 n 1 r an D g.re{  /  e{ n  dI 2 2 0 or, equivalently, an D

1 

Z

2 0

g.re{  / d ; for n  1: .re{  /n

Thus, for jwj < r, we obtain from (9.4)

9.2 Integral Representation of Harmonic Functions

1 f .w/ D 2

2

Z

2 0

But 1C2 and thus 1 f .w/ D 2

233

Z

3 X  w n 5 d : g.re{  /  41 C 2 re{  n1

X  w n re{  C w ; D re{  re{   w n1

2

g.re{  /  0

re{  C w d for jwj < r : re{   w

The last formula gives a representation of a holomorphic function f in terms of its real part g, when the function f is real at 0. Taking the real part of both sides and renaming the variable w to z we obtain equation (9.3), the Poisson formula. t u The function   { r 2  jzj2 re C z D < ˇ ˇ re{   z ˇre{   zˇ2

(Poisson kernel)

is known as the Poisson kernel. Note that setting z D 0 in formula (9.3) we obtain again the MVP for harmonic functions. The original derivation of formula (9.3) assumed that g was harmonic in the closed disc fjzj  rg. However, the result remains true for jzj < r under the weaker assumption that g is harmonic in the open disc fjwj < rg and continuous on its closure. In this case, fix t with 0 < t < 1 and look at the function of z given by g.tz/. It is harmonic on the closed disc fjwj  rg and hence, by the already proven formula (9.3), Z 2 2 1 r  jzj2 { g.tz/ D ˇ  g.t re / d: ˇ 2 0 ˇre{  zˇ2 Since the function g is uniformly continuous on the closed disc, we know that g.tz/ approaches g.z/ uniformly on the circle fjwj D rg as t approaches 1. Hence both sides of the last equation converge to the expected quantities. As a special case we apply the Poisson formula to the function g which is identically 1 and obtain Z

2 0

r 2  jzj2 ˇ ˇ d D 2 for all z 2 C with jzj < r: ˇre{   zˇ2

(9.6)

Definition 9.10. A harmonic conjugate of a real-valued harmonic function u is any real-valued function v such that u C { v is holomorphic.

234

9 Harmonic Functions

Harmonic conjugates always exist locally, and globally on simply connected domains. They are unique up to additive real constants. In fact, it is easy to see that they are given locally as follows. Proposition 9.11. If g is harmonic and real-valued in jzj <  for some  > 0, then the harmonic conjugate of g vanishing at the origin is given by 1 2{

Z

2

g.re{  /  0

re{  z  re{  z d; ˇ ˇ ˇre{   zˇ2

for jzj < r < :

The following result is interesting and useful. Theorem 9.12 (Harnack’s Inequalities). If g is a positive harmonic function on jzj < r that is continuous on jzj  r, then r C jzj r  jzj  g.0/  g.z/   g.0/ ; for all jzj < r: r C jzj r  jzj Proof. Our starting point is (9.3). We use elementary estimates for the Poisson kernel: r 2  jzj2 r  jzj r 2  jzj2 r 2  jzj2 r C jzj D :   D ˇ ˇ ˇre{   zˇ2 r C jzj .r C jzj/2 .r  jzj/2 r  jzj Multiplying these inequalities by the positive number g.w/ D g.r e{  / and then averaging the resulting function over the circle jwj D r, we obtain r  jzj 1  r C jzj 2

Z

2

g.re{ / d  0



1 2

Z

2 0

r 2  jzj2 g.re{ /  ˇ ˇ d ˇre{   zˇ2

r C jzj 1  r  jzj 2

Z

2

g.re{ / d: 0

The middle term in the above inequalities is g.z/ as a consequence of (9.3), while the extreme averages are equal to g.0/ by the MVP. t u Remark 9.13. Exercise 9.6 gives a remarkable consequence of Harnack’s inequalities that we use in establishing our next result. Theorem 9.14 (Harnack’s Convergence Theorem). Let D be a domain and let fuj g be a nondecreasing sequence of real-valued harmonic functions on D. Then (a) Either lim uj .z/ D C1 for all z 2 D j !1

(b) The function on D defined by U.z/ D lim uj .z/ is harmonic in D. j !1

9.3 The Dirichlet Problem

235

Proof. Since a nondecreasing sequence of real numbers converges if and only if it is bounded, the assumption that lim uj .z/ is not C1 for all z 2 D allows us to j !1

conclude that there exist z0 in D and a real number M such that uj .z0 / < M for all j . Then lim uj .z0 / exists, and it equals the value of the series j !1

u1 .z0 / C

1 X

ŒunC1 .z0 /  un .z0 / ;

nD1

which is therefore convergent. Let K denote a compact subset of D. By enlarging K if necessary, we may assume that z0 2 K. It follows from Harnack’s inequalities (see Exercise 9.6) that there exists a real constant c such that 0  unC1 .z/  un .z/  c ŒunC1 .z0 /  un .z0 / for all z in K and all n in N. It follows immediately that the series u1 .z/ C

1 X

ŒunC1 .z/  un .z/ converges

nD1

uniformly on K; that is, uj converges uniformly to a function U on compact subsets of D. It is now easy to show that U is harmonic in D. t u

9.3 The Dirichlet Problem Let D be a bounded region in C and let f 2 C.@D/. The Dirichlet problem is to find a continuous function u defined on the closure of D that agrees with f on the boundary of D and whose restriction to D is harmonic. We will consider, for the moment, only the special case where D is a disc; without loss of generality we may assume that the disc has radius one and center at zero. For a piecewise continuous function u on S 1 and z 2 C with jzj < 1, we define (compare with (9.3)) 1 P Œu.z/ D 2

Z

2



e{  C z u.e /  < {  e z {

0

 d

(9.7)

or, equivalently, 1 P Œu.z/ D 2

Z

2 0

1  jzj2 u.e{  /  ˇ ˇ d: ˇe{   zˇ2

(9.8)

236

9 Harmonic Functions

a

b

C1

C1 0

u C2

0

C2

u

Fig. 9.1 u1 and u2 . (a) The function u1 . (b) The function u2 .

The following properties of the operator P are easily established: 1. P Œu is a well-defined function on the open unit disc. Hence we view P as an operator that assigns the function P Œu, on the open unit disc to each piecewise continuous function u on the unit circle. 2. P Œu C v D P Œu C P Œv and P Œcu D c  P u for all piecewise continuous functions u and v on S 1 and every constant c (thus P is a linear operator). 3. If u is a real nonnegative piecewise continuous function on S 1 , then P Œu is a real-valued nonnegative function on the open unit disc. 4. P Œu is harmonic in the open unit disc. To establish this claim we may assume (by linearity of the operator P ) that u is real-valued. In this case, P Œu is obviously the real part of an analytic function on the disc. 5. For all constants c, P Œc D c; as follows from (9.6) (or directly because constant functions are harmonic). 6. Properties 5 and 3 imply that any bound on u yields the same bound on P u. For example, for real-valued function u satisfying m  u  M for some real constants m and M , we have m  P Œu  M . We now establish the solvability of the Dirichlet problem for discs. Theorem 9.15 (H. A. Schwarz). If u is a piecewise continuous function on the unit circle S 1 , then the function P Œu is harmonic on fjzj < 1g; furthermore, for 0 2 R, its limit as z approaches e{ 0 is u.e{ 0 / provided u is continuous at e{ 0 . In particular, the Dirichlet problem is solvable for discs. Proof. We only have to study the boundary values for P Œu. Let C1 and C2 be complementary arcs on the unit circle. Let u1 be the function which coincides with u on C1 and vanishes on C2 ; let u2 be the corresponding function for C2 (see Fig. 9.1). Clearly P Œu D P Œu1  C P Œu2 . The function P Œu1  can be regarded as an integral over the arc C1 ; hence it is harmonic on C  C1 . The expression

9.3 The Dirichlet Problem

237

 <

e{  C z e{   z



1  jzj2 Dˇ ˇ ˇe{   zˇ2

vanishes on jzj D 1 for z ¤ e{  . It follows that P Œu1  is zero on the one-dimensional interior of the arc C2 . By continuity P Œu1 .z/ approaches zero as z approaches a point in the interior of C2 . In proving that P Œu has limit u.e{ 0 / at e{ 0 , we may assume that u.e{ 0 / D 0 (if not replace u by u  u.e{ 0 /). Under this assumption, given an  > 0, we can find complementary arcs C1 and C2 on the unit circle such that e{ 0 is an interior point ˇ ˇ  ˇ ˇ  of C2 and ˇu.e{  /ˇ < for e{  2 C2 . This last condition implies that ˇu2 .e{  /ˇ < 2 2  for all e{  , and hence jP u2 .z/j < for all jzj < 1. 2 But we also have that u1 is continuous and vanishes at e{ 0 . Since P Œu1  is continuous at e{ 0 and agrees with u1 there, there exists a ı > 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
9.6 Subharmonic Functions Definition 9.23. Consider a domain D in C. A continuous real-valued function u on D is subharmonic if whenever G is a bounded subdomain of D such that @G  D and ' is a continuous real-valued function in G [ @G such that

246

9 Harmonic Functions

(i) ' is harmonic in G (ii) u.z/  '.z/ for all z in @G then u.z/  '.z/ for all z in G:

(9.14)

A function u is superharmonic if the function u is subharmonic. Remark 9.24. The analogous of subharmonic (superharmonic) functions in one real variable are convex (concave) functions. We leave it to the reader to translate results that hold for subharmonic functions to the analogous results for superharmonic functions. We discuss a number of key properties of the class of subharmonic functions: 1. A continuous real-valued function is harmonic if and only if it is both subharmonic and superharmonic. This local property is an immediate consequence of the definitions. 2. Harmonic functions are characterized by the mean value property (Theorem 9.20); that is, if u is continuous in D, then u is harmonic in D if and only if for all z0 2 D and all r0 > 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
264

9 Harmonic Functions

fz 2 CI jzj < 1g and fjS 1 D g if and only if Z g./  n d D 0 for n D 0; 1; 2; : : : : jjD1

9.14. Does there exist a function f holomorphic on jzj < 1 such that lim f .z/ D  C  1

z!

for all  with jj D 1‹

9.15. Let f 2 C2 .D/. Show that f is holomorphic or anti-holomorphic on D if and only if f and f 2 (the square of f ) are harmonic on D. 9.16. Assume h is harmonic in a domain D. Prove that so are hz , hzN , hx , and hy . The rest of this exercise concerns universal bounds for harmonic functions and assume that jh.z/j  M for all z 2 D. • Let D D U.z0 ; R/ for some z0 2 C and some R > 0. Show that there exists a universal constant c > 0 such that the absolute values of each of the four partial cM . derivatives evaluated at z D 0 are bounded by R • Construct examples to show that the above estimates do not hold for arbitrary z 2 U.z0 ; R/. • Let D be arbitrary again. Show that the absolute values of each of the four partial cM derivatives evaluated at z 2 D are bounded by , where R is the distance R  jzj from z to the boundary of D. 9.17. If h is harmonic and jhj is bounded for 0 < jz  z0 j < R, show that h extends to be harmonic in jz  z0 j < R. 9.18. Let u be a nonconstant real-valued harmonic function on a domain D. Show that the set of critical values of du D ux dx C uy dy (i.e., the set of points in D at which both ux and uy vanish) is discrete. 9.19. Show that if u is real valued and continuous in D, then u is superharmonic in D if and only if for all  2 D, and r0 > 0 sufficiently small so that cl U.; r0 /  D, u satisfies Z 2 1 u. C r e{ / d for all 0  r < r0 : u./  2 0 Conclude that superharmonic functions satisfy the minimum principle. 9.20. Show that if u is real valued and continuous in D, then u is subharmonic in D if and only if for all  2 D, and r0 > 0 sufficiently small so that cl U.; r0 /  D, u satisfies the Area Mean Value Inequality 1 u./  2{r 2

“ u.z/ dz dz: jzjr

Exercises

265

9.21. (a) Prove that there are no nonconstant subharmonic functions on C that are bounded from above. (b) Generalize the above result to the finitely punctured plane. 9.22. (a) Show that z 7!  log jzj with 0 < jzj < 1 is the Green’s function for the unit disc with singularity at the origin. (b) What is the Green’s function for the unit disc with a singularity at an arbitrary point c of the disc? 9.23. (a) Let D1 and D2 be domains with c 2 D1  D2 . Show that for all z 2 D1 , gD1 .z; c/ < gD2 .z; c/: (b) Let 0 < r < R and assume that D is a domain with fjzj < rg  D  fjzj < Rg: Show that for all 0 < jzj < r, log r  log jzj < gD .z; 0/ < log R  log jzj : 9.24. Let D be a proper simply connected subdomain of C and c 2 D. Let f W D ! D be the Riemann map with f .c/ D 0 and f 0 .c/ > 0. Show that z 7!  log jf .z/j defines the Green’s function for D with singularity at c. Is the normalization f 0 .c/ > 0 needed? 9.25. Show that the Green’s function is a conformal invariant in the following sense: If f W D1 ! D2 is a conformal map between plane domains, z 2 D1 , and g is the Green’s function on D2 with singularity at f .z/, then g ı f is the Green’s function on D1 with singularity at z. 9.26. Extend the discussion of the Green’s function to include domains in the Riemann sphere C [ f1g.

Chapter 10

Zeros of Holomorphic Functions

There are certain (classical families of) functions of a complex variable that mathematicians have studied frequently enough for them to acquire their own names. These are, of course, functions that arise naturally and repeatedly in various mathematical settings. Many of these functions are defined by infinite products. Examples of such named functions include Euler’s -function, the Riemann function, and the Euler ˆ-function. We will study only the first of these, in Sect. 10.4. There is a long history of synergy between the understanding of such functions and the development of complex analysis. Indeed, motivation for much of the theory and techniques of complex analysis was the desire to understand specific functions. In turn, the understanding of these functions has fed and continues to feed the development of the theory of complex variables. Holomorphic functions in general, and these classically studied functions in particular, are often understood by their zeros. In this chapter we develop techniques to study the zeros of holomorphic functions. We show that one can always construct a meromorphic function with prescribed zeros and poles. To do so, we begin by developing the theory for infinite products of holomorphic functions in the first section of this chapter. This theory is applied in Sects. 10.2 and 10.6 to study, respectively, meromorphic functions on an arbitrary proper domain in the complex plane with prescribed zeros and poles and bounded analytic functions on the unit disc with prescribed zeros. In Sect. 10.5 we introduce the notion of divisors on a plane domain and discuss some of their properties; they will prove useful in the following section. Recall that we discussed finite Blaschke products in an earlier chapter, as opposed to the infinite Blaschke products that we study here in Sect. 10.6. This in turn leads us to begin the study of bounded analytic functions on D. In Sect. 10.3 we prove Bers’s theorem that a plane domain is determined by its ring of holomorphic functions; the beginning of the interplay between the geometry of a plane domain and its ring of holomorphic functions.

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 10, © Springer Science+Business Media New York 2013

267

268

10 Zeros of Holomorphic Functions

10.1 Infinite Products We begin with some language needed to discuss infinite products. We then develop lemmas that lead to Theorem 10.5, which relates the uniform convergence of certain infinite sums to the uniform convergence (to a holomorphic function) of corresponding infinite products. Definition 10.1. Let un 2 C for each n 2 Z>0 and set pn D .1 C u1 /.1 C u2 /    .1 C un /: If lim pn exists and equals p, we write n!1

pD

1 Y

.1 C un /:

nD1

We call pn the partial product of the infinite product p. We say that the infinite 1 Y .1 C un / converges if fpn g does. product nD1

Lemma 10.2. Let fun g1 nD1  C and set pN D

N Y

.1 C un / and pN D

nD1

Then

N Y

.1 C jun j/:

nD1

pN  eju1 jCCjuN j

and

jpN  1j  pN  1:

Proof. We know that x > 0 implies that ex  1 C x. Therefore, 1 C jun j  ejun j so that pN  eju1 jCCjuN j . The second statement is proved by induction on N . For N D 1, jp1  1j D ju1 j D p1  1. For N  1, jpN C1  1j D jpN .1 C uN C1 /  1j D j.pN  1/.1 C uN C1 / C uN C1 j  jpN  1j  j1 C uN C1 j C juN C1 j and by induction this expression is  .pN  1/  .1 C juN C1 j/ C juN C1 j D pN C1  1:

t u

10.1 Infinite Products

269

Theorem 10.3. If fun g is a sequence of bounded functions on a set S such that P jun j converges uniformly on S , then: (1) f .z/ D (2) If J W

1 Y

.1 C un .z// converges uniformly on S .

nD1 Z>0 !

Z>0 is any bijection, then f .z/ D

1 Y

.1 C uJ.k/ .z//:

kD1

(3) f .z0 / D 0 if and only if un .z0 / D 1 for some n 2 Z>0 . P Proof. By uniform convergence of jun j on S , there exists c in R>0 such that sup

X

jun .z/j  c:

z2S

Let pN .z/ D

N Y

.1 C un .z// and qM .z/ D

nD1

M Y

.1 C uJ.k/ .z//:

kD1

We know that jpN j  jpN  1j C 1  pN  eju1 jCCjuN j  ec : Choose  with 0 <  < 14 . Then there exists an N0 2 Z>0 such that 1 X

jun .z/j <  for all z 2 S I

nDN0

in particular, jun .z/j <  <

1 4

for all z 2 S and all n > N0 . Choose M0 such that

f1; 2; : : : ; N0 g  fJ.1/; J.2/; : : : ; J.M0 /g: If M; N > maxfM0 ; N0 g, then we can write  Q 1 .1 C un .z// 1 : qM .z/  pN .z/ D pN .z/ Q 2 .1 C un .z// Q Q Here the symbols 1 and 2 denote products takenQover appropriate disjoint Q indices. For our purposes the important facts about 1 and 2 are that only indices n > N0 appear and that the indices that appear in the two products are disjoint.

270

10 Zeros of Holomorphic Functions

Let us define e un .z/ by 8 Q < 1 C un .z/ if n appears in 1 Q 1 1 Ce un .z/ D if n appears in 2 : 1 C un .z/ and let I be the union of the indexing sets in

Q 1

and

Q

2.

Now if a and b 2 C and ı 2 R>0 satisfy jaj < ı <

1 2

jbj < 2ı. Therefore

Note that I  Z>0 . 1 D 1 C b, then and 1Ca

ˇ ˇ ˇY ˇ ˇ ˇ un .z//  1ˇ jqM .z/  pN .z/j D jpN .z/j ˇ .1 Ce ˇ ˇ n2I

Y

 jpN .z/j

! .1 C je un .z/j/  1

n2I

X

 jpN .z/j exp 0

! un .z/j  1 je

n2I

0

 jpN .z/j @exp @

!

1 X

1

1

un .z/jA  1A je

nDN0

 jpN j .e2  1/: We now claim that ex  1  2x for 0  x  12 . This claim may be verified as follows. Define F .x/ D ex  1  2x for x 2 R. Observe that F .0/ D 0 and 1 F 0 .x/ D ex  2. Since the (real) exponential function is increasing, F 0  e 2  2 on Œ0; 12 . But 1

. 1 /2 . 1 /3 1 C 2 C 2 C 2 2Š 3Š  2 1 1 1 <1C C C D 2 2 1

e2 D 1 C

1 2

D 2;

and thus F is nonpositive on Œ0; 12 . Therefore jqM .z/  pN .z/j  ec 4 for all z 2 S and we can conclude as follows: 1. If we let J be the identity map, then pN .z/ ! f .z/ uniformly for z 2 S . 2. For arbitrary J; we conclude that qM .z/ ! f .z/ uniformly for z 2 S .

10.1 Infinite Products

271

3. Since pN0 D pM  .pM  pN0 /, we have that, for sufficiently large M , jpN0 j  jpM j C jpM  pN0 j  jpM j C .e2  1/ jpN0 j  jpM j C 4 jpN0 j or, equivalently, that jpM .z/j  .1  4/ jpN0 .z/j : Therefore f .z/ D 0 if and only if pN0 .z/ D 0.

t u

Theorem 10.4. Assume 0  un < 1. (1) If (2) If

1 X nD1 1 X

un < 1, then 0 < un D C1, then

nD1

1 Y

.1  un / < 1.

nD1 1 Y

.1  un / D 0.

nD1

Proof. The first claim is a consequence of the previous theorem. To prove the second claim, we start with the observation that 1  x  ex for 0  x  1: Let pN D .1u1 /.1u2 /    .1uN /. Since p1  p2      pN  0, lim pN D N !1

exists. We call it p. Now 0  p  pN D

N Y

.1  un /  e.u1 CCuN / :

nD1

Since lim e.u1 CCuN / D 0, the theorem follows.

t u

N !1

Finally, we can establish when an infinite product is holomorphic. Theorem 10.5. Let D be a domain in C and suppose that ffn g is a sequence in H.D/ with fn not identically 0 for all n.

(a) If

1 X

j1  fn j converges uniformly on compact subsets of D, then

nD1

1 Y nD1

converges uniformly on compact subsets of D to a function f in H.D/ and z .f / D

1 X nD1

z .fn / for all z 2 D:

fn

272

(b) If

10 Zeros of Holomorphic Functions 1 X

j1  fn j diverges pointwise on D, then

nD1

1 Y

fn converges to 0 on D.

nD1

Proof. For part (a), we only have to verify the formula for the order of z. We note that the sum in that formula is finite (i.e., all but finitely many summands are zero). Let z0 2 D and let K  D be a compact set containing a neighborhood of z0 . There is an N in Z>0 such that j1  fn .z/j < 12 for all z 2 K and all n  N: Therefore, fn .z/ ¤ 0 for all z 2 K and for all n  N . Thus z0 .f / D z0

N 1 Y

! fn C z0

nD1

1 Y

! fn

nDN

D

N 1 X

z0 .fn / C 0:

nD1

t u

Part (b) is easily verified.

10.2 Holomorphic Functions with Prescribed Zeros Our goal is to construct a holomorphic function with arbitrarily prescribed zeros (at a discrete set of points in any given domain). To this end we begin by defining the elementary functions, first introduced by Weierstrass. We investigate some of their properties and then use them along with Theorem 10.5 to construct the required holomorphic functions. Definition 10.6. The Weierstrass elementary functions are the entire functions Ep , for p 2 Z0 , defined as follows. Let z 2 C and set E0 .z/ D 1  z; and, for p 2 Z>0 ,   zp z2 Ep .z/ D .1  z/ exp z C C    C : 2 p Note that, for all nonnegative integers p, Ep .0/ D 1 and Ep .z/ D 0 if and only if z D 1. Furthermore, the unique zero of Ep is simple. ˇ ˇ Lemma 10.7. If jzj  1, then ˇ1  Ep .z/ˇ  jzjpC1 for all nonnegative integers p. Proof. The statement is clearly true if p D 0. If p  1, we have z2

zp

z2

zp

Ep0 .z/ D .1  z/ezC 2 CC p Œ1 C z C    C zp1   ezC 2 CC p z2

zp

D zp ezC 2 CC p :

10.2 Holomorphic Functions with Prescribed Zeros

273

We therefore conclude that 0 .Ep0 / D p. Further, z2

zp

Ep0 .z/ D zp ezC 2 CC p D zp

 n X 1 X z2 zp 1 zC CC D bn zn ; nŠ 2 p np nD0

with bp D 1 and bn > 0 for all n  p. Therefore 1  Ep .z/ D

X bn znC1 : n C 1 np

Set

1  Ep .z/ ; zpC1 X and observe that  2 H.C/ and that .z/ D an zn , with an > 0 for all n 2 Z0 . .z/ D

n0

For jzj  1, we have

ˇ ˇ ˇ X ˇX X ˇ ˇ nˇ ˇ an z ˇ  an jzn j  an D .1/ D 1I j.z/j D ˇ ˇ n0 ˇ n0 n0 ˇ ˇ thus ˇ1  Ep .z/ˇ  jzjpC1 for jzj  1.

t u

Theorem 10.8 (Weierstrass Theorem). Assume that fzn g is a sequence of nonzero complex numbers with lim jzn j D 1. n!1

If fpn g  Z0 is a sequence of nonnegative integers with the property that for all positive real numbers r we have  1  X r 1Cpn < 1; jzn j nD1

(10.1)

then the infinite product P .z/ D

1 Y nD1

 Epn

z zn

 ; z2C

defines an entire function whose zero set is fz1 ; z2 ; : : :g. More precisely, if z D c appears   0 times in the above sequence of zeros, then c .P / D . Furthermore, condition (10.1) is always satisfied for pn D n  1. Thus any discrete set in C is the zero set of an entire function.

274

10 Zeros of Holomorphic Functions

Proof. We first show that (10.1) holds for pn D n 1. In this case we have to show P 1 r n . But jan j n ! 0 as n ! 1, convergence of the series an , with an D jzn j and the root test allows us to conclude that  1  X r n < 1: jzn j nD1 Now let fpn g be any sequence of nonnegative integers satisfying condition (10.1) for all r > 0; fix r > 0 and assume that jzj  r. From Lemma 10.7 we conclude that ˇ  ˇ ˇ ˇpn C1   ˇ z ˇˇ ˇˇ z ˇˇ r pn C1 ˇ1  E p   : n ˇ zn ˇ ˇ zn ˇ jzn j   Q Therefore we can apply Theorem 10.5 to conclude that Epn zzn converges uniformly on all compact subsets of C to an entire function that has the required zero set. t u We next prove a generalization of two consequences of the fundamental theorem of algebra, which has been proven previously. The first of these algebraic consequences is that for every finite sequence fz1 ; : : : ; zn g of points in the complex plane (that may contain repeated points) there is a polynomial vanishing precisely at the points of that sequence. The second is that every nonzero complex polynomial p has a factorization n Y .z  zj / for all z 2 C; p.z/ D c j D1

where c is a nonzero constant and fzj g are the zeros of p, repeated according to their multiplicity. We need the analytic tools that were developed to handle infinite sequences. Theorem 10.9 (Weierstrass Factorization Theorem). Let f be in H.C/  f0g, and set k D 0 .f /. Let fzn I n 2 I g denote the zeros of f in Cf0g, listed according to their multiplicities. There exist a g 2 H.C/ and a sequence of nonnegative integers fpn I n 2 I g such that   Y z f .z/ D zk eg.z/ Epn zn n2I for all z in C. Proof. Observe that I  N may be finite (including the possibility that I is empty) or countable. In the finite case the theorem has already been established. In any case, we can choose any sequence fpn I n 2 I g of nonnegative integers such that (10.1) holds for all r > 0 and set

10.2 Holomorphic Functions with Prescribed Zeros

P .z/ D

Y

 Epn

n2I

z zn

275

 and G.z/ D

f .z/ : zk P .z/

Then G 2 H.C/ and G.z/ ¤ 0 for all z 2 C. Since G is a nonvanishing entire function, there is a g 2 H.C/ with eg.z/ D G.z/ for all z 2 C. t u Theorem 10.10. Let D be a proper subdomain of b C. Let A be a subset of D that has no limit point in D, and let  be a function mapping A to Z>0 . Then there exists a function f 2 H.D/ with z .f / D .z/ for all z 2 A, whose restriction to D  A has no zeros. Proof. To begin, we make the following observations: 1. A is either finite or countable. 2. Without loss of generality, we may assume that 1 2 D  A and that A is nonempty. 3. If A is finite, let A D fz1 ; : : : ; zn g. Set j D .zj /, for all 1  j  n, and choose z0 2 C  D. In this case we set f .z/ D

.z  z1 /1    .z  zn /n ; .z  z0 /1 CCn

and note that f is holomorphic on D, and does not vanish on b C  fz1 ; : : : ; zn g since f .1/ D 1. Since D  A  b C  fz1 ; : : : ; zn g, we have thus established the theorem for finite sets A. To prove the theorem for infinite sets A, let K D b C  D. Note that K is a nonempty compact subset of C. Let f˛n gn2N be a sequence whose terms consist of all ˛ 2 A, where each ˛ is repeated .˛/ times. We first claim that, for each positive integer n, we can choose a ˇn 2 K such that jˇn  ˛n j  jˇ  ˛n j for all ˇ 2 K. To see that this can be done, note that, for each positive integer n, the function z 7! ln .z/ D jz  ˛n j is continuous on K and, therefore, achieves a minimum at some ˇn 2 K. The function f we are seeking (whose existence must be established) is f .z/ D

1 Y nD1

 En

˛n  ˇn z  ˇn

 :

We show next  that ˇ the product on the RHS converges in D, by proving that P ˇˇ ˛n ˇn ˇ ˇ1  En zˇn ˇ converges uniformly on compact subsets of D. For this we first prove that lim jˇn  ˛n j D 0. If we assume jˇn  ˛n j  ı for some ı > 0 and n!1

infinitely many n, then for some subsequence f˛nj g of f˛n g, ˇ ˇ ˇz  ˛n ˇ  ı for all z 2 K: j

(10.2)

276

10 Zeros of Holomorphic Functions

But a subsequence of this subsequence converges to some point ˛ in b C. From (10.2) we conclude that ˛ … K. Thus we arrive at the contradiction that ˛ 2 D and is a limit point of A. Next, we put rn D 2 j˛n  ˇn j and observe that frng converges to zero. Let K0 be any nonempty compact subset of D; since K and K0 are disjoint compact subsets of b C, the distance between them must be positive. Therefore, the fact that rn ! 0 implies there is an N 2 Z>0 such that jz  ˇn j > rn for all z 2 K0 and all n > N . Thus ˇ ˇ ˇ ˛n  ˇn ˇ rn 1 ˇ ˇ ˇ z  ˇ ˇ  2r D 2 for all n > N and all z 2 K0 ; n n and hence

ˇnC1  nC1 ˇ  ˇ ˇ ˇ ˇ ˇ ˇ 1 ˇ1  En ˛n  ˇn ˇ  ˇ ˛n  ˇn ˇ  ˇ ˇ ˇ zˇ zˇ ˇ 2 n

n

for all n > N and all z 2 K0 , where the first inequality follows from Lemma 10.7. By Theorem 10.5, the infinite product defining f converges andf 2 H.D/.  Finally, ˛n  ˇn D 0 for it follows from Lemma 10.7 that f .z/ D 0 if and only if En z  ˇn some n 2 Z>0 if and only if z D ˛n for some n 2 Z>0 . t u As an immediate corollary we obtain the following result. Theorem 10.11. If D is a nonempty proper subdomain of b C, then M.D/ is the field of fractions of the integral domain H.D/; that is, for every f 2 M.D/ there exist g g in H.D/ and h in H.D/  f0g such that f D . h

10.3 The Ring H.D/ Let D1 and D2 be nonempty proper subdomains of b C, and assume there exists a conformal map F W D1 ! D2 between these domains. The map F induces a ring isomorphism F  W H.D2 / ! H.D1 / defined by F  .f /.z/ D f .F .z//; f 2 H.D2 /; z 2 D1 : Similarly, an anti-conformal map F of D1 onto D2 induces a ring isomorphism F  W H.D2 / ! H.D1 / defined by F  .f /.z/ D f .F .z//; f 2 H.D2 /; z 2 D1 : We have thus defined a map  that sends a conformal or anti-conformal map F between two domains in the extended complex plane to an isomorphism F  between their rings of holomorphic functions. It should be observed that for the identity map F , F  is also the identity, and for any F , .F 1 / D .F  /1 .

10.3 The Ring H.D/

277

While if J W D ! J.D/ D D is the anti-conformal conjugation map (J.z/ D z), then J  W H.D/ ! H.D/ is defined by J  .f /.z/ D f .z/; f 2 H.D/, and z 2 D. We also observe that for any two such maps F1

F2

D1 ! D2 ! D3 ; .F2 ı F1 / D F1 ı F2 : The main purpose of this section is to obtain a theorem due to Bers that is the converse to these facts; essentially that the map  is an isomorphism between appropriate categories. We shall freely use the results that follow from the exercises to this chapter. Theorem 10.12. Let D1 and D2 be proper subdomains of b C. If ' W H.D2 / ! H.D1 / is a ring isomorphism, then '.{/ D ˙{: Furthermore: (a) If '.{/ D {, then there exists a unique conformal map F of D1 onto D2 such that ' D F  . (b) Similarly, if '.{/ D {, there exists a unique anti-conformal map F of D1 onto D2 such that ' D F  . Proof. The proof proceeds in a number of steps. (1) It is immediate that '.{/ D ˙{, since the two constant functions ˙{ are the only elements of the two rings H.Dj / whose squares are the constant function 1. So we start by proving (a). (2) We now show that we may assume that neither D1 nor D2 contain the point 1. If D  b C is any proper domain in the sphere and 1 2 D, then there exists 1 maps D onto some c 2 C that is not in D. The conformal map Tc W z 7! zc a subdomain of C. Let us assume for the moment that both domains1 D1 and D2 contain 1 and that c … D1 and d … D2 . In this case, .Tc /1 ı ' ı Td W H.Td .D2 // ! H.Tc1 .D1 // is a ring isomorphism preserving {. If we can establish the existence of a conformal map G W Tc .D1 / ! Td .D2 / such that G  D .Tc /1 ı ' ı Td , then ' D .Td1 ı G ı Tc / . (3) '.c/ 2 C for all constant functions c 2 H.D2 /. If c is a complex rational constant (i.e., if c D a C b { with a and b 2 Q), then the assertion follows from (1) and the easily proved fact that '.x/ D x for every x 2 Q. For a nonrational constant, the conclusion follows from the observation that nonrational

1

The argument is, of course, simplified if only one of these contains 1.

278

10 Zeros of Holomorphic Functions

constants c are characterized by the existence of the multiplicative inverse of c  r for every rational constant r. Observe that the same argument applied to ' 1 shows that ' defines (by restriction) a ring isomorphism of C onto itself that fixes all complex rational constants. (4) Let us define, for c 2 D2 , Ic D ff 2 H.D2 /I f .c/ D 0g and, similarly, for c 2 D1 , Ic D ff 2 H.D1 /I f .c/ D 0g: These are obviously both principal (with generator z 7! z  c) and maximal ideals in their respective rings. They are the only such ideals in these rings (as c varies over the points in D2 and D1 , respectively). (5) We are now ready to define the map F W D1 ! D2 . Let c 2 D1 . Then Ic is a principal maximal ideal in H.D1 /, and it follows that this fact holds if and only if ' 1 .Ic / is a principal maximal ideal in H.D2 /, thus if and only if ' 1 .Ic / D IF .c/ for a unique point F .c/ 2 D2 . The map we have defined is injective and surjective. We leave it to the reader to verify these two claims. (6) The definition of F yields Ic D '.IF .c/ /. We want to conclude from this fact that for all f 2 H.D2 / and all c 2 D1 we have '.f /.c/ D f .F .c//:

(10.3)

So let f 2 H.D2 / and c 2 D1 . Then the function f  f .F .c// belongs to IF .c/ , and it follows that '.f  f .F .c/// D '.f /  '.f .F .c/// belongs to Ic : Thus '.f /.c/ D '.f .F .c///.c/ D '.f .F .c///; and it remains to show that '.d / D d for all d 2 C. (7) Let fbn D F .an /g be an infinite and discrete sequence of distinct points in D2 . By the Mittag-Leffler theorem (see Exercise 10.10), there exists f 2 H.D2 / such that f .bn / D n for all positive integers n. Thus '.f /.an / D '.f .F .an /// D '.f .bn // D '.n/ D n; and we conclude that fan g is an infinite and discrete sequence of distinct points in D1 . (8) As a consequence of the last assertion, the closure of F .K/ is compact in D2 whenever K is compact in D1 .

10.3 The Ring H.D/

279

(9) For all ˇ 2 C, there exist f 2 H.D2 / and c 2 D1 such that '.f /.c/ D ˇ and '.f / is univalent in a neighborhood of c. To verify this claim, we note that there certainly exist nonconstant functions in H.D2 /; in particular, it is convenient for future use to consider the identity function on D2 defined by I.z/ D z. Since '.I / 62 C (by (3)), there exists c 2 D1 such that '.I / is univalent in a neighborhood of c. If '.I /.c/ D ˛, then f D I C ' 1 .˛  ˇ/ has the required properties. (10) We already know from (3) that the isomorphism ' defines a ring isomorphism  of C onto itself that fixes all complex rational constants. To show that  is the identity on C, it suffices to show that  is continuous (see Exercise 2.19(c)), and by the additivity property of  all we have to show is that it is bounded in a neighborhood of the origin; we will actually show the equivalent condition that  1 is bounded in a neighborhood of the origin. To do so, we choose f 2 H.D2 / and c 2 D1 such that '.f /.c/ D 0 and '.f / is univalent in a neighborhood V  D1 of c. We may and do assume that the closure K of V in D1 is compact. Since '.f /.V / is an open set and 0 2 '.f /.V /, there exists ı > 0 such that the disc U.0; ı/  '.f /.V /. If  1 were not bounded in a neighborhood of the origin, we could choose a sequence of distinct complex numbers f˛n g with each j˛n j < ı, ˛n ! 0, and  1 .˛n / ! 1. Thus there would be a sequence fcn g  U such that '.f /.cn / D ˛n for all positive integers n. If we set dn D F .cn /, then the sequence fdn g is contained in the compact set F .K/, and thus ff .dn /g is a bounded sequence, but f .dn / D f .F .cn // D  1 .'.f /.cn // D  1 .˛n / ! 1: (11) The fact that  D 'jC is the identity map on C finally implies that Equation (10.3) holds: '.f /.c/ D f .F .c// for all f 2 H.D2 / and all c 2 D1 . It is quite easy to identify the function F we have defined with an analytic function on D1 : apply the last equation to the function I and conclude that '.I / D F . Thus F is in fact in H.D1 / and is conformal onto D2 . (12) The uniqueness of F is now clear. (13) To establish (b), we follow ' by the map .J  /1 W H.D1 / ! H.J.D1 //; where J is the anti-conformal involution, observe that ..J  /1 ı '/.{/ D { and apply the previous case.

t u

280

10 Zeros of Holomorphic Functions

Corollary 10.13. Let D1 and D2 be proper subdomains of b C, and let ' W H.D2 / ! H.D1 / be a ring isomorphism. Then ' is either a C-linear or a conjugate-C-linear algebra isomorphism.

10.4 Euler’s -Function In this section we introduce an important function among whose remarkable properties is the fact that it extends the factorial function on the integers to an entire function. Our development is brisker than in previous sections.

10.4.1 Basic Properties Define, for z 2 C, G.z/ D

1  Y z z 1C e n: n nD1

The infinite product converges to an entire function with a simple zero at each negative integer. We claim that the entire function h defined for all z 2 C by h.z/ D z G.z/ G.z/ satisfies h.z/ D

sin z :

(10.4)

Simple calculations show that h.z/ D z

 1  Y z2 1 2 ; n nD1

and hence, using (7.3) of Chap. 7 for the last equality, 1

h0 .z/ d 1 X 2z D log h.z/ D  D cot z: h.z/ dz z nD1 n2  z2 It follows that h.z/ D c sin z for some nonzero constant c. To evaluate c, we note that

10.4 Euler’s -Function

281

sin z h.z/ D ; and lim D 1; z!0 z z 1 and thus we conclude that c D . The function z 7! G.z  1/ is entire and has a simple zero at each nonpositive integer and no other zeros. It follows that lim

z!0

G.z  1/ D ze .z/ G.z/

(10.5)

for some entire function . We proceed to determine this function. Differentiating logarithmically both sides of the last equation, we obtain 1  X nD1

1 1  z1Cn n



 1  X 1 1 1 0 D C .z/ C :  z zCn n nD1

Since 1  X nD1

1 1  z1Cn n

 D

 1  X 1 1  zCn nC1 nD0

 1  X 1 1 1 ;  D 1C z zCn nC1 nD1 we conclude that

0 .z/ D

1  X 1 nD1

n



1 nC1

  1 D 0:

Hence the function is constant; it is known as Euler’s constant. To determine , we return to our function G and observe that if we set z D 1 in (10.5) we obtain that 1 D G.0/ D e G.1/, and hence e D G.1/ D

 1  Y 1 1 e n: 1C n nD1

Thus 1

1

e D lim .n C 1/e.1C 2 CC n / ; n!1

and

   1 1 :  D lim log.n C 1/  1 C C    C n!1 2 n   1 Since lim .log.n C 1/  log n/ D lim log 1 C D 0, we obtain n!1 n!1 n

282

10 Zeros of Holomorphic Functions

   1 1  log n : 1C CC

D lim n!1 2 n Next set, for z 2 C,

(10.6)

H.z/ D e z G.z/;

and compute that H.z  1/ D e z e G.z  1/ D e z zG.z/ D zH.z/: We can now introduce Definition 10.14. Euler’s -function is defined by .z/ D

1 e z Y  1 z 1 z D e n ; z 2 C: 1C zH.z/ z nD1 n

(10.7)

Note that  is a meromorphic function on C, with simple poles at z D 0; 1; 2; : : :, and that it has no zeros. The -function satisfies a number of useful functional equations. We derive some of these, which will lead us to (10.12). The first of these functional equations is .z C 1/ D

1 1 D D z.z/: .z C 1/H.z C 1/ H.z/

(10.8)

Further, it follows from (10.4) that .z/.1  z/ D

: sin z

A simple calculation shows that .1/ D e

 1  Y 1 1 1 1C e n D 1: n nD1

Together with (10.8), this implies that .n/ D .n  1/Š; for all n 2 Z>0 : Also, it follows from (10.9) that .

 since 

1 2

> 0 from (10.7).

1 2 2 / D

D , and hence sin 2

  p 1 D ; 2

(10.9)

10.4 Euler’s -Function

283

We derive some other properties of Euler’s -function that we will need. We start with a calculation, from (10.7): d  0 .z/ d d D .log..z// dz .z/ dz dz

! 1  d 1 X 1 1 D     dz z nD1 z C n n 2 1  X 1 : D zCn nD0

Both functions

(10.10)

  1 z 7! .2z/ and z 7! .z/ z C 2

have simple poles precisely at the points 0; 1; 2; : : : and  12 ;  32 ; : : :. The ratio of the two functions is hence entire without zeros. The next calculation will show more: !   1 1 X X d  0 .z C 12 / d  0 .z/ 1 1 C C D 1 2 dz .z/ dz .z C 2 / .z C n/ .z C n C 12 /2 nD0 nD0 ! 1 1 X X 1 1 D4 C .2z C 2n/2 nD0 .2z C 2n C 1/2 nD0 ! 1 X 1 D4 .2z C m/2 mD0   d  0 .2z/ : D2 dz .2z/ Therefore, there exists a constant A such that 2

 0 .z C 12 /  0 .2z/  0 .z/ D C A .2z/ .z/ .z C 12 /

or, equivalently,    d 1 d log .2z/ D log .z/ z C  A: dz dz 2

284

10 Zeros of Holomorphic Functions

Thus there exists a constant B such that    1  Az  B Log .2z/ D Log .z/ z C 2 or, equivalently,

  1 .2z/ eAzCB D .z/  z C : 2

(10.11)

Next work backward to determine A and B. Setting z D 12 in (10.11), we obtain p 1 1  e 2 ACB D  1; that is, 12 A C B D 12 log . Setting z D 1 in (10.11), we p obtain eACB D 12 ; that is, A C B D 12 log  log 2. Thus A D 2 log 2 and B D 12 .log / C log 2, and we have established 10.4.1.1 Legendre’s Duplication Formula   p 1 : .2z/ D 22z1 .z/  z C 2

(10.12)

10.4.2 Estimates for .z/ The estimate of .z/ for large values of jzj that is found in this section is known as Stirling’s formula. To derive this formula, we first express the partial sums 2 n  X 1 d  0 .z/ in (10.10) as a convenient line integral. View z D x C {y of zCk dz .z/ kD0 as a (fixed) parameter and  D  C { as a variable, and define ˆ./ D

cot  ; for  2 C: .z C /2

The function ˆ has singularities at  D z and at  2 Z; if z 62 Z, ˆ has a double pole at z and simple poles at the integers. Let Y be a positive real number, n be a nonnegative integer, and K be the rectangle in the -plane described by Y   Y and 0    n C 12 ; see Fig. 10.1. Then the residue theorem yields the next result. Lemma 10.15. If 0, then 1 pr: v: 2 {

Z

X 1 1 C ; 2 2z .z C /2 D0 n

ˆ./ d D  @K

10.4 Euler’s -Function

285

Fig. 10.1 The rectangle K

where as usual @K is positively oriented. We plan to let Y ! 1 and n ! 1. We thus have to study several line integrals, as follows. 10.4.2.1 The Integral Over the Horizontal Sides:  D ˙Y On the horizontal sides D ˙Y , cot  converges uniformly to ˙{ as Y goes cot  converges to 0 on each of the line segments   0; D ˙Y to 1. Thus .z C /2 as Y ! 1. We need to show that Z lim lim

n!1 Y !1 0

nC 12

cot . ˙ {Y / d D 0: .z C  C {Y /2

Since we are able to control the speed with which Y and n approach infinity, this presents a small challenge; for example, we can set Y D n2 and then let n ! 1. 10.4.2.2 The Integral Over the Vertical Side  D n C

1 2

On the vertical line,  D n C 12 , cot  is bounded since cot is a periodic function. Thus we conclude that for some constant c, ˇZ ˇ Z ˇ ˇ d ˇ ˇ ˆ./d ˇ  c : ˇ 1 j C zj2 ˇ DnC 12 ˇ DnC 2 On  D n C 12 , we have  D n C 12  { D 2n C 1  . We then use residue calculus (as in case 1 of Sect. 6.6) to conclude that

286

10 Zeros of Holomorphic Functions

1 {

Z

1 D 2 { j C zj d

DnC 12

Z DnC 12

Therefore,

2 d D : . C z/.2n C 1   C z/ 2n C 1 C 2x

Z

d

lim

n!1 DnC 1 2

j C zj2

D 0:

10.4.2.3 The Integral Over the Imaginary Axis We now turn to the computation of the principal value of the integral over the imaginary axis, which may be written as follows. 1 2

Z

1 0

 Z 1 1 1 2 z d D  cot {  cot h 2 d : .{ C z/2 .{  z/2 . C z2 /2 0 

We now return to the main task. We let Y and n tend to 1 in Lemma 10.15 to conclude that 1

X d  0 .z/ 1 D dz .z/ .z C n/2 nD0 Z 1 1 D 2 C pr. v. ˆ./ d 2z C1 Z 1 2 z 1 cot h 2 d : D 2C 2z . C z2 /2 0 Replacing cot h by 1 C

2 in the above expression and noting that e2  1 Z 1 2 z 1 d D ; 2 2 2 . C z / z 0

we obtain d  0 .z/ 1 D 2C dz .z/ 2z

Z

1 0

1 1 D C 2C z 2z

 1C Z

2 e2  1

1 0

. 2



2 z d . 2 C z2 /2

4 z d : 2 2 2 Cz / e 1

Note that we have restricted our study to values of z with positive real parts. We can integrate with respect to z and change the order of integration in the last integral to conclude that Z 1  0 .z/ 2 d e C Log z  1  DC : 2 2 2 .z/ 2z Cz e 1 0

10.4 Euler’s -Function

287

Using integration by parts, we see that Z

1

 0

Z

1 2 d D 2 C z2 e2  1

1

z2  2 log.1  e2 / d I . 2 C z2 /2

0

therefore  0 .z/ e C Log z  1 C 1 DC .z/ 2z

Z

1 0

z2  2 log.1  e2 / d ; . 2 C z2 /2

and we conclude that   1 Log z C J.z/; Log .z/ D C C C z C z  2 0

where J.z/ D

1

Z

1

0

. 2

(10.13)

1 z log d : 2 2 Cz / 1  e2

If z ! 1 and z stays away from {R, then J.z/ ! 0. We have almost established the next result.

10.4.3 The Formulae for the Function Theorem 10.16 (Stirling’s Formula). For 0, .z/ D

p 1 2 zz 2 ez eJ.z/ :

(10.14)

Proof. We know from (10.13) that .z/ D eC

0 CC z

1

zz 2 eJ.z/ ;

(10.15)

and we only need to determine the constants C 0 and C , which we do using (10.15) and the two functional equations (10.8) and (10.9) already derived for . Replacing .z/ by the RHS of (10.15) (and also .z C 1/ by the corresponding value) in .z C 1/ D z.z/, we obtain   1 1 C D .z C / Log 1 C C J.z/  J.z C 1/; 2 z and letting z tend to 1 we conclude that C D 1. To obtain C 0 one can proceed in a similar manner, replacing .z/ and .1  z/ by the corresponding RHS of (10.15) , with z D 12 C { y. We leave the details to the reader in .z/.1  z/ D sin z (Exercise 10.9). t u

288

10 Zeros of Holomorphic Functions

Corollary 10.17. nŠ lim p  n D 1: n!1 2 n ne Proof. Note that .n C 1/

nC 12

nC 12

Dn

  1  1 n 1 2 1C 1C ; n n

and therefore 1

.n C 1/nC 2

D 1: 1 e nnC 2 Applying Stirling’s formula (10.14) with z D n C 1, we obtain lim

n!1

.n C 1/ D

(10.16)

p 1 2 .n C 1/nC 2 e.nC1/ eJ.nC1/ :

Since we already know that lim J.n C 1/ D 0, the claim is proved. n!i 1

t u

With some further work we can prove the following integral expression for the -function. Z 1 et t z1 dt; for 0:

.z/ D

(10.17)

0

Proof. Denote the RHS of (10.17) by F .z/. It certainly defines a holomorphic function whenever the integral converges, that is, in the right half-plane 0. Since Z 1 Z 1 t z F .z C 1/ D e  t dt D z et  t z1 dt D zF .z/ for 0; 0

0

we see that F and  satisfy the same functional equation (10.8). Thus F .z/ F .z C 1/ D for 0; .z C 1/ .z/ and the function F can be extended to be defined on all of C. For z with 1 
1

jF .z/j 

et  t
0

and since F is a continuous function on the closed interval Œ1; 2, it is bounded there. The last estimate shows that F is bounded on the strip 1 
10.5 Divisors and the Field of Meromorphic Functions

289

  1 1 log jzj  =z Arg z C <J.z/: log j.z/j D log 2 
10.5 Divisors and the Field of Meromorphic Functions It is convenient to introduce the following. Definition 10.18. Let D be a nonempty plane domain. A divisor on D is a formal expression: Y  zi i ; i

where the fzi g form a discrete set in D and the i are integers. We will also write a divisor D as Y DD zz .D/ ; z2D

with the understanding that z .D/ 2 Z for all z 2 D and z .D/ D 0 for all z not in a discrete subset of D (that depends on the divisor, of course). If z D 0 for all z 2 D, the divisor is denoted by I . There is a commutative law for divisors, where I is the unit Y multiplication Y zz .D1 / and D2 D zz .D2 / are two divisors, then element, and if D1 D z2D

z2D

D1  D2 D

Y

zz .D1 /Cz .D2 / :

z2D

The set of all divisors on D with this operation becomes a commutative group, denoted by Div.D/. An interesting and important subset of Div.D/ is obtained from the divisors naturally associated to meromorphic functions on D. To each not identically zero meromorphic function f on D we can associate its divisor .f / defined by

290

10 Zeros of Holomorphic Functions

.f / D

Y

zz .f / ;

z2D

where z .f / denotes the order of the function f at the point z in D (see Sect. 3.5). In particular, to any constant (nonzero) function in D, we associate the divisor I . A divisor of a nonzero meromorphic function in D is called a principal divisor. The set of principal divisors is a subgroup of Div.D/, and the function ./ that associates to each nonzero meromorphic function f on D its divisor .f / is a homomorphism from the multiplicative group M.D/  f0g to Div.D/, whose image is the subgroup of principal divisors. Much more is true as shown next. Theorem 10.19. Let D be a nonempty domain in C. The map ./ W M.D/  f0g ! Div.D/ is a surjective homomorphism, whose kernel is the set of nowhere vanishing holomorphic functions on D. Proof. Let D 2 Div.D/. Since I is clearly in the image of the map ./, we assume D 6D I . We can then write D D D1 D21 , where D1 and D2 are relatively prime integral divisors (see Exercise 10.5), and use Theorem 10.10 to conclude the proof. t u

10.6 Infinite Blaschke Products Let DD

Y

zi i

i

be an integral divisor (a divisor such that i  0 for all i , see Exercise 10.5) on the unit disc D. It is clear that the definitions of Sect. 8.5 associate a Blaschke product BD to the divisor D. We end this book with some applications of Blaschke products to the study of bounded analytic functions on D. Y  Theorem 10.20. Let I 6D D D ai i be an integral divisor on the unit disc D. The Blaschke product BD D function on D if and only if

Y

i

Baii converges to a nonconstant bounded analytic

i

X

i .1  jai j/ < 1:

(10.18)

i

P If i i .1  jai j/ D 1, then the Blaschke product converges to the constant zero function on the disc.

10.6 Infinite Blaschke Products

291

Proof. Without loss of generality, we assume that ai ¤ 0 for all i . It is convenient Y aj , with each ai appearing in this product i times. to write the divisor D D j

Assume (10.18) translated to

X

ˇ ˇ .1  ˇaj ˇ/ < 1. By Theorem 10.5, it suffices to

ˇj ˇ Xˇ ˇ j z  a ja j j ˇ converges uniformly on compact subsets jzj  r ˇ1 C show that ˇ aj 1  aNj z ˇ j

of D, with 0 < r < 1. For jzj D r, we estimate

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇaj ˇ z  aj ˇˇ ˇˇ aj  jaj j2 z C jaj jz  ˇaj ˇ aj ˇˇ ˇ ˇDˇ ˇ ˇ1 C ˇ ˇ aj 1  aNj z ˇ ˇ aj .1  aNj z/ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ a .1  ˇa ˇ/ C ˇa ˇ z.1  ˇa ˇˇ/ ˇ j j j ˇ ˇ j Dˇ ˇ ˇ ˇ aj .1  aNj z/  ˇ ˇ ˇ ˇ ˇ ˇ ˇ .1  ˇaj ˇ/ 1 C jaaj j z ˇ j ˇ ˇ ˇ D ˇˇ ˇ 1  aNj z ˇ ˇ ˇ ˇ ˇ ˇ .1  ˇaj ˇ/.1 C r/  : 1r By the maximum principle for holomorphic functions, the same estimate is valid for jzj  r. Hence on this compact set Xˇ X ˇ ˇ1  Ba .z/ˇ  1 C r .1  jaj j/ j 1r j j is finite, and the infinite Blaschke product converges there. The infinite product BD D

Y

Baii

i

is bounded by 1 on D, because each factor is. We already know that this bounded analytic ˇ is ˇ nonconstant, since ai .BD / D i . P function If j .1  ˇaj ˇ/ D C1, the infinite product converges to the zero function by the second part of Theorem 10.4. t u Let F be a nowhere vanishing holomorphic function defined on a neighborhood of D. Then log jF j is harmonic there, and the mean value property for such functions tells us that Z 2 ˇ ˇ 1 (10.19) log ˇF .re{ /ˇ d ; log jF .0/j D 2 0 for 0  r  1.

292

10 Zeros of Holomorphic Functions

An important generalization of this simple formula that relates the values of F on the boundary of D to its value at the origin is the next result. Theorem 10.21 (Jensen’s Formula). Let f be a holomorphic function in a neighborhood V of D, and assume that f .0/ 6D 0. For fixed 0 < r < 1, let fa1 ; : : : ; ak g denote the zeros of f in the disc U.0; r/ listed with multiplicity, and assume that f does not vanish on the circle fjzj D rg. Then log jf .0/j C

k X j D1

ˇ Z 2 ˇ ˇD 1 log jf .re{ /jd : ˇ 2 0 j

ˇ ˇr log ˇˇ a

Proof. For a 2 D, the modified Blaschke factor ba .z/ D z  a ; z 2 V; B 1  az N has been encountered many times before. Its properties readily imply that F .z/ D

f .z/ ; z 2 V; k z Y b aj B r r j D1

defines a function that is holomorphic and free of zeros in V . Thus, by (10.19), Z 2 " ˇ ˇ 1 ˇb a ˇ log jf .0/j  log jf .re{ /j log ˇB j .0/ˇ D r 2 0 j D1 3 k ˇ ˇ X ˇb a { ˇ5  log ˇB d : j .e /ˇ k X

j D1

r

Evaluating

ˇ ˇ b aj .e{ /ˇˇ D 1 b aj .0/ D  aj and ˇˇB B r r r leads us to the required formula.

t u

As a consequence we have the following reflection of the subharmonicity of log jf j.2

2

In more general definitions of subharmonic functions than used in this text, they are allowed to assume the value 1.

10.6 Infinite Blaschke Products

293

Corollary 10.22 (Jensen’s Inequality). Under the hypothesis of the theorem, log jf .0/j 

1 2

Z

2

ˇ ˇ log ˇf .re{ /ˇ d :

0

ˇ ˇr Proof. For each j , log ˇˇ a

ˇ ˇ ˇ  0. t u ˇ j Y  Theorem 10.23. Let D D ai i be an integral divisor on the unit disc D. There i

exists X a bounded analytic function f on D with divisor equal to D if and only if i .1  jai j/ < 1. Furthermore, the function f is then of the form f D BD F , i

with F holomorphic, free of zeros, and bounded on D. Proof. Let f 2 H.D/ be a bounded function with .f / D D; choose M 2 R>0 such that jf j  M . Assume for the moment that aj 6D 0 for allˇ j .ˇ We can certainly find a sequence of numbers 0 < ri < 1 with lim ri D 1 and ˇaj ˇ 6D ri for all i and j . By Jensen’s formula applied with r D ri , X

log jf .0/j C

jaj j
i !1

ˇ ˇ Z 2 ˇr ˇ ˇ ˇ 1 j log ˇˇ ˇˇ D log ˇf .re{ /ˇ d  log M: aj 2 0

Letting i ! C1 (thus ri ! 1), we conclude that X j

X ˇ ˇ 1 j .1  ˇaj ˇ/  j log ˇ ˇ  log M  log jf .0/j : ˇaj ˇ j

f .z/ . Then g is a bounded z analytic function on D with g.0/ ¤ 0, and the may be applied to Xprevious ˇargument ˇ ˇ ˇ g to conclude that in this case we also have j .1  aj / < 1. If  D D .0/ > 0 (i.e., if f .0/ D 0), set g.z/ D

Conversely, if

X i

j

i .1  jai j/ < 1, it follows from Theorem 10.20 that BD is a

bounded analytic function on D satisfying .BD / D D. Furthermore, if f is analytic f 2 H.D/ is zero-free and bounded and bounded on D with .f / D D, then F D BD on D. t u

294

10 Zeros of Holomorphic Functions

Exercises  1  Y 1 10.1. Calculate 1 n : 2 nD2 10.2. Let fan g be a sequence in C¤1 . The infinite product 1 Y .1 C an / is said to be absolutely convergent if the corresponding series nD1 1 X

Log.1 C an / converges absolutely.

nD1

1. Show that

1 Y

.1 C an / converges absolutely if and only if

nD1

1 X

jan j converges.

nD0

2. Show that the value of an absolutely convergent product does not change if the factors are reordered. 1 X 3. Find examples that show that the convergence of the series an is neither nD0

necessary nor sufficient for the convergence of the infinite product

1 Y

.1 C an /.

nD1 1

10.3. Show that . 16 / D 2{ . 3 / 2  .. 31 //2 . 10.4. Find the residues of  at the poles z D n, n  1. 10.5. A divisor D on a domain D is integral if z .D/  0 for all z 2 D. Define appropriately the greatest common divisor gcd.D1 ; D2 / and the least common multiple lcm.D1 ; D2 / of two integral divisors D1 and D2 . Show that both the gcd and the lcm exist, and obtain formulae for them. The integral divisors D1 and D2 are relatively prime if their gcd is the empty set. 10.6. Show that two nonzero meromorphic functions f and g in D give rise to the same divisor if and only if there exists h in H.D/ that does not vanish in D and such that f D gh. 10.7. Show that a principal divisor on D is integral if and only if it is the divisor of an analytic function on D. 10.8. Show that if f and g are analytic functions in D (where at least one is not the zero function), then there exists a function h 2 H.D/ such that h is a greatest common divisor for f and g. That is, h divides f and g (in H.D/), and it is divisible by every holomorphic function dividing both f and g. If neither f nor g are the zero function, show that .h/ D gcd..f /; .g//:

Exercises

295

Hint: Apply Theorem 10.10. 10.9. Replace .z/ and .1 z/ by the corresponding RHS of (10.15) in .z/.1  , with z D 12 C { y in the proof of Stirling’s formula (Theorem 10.16) to z/ D sin z obtain C 0 in a similar manner to that used to obtain C . Give full details. 10.10. Prove the following form of the Mittag-Leffler Theorem. Let fan g be a discrete sequence of distinct points in a domain D  C, and let fbn g be an arbitrary sequence of complex numbers. Then there exists a holomorphic function f on D such that f .an / D bn for all n. Let D  C be a domain with H.D/ its ring of holomorphic functions. The following six exercises study the ideal structure of this ring; in particular, they establish that every finitely generated ideal in H.D/ is a principal ideal. 10.11. Show that every nonempty collection of holomorphic functions in D, except for the set consisting of the single function zero, has a greatest common divisor. 10.12. Show that if h is a greatest common divisor for f1 ; : : : ; fn in H.D/, then there exist g1 ; : : : ; gn in H.D/ such that f1 g1 C    C fn gn D h: Hint: First consider the case when n D 2 and h D 1. 10.13. Let f1 ; : : : ; fn be holomorphic functions in D and consider the ideal I generated by them in H.D/: I D ff1 g1 C    C fn gn I gj 2 H.D/g: Prove that I is a principal ideal; that is, there exists a function f in H.D/ such that I D ff g I g 2 H.D/g: 10.14. Show that there exist nonprincipal maximal ideals in H.D/. 10.15. Let ' W H.D/ ! C be a C-algebra homomorphism. Show that there exists a unique c 2 D such that '.f / D f .c/ for all f 2 H.D/: 10.16. Characterize the principal maximal ideals in the ring of holomorphic functions on a plane domain. Do the concepts of “principal maximal” and “maximal principal” ideals coincide? 10.17. Complete the proof of Theorem 10.12 by supplying all missing details.

Bibliographical Notes

The references required for proofs and definitions were included in the body of the book. The purpose of these notes is to list a number of basic books, not necessarily the latest edition or reprint, on complex analysis that the authors of this volume have found useful and interesting. Although we tried to write a book following a path that Bers might have chosen, certain portions show the influences of the authors in our list, particularly those of Lars Ahlfors and Henri Cartan. This is an incomplete list, arranged as four categories, reflecting the tastes and limited knowledge of the authors. 1. Undergraduate texts: The texts by Churchill and Brown [6], Derick [8], Fisher [9], and Silverman [33] are each very appropriate for an undergraduate course that is centered around applications and uses the Ahlfors and Bers approach to complex integration and analytic continuation along a path; Palka [27] is a very thorough and careful undergraduate text that is also frequently used in graduate courses, and uses the Ahlfors and Bers approach to complex integration and analytic continuation along a path; Marsden [23] is more theoretical while Bak– Newman [2] is the only undergraduate text on the subject that ends up with a treatment of the prime number theorem. 2. Graduate texts: Ahlfors [1] is among the outstanding mathematics books in any field. It clearly influenced almost all books on the subject that followed, including, of course, our own. Another classical book is the lovely treatment by Nevanlinna–Paatero [26]; Narasimhan [24] is an outstanding very modern treatment. The beautiful text by Cartan [5] starts with a treatment of formal power series; it is reflected in and is an inspiration for our chapter on convergent power series. The first of the two Hille volumes [13] is a standard introduction to the subject; the second [14] deals with many interesting special topics. The book by Heins [12] covers many prerequisites currently dealt with in other courses and some advanced topics. Conway’s book [7] is very concise yet complete. It includes the big Picard theorem. Berenstein and Gay [3] is in tune with more recent, modern developments in complex variables. The Green–Krantz text [11] is a treatment of complex variables as an outgrowth of real multivariate calculus. 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, © Springer Science+Business Media New York 2013

297

298

10 Bibliographical Notes

The Remmert volume [28] deals with some number theoretic topics that we have omitted and contains short biographical notes on some of the principal contributors to classical function theory. It also contains a discussion of the origins of some of the basic concepts in function theory. The text by Boas [4] includes a discussion of Abel summability. Another standard reference is Lang [21]. Lang and Bers probably influenced each other’s view of complex variables as a result of long discussions in the Columbia mathematics department fifth floor lounge while Lang was writing his book and Bers was teaching Complex Variables I and II during the 1966–1967 academic year. The first chapter of H¨ormander [15] serves as a thorough review for those who have learned one variable complex analysis and are interested in seeing how it leads naturally to a study of several variables. Rudin [30] is an integrated treatment of real and complex analysis, while the very recent volume by Lax-Zalcman [22] applies complex analysis to provide efficient proofs of real variable theorems. The text by Needham [25] is intended mainly for beginners, undergraduates perhaps. It is full of theory and applications, and all the material is presented from a wonderful geometric perspective. The volume by Stein– Shakarchi [34] treats complex analysis as part of the wider analytic universe. 3. Problem books: There are paperback books that contain problem sets and their solutions. Classical is a five-volume set by Konrad Knopp. Whenever Bers taught the yearlong graduate complex analysis course at Columbia, he taught without a text and simply told his students to read and work through all of the problems in Knopp’s books on the theory of functions [16–20] (Bers translated one of these five volumes from the German [18]). Knopp’s problem books are still eminently relevant and useful. A paperback by Shakarchi [32] includes solutions to all of the undergraduate level problems from Lang’s graduate text (problems from the first eight chapters) and solutions to selected problems from the more advanced chapters. 4. History: Although not primarily a book on the history of mathematics, [28] belongs on this list. Roy [29] is a scholarly and well-researched treatment of sources in the development of infinite series and products. His appendix to our first chapter is indicative of the breadth of the book. The Freitag–Busam text [10] also contains interesting historical remarks. A volume by Sandifer [31] contains a thorough history of part of the work of a mathematical giant.

References

1. Ahlfors, L.V.: Complex Analysis, 3rd edn. McGraw-Hill, New York (1979) 2. Bak, J., Newman, D.J.: Complex Analysis. Springer, Berlin (1982) 3. Berenstein, C.A., Gay, R.: Complex variables, an introduction. In: Graduate Texts in Mathematics, vol. 125. Springer, Berlin (1991) 4. Boas, R.P.: Invitation to Complex Analysis. Random House, New York (1987) 5. Cartan, H.: Elementary Theory of Analytic Functions of One or Several Complex Variables. Addison-Wesley, Reading (1963) 6. Churchill, R.V., Brown, J.W.: Complex Analysis and Applications, 5th edn. McGraw-Hill, New York (1990) 7. Conway, J.B.: Functions of One Complex Variable, 2nd edn. Springer, Berlin (1978) 8. Derrick, W.R.: Complex Analysis and Applications, 2nd edn. Wadsworth International Group, New York (1982) 9. Fisher, S.D.: Complex Variables, 2nd edn. Dover Publications, New York (1999) 10. Freitag, E., Busam, R.: Complex Analysis. Springer, Berlin (2005) 11. Greene, R.E., Krantz, S.G.: Function Theory of one Complex Variable. John Wiley & Sons Inc., New York (1997) 12. Heins, M.: Complex Function Theory. Academic Press, New York (1968) 13. Hille, E.: Analytic Function Theory, vol. I. Blaisdell, New York (1959) 14. Hille, E.: Analytic Function Theory, vol. II. Blaisdell, Waltham (1962) 15. H¨ormander, L.: An Introduction to Complex Analysis in Several Variables. Van Nostrand, Princeton (1966) 16. Knopp, K.: Theory of Functions I. Elements of the General Theory of Analytic Functions. Dover Publications, New York (1945) 17. Knopp, K.: Theory of Functions II. Applications and Continuation of the General Theory. Dover Publications, New York (1947) 18. Knopp, K.: Problem Book in the Theory of Functions: Problems in the Elementary Theory of Functions, , vol. 1. Dover Publications, New York (1948) (Translated by Lipman Bers) 19. Knopp, K.: Elements of the Theory of Functions. Dover Publications Inc., New York (1953) (Translated by Frederick Bagemihl) 20. Knopp, K.: Problem Book in the Theory of Functions: Problems in the Advanced Theory of Functions, vol. II. Dover Publications, New York, NY (1953) (Translated by F. Bagemihl) 21. Lang, S.: Complex analysis, 4th edn. In: Graduate Texts in Mathematics, vol. 103. Springer, Berlin (1999) 22. Lax, P.D., Zalcman, L.: Complex Proofs of Real Theorem. American Mathematical Society University Lecture Series, AMS, New York (2012) 23. Marsden, J.E.: Basic Complex Analysis. W. H. Freeman and Company, New York (1973)

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, © Springer Science+Business Media New York 2013

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300

References

24. Narasimhan, R.: Complex analysis in One Variable. Verlag, Birkh˝auser (1985) 25. Needham, T.: Visual Complex Analysis. Oxford University Press, Oxford (2004) 26. Nevanlinna, R., Paatero, V.: Introduction to Complex Analysis. Addison-Wesley, New York (1964) 27. Palka, B.: An Introduction to Complex Function Theory. Springer, Berlin (1991) 28. Remmert, R.: Theory of Complex Functions. Springer, Berlin (1991) (Translated by R. B. Burckel) 29. Roy, R.: Sources in the Development of Mathematics. Cambridge University Press, Cambridge (2011) 30. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) 31. Sandifer, C.E.: The Early Mathematics of Leonhard Euler. The Mathematical Association of America, Washington (2007) 32. Shakarchi, R.: Problems and Solutions for Complex Analysis. Springer, Berlin (1999) 33. Silverman, R.A.: Complex Analysis with Applications. Prentice-Hall (1974) 34. Stein, E.M., Shakarchi, R.: Complex Analysis. Princeton Lectures in Analysis. Princeton University Press, Princeton (2003)

Index

A Abel’s lemma, 43 limit theorem, 52 Abel, N. H., 43, 52 absolutely convergent product, 294 Ahlfors, L. V., 108, 237 analytic continuation, 68 function, 29 annulus, 100 anti-conformal map, 199 Argz, 59 arg z, 59 Argument, 59 Principle, 150 Aut D, 200 automorphism, 200

B ball, 24 barrier, 251 Bers, L., 1, 5, 108, 267, 277 Bessel, F. W., 7 Blaschke factor, 292 product, 221, 290 finite, 221 infinite, 290 Blaschke, W. J. E., 199, 292 Bolzano, B., 25 Bolzano–Weierstrass Theorem, 25 bounded in the strong sense, 183

bounded subset of C or R, 24 of C.D/, 183

C

b C, 16 C.D/ metric, 174 Cp .D/, 87 Cantor diagonalization method, 185 Cantor, G., 185 Carath´eodory, C., 13 Casorati, F., 13, 145, 146 Casorati–Weierstrass Theorem, 145 Cauchy sequence, 23 Cauchy’s inequalities, 125 Cauchy’s integral formula, 121, 147 extended version, 130 for smooth functions, 134 general form, 127 generalized, 124, 191 Cauchy’s Theorem, 106 for a rectangle, 104 general form, 127 Cauchy, A. L., 3, 4, 6, 7, 9–12, 14, 47, 82, 103, 107, 115, 119, 121, 127, 129, 130, 134, 139, 140, 147, 148, 162, 172, 184, 185, 191, 192 Cauchy–Hadamard, 47 Cauchy–Riemann equations complex form, 32 real form, 30 circle, 19 C, 203 circle in b

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, © Springer Science+Business Media New York 2013

301

302

Index

C, 77 circle in b closed curve, 83 Collins, J., 12 Compactness Theorem, 185 complex number argument, 21 conjugate, 16 modulus, 18 norm, 18 root, 22 complex sphere, 16 conformal automorphism, 200 equivalence, 200 map, 199 conjugate complex, 16 harmonic, 233 contour integral, 84 convergence in -metric, 176 locally uniform, 172 of meromorphic functions, 178 of number sequence, 23 uniform on compact subsets, 172, 176 Conway, J. B., 145, 188 coordinates polar, 21 rectangular, 16 CR, 30 complex form, 3, 32 real form, 3 cross ratio, 202 curve, 83 regular, 241 cycle, 126 homologous cycles, 128 homologous to zero, 126 index with respect to a point, 126

diffeomorphism, 151 differential closed, 87 conjugate, 240 exact, 87 form, 83 of class Cp , 87 Dirichlet problem, 235, 236, 251 proper solution, 252 regularity, 251 Dirichlet, L., 6, 235, 251 disc, 24 of convergence, 44 C, 204 discs in e divisor, 289 integral, 289, 294 of bounded holomorphic functions on D, 293 of holomorphic function, 275 principal, 289, 290 domain, 27 elliptic, 248 hyperbolic, 211, 248 parabolic, 248 simply connected, 99

D df , 87 D, 204 d z, 87 d z, 87 de Moivre’s formula, 22 de Moivre, A., 22 definition of , 58 @f , 87 @f , 87 derivative, 28 diagonalization, 185 Dieudonn´e, J., 129

F f 0 , 28 f .n/ , 29 Folland, G. B., 91 fractional linear transformation, 30, 200 Frobenius Theorem, 38 Fubini, G., 169 function adherent, 187 algebraic, 54 analytic, 29 anti-holomorphic, 29 complex differentiable, 28

E elementary functions Weierstrass, 272 entire function, 29 Euler’s -function, 267 ˆ-function, 267 constant, 281 formula, 56 identity, 57 Euler, L., 7, 9, 56, 267 extended complex plane, 16

Index complex trigonometric, 56 conformal, 199 differential of, 87 entire, 29 estimates for .z/, 284 exponential, 54 Gamma, 280 , 282 Green’s, 256 harmonic, 230 holomorphic, 29 at 1, 70 at a point, 29 bounded, 290 on a set, 29 schlicht, 174 simple, 174 univalent, 174 hyperbolic trigonometric, 75 hypergeometric, 69 meromorphic, 72, 289 of class Cp , 87 order at a point, 71 order of a zero, 69 periodic, 58 pole, 70 order, 70 simple, 70 rational, 30 smooth, 87 subharmonic, 245 superharmonic, 246 total differential of, 87 transcendental, 54 Weierstrass, 272 zero, 69 simple, 69 Fundamental Theorem of Algebra, 37, 125, 165

G -function, 280, 282 Gauss, C. F., 6, 7 geodesic, 214 path, 215 segment, 215 geometric series, 42 Goursat’s Theorem, 103 for triangles, 109 strengthened version, 119 Goursat, E., 10, 82, 103, 104, 108, 109, 119, 191 Grabiner, S., 188

303 Green’s function, 256 singularity, 256 theorem, 90 version 2, 91 Green, G., 10, 82, 90 Gregory, J., 12

H H2 , 204 H.D/, 72, 125, 276 Hadamard, J. S., 47 half-plane, 19 harmonic conjugate, 233 existence of conjugates, 234 function, 230 measure, 256 harmonization, 249 Harnack’s Convergence Theorem, 234 inequalities, 234 Hilbert, D., 13 holomorphic function, 29 prescribed zeros, 272 ring, 276 homologous, 126 homotopy, 97 Hurwitz’s Theorem, 174 Hurwitz, A., 173 hyperbolic distance, 213 domain, 211 isometry, 214 length, 213 line, 214

I identity principle, 62, 66 index of a closed curve with respect to a point, 96 of a cycle with respect to a point, 126 I.; c/, 96 inf, 43 infimum, 43 infinite products, 268 integral along a pdp, 85 contour, 84 line, 84 of closed form on continuous path, 95

304 integral (cont.) of form on differentiable path, 84 path, 84 isolated singularity, 143 isometry, 214 hyperbolic, 214 J Jacobi, C., 6 Jacobian, 156 Jensen’s formula, 292 inequality, 293 Jensen, J., 6, 292 Jordan curve, 83, 129 orientation, 129 Jordan, C., 129, 149, 152 K Kelley, J. L., 15 Koebe, P., 13 L l’Hopital’s rule, 79 Landau, E., 6, 13 Laplace operator, 230 Laplace’s equation complex coordinates, 230 polar coordinates, 230 rectangular coordinates, 230 Laplace, P.-S., 7 Laplacian, 230 Laurent, P. A., 12, 139, 141, 143, 144, 148 Legendre’s duplication formula, 284 Legendre, A.-D., 7 limit of a function, 25 limit point, 25 line, 19 line integral, 84 Liouville’s Theorem, 125 Liouville, J., 115, 211 Lobachevsky, N. I., 212 locally, 231 locally uniform convergence, 172 Log z, 59 log z, 59 logarithm, 59 branch, 60 M M¨obius group, 201

Index M¨obius transformation, 30, 200 fixed points, 202 trace, 202 M¨obius, A. F., 200, 202 M.D/, 72, 125, 276 Maclaurin, C., 11, 124 map anti-conformal, 199 conformal, 199 Riemann, 207 Maximum Modulus Principle for analytic functions, 133 Maximum Modulus Principle (MMP), 131 Maximum Principle, 133 for subharmonic functions, 247 for harmonic functions, 231 mean value inequality, 246 Mean value property harmonic functions, 233 mean value property (MVP), 131 area form, 243 for harmonic functions, 231, 243 metric Poincar´e, 213 Minimum Principle, 133 for harmonic functions, 231 for superharmonic functions, 264 Mittag-Leffler Theorem, 295 modulus, 18 Monodromy, 68 Montel’s Theorem, 187 Morera’s Theorem, 124 Morera, G., 4, 14, 115, 124, 172 Munkres, J. R., 15, 129 N Newton, I., 12 normal derivative, 241 notation, 15 number, 15 algebraic, 54 transcendental, 54 O order of a function at a point, 71 pole, 70 zero, 69 P pairing, 88 parallelogram law, 19

Index partition of an interval, 84 path, 83 closed, 83 simple, 129 continuous, 83 differentiable, 83 geodesic, 215 homologous, 128 homotopic to a point, 97 homotopic as closed paths, 97 homotopic with fixed end points, 97 index of, 96 integral, 84 parameterized, 83 piecewise differentiable, 85 range, 83 reparameterization, 84 simple, 83 traversed backward, 85 winding number, 96 pdp: piecewise differentiable path, 85 period of a function, 58 Perron family, 249 Perron’s principle, 250 Perron, O., 249, 250 Picard’s Theorem, 145 Picard, C. E., 6, 13, 145 Pick, G., 135 Poincar´e metric, 212 Poincar´e, H., 13, 199 point at infinity, 16 Poisson formula, 232 classical reformulation, 240 Fourier series interpretation, 239 geometric interpretation, 237 kernel, 233 pole simple, 70 pr. v., 158, 162, see principal value primitive, 87 along continuous function, 98 along path, 93 principal part, 144 at 1, 144 principal value, 158, 162 product absolutely convergent, 294 product of two pdp’s, 85 PSL.2; C/, 201 PSL.2; R/, 206 punctured disc, 100

305 R

b R, 204 radius of convergence, 44 range of a path, 83 ratio test, 46 rational function, 30 region, 27 .xy/-simple, 90 convex, 100 simply connected, 99 symmetric, 244 removable singularity, 70 residue, 147 at infinity, 149 Residue Theorem, 148, 164 , 176 Riemann map, 207 sphere, 16 Riemann -function, 267 Riemann Mapping Theorem, 207, 256, 259 Version 2, 259 Riemann, B., 3, 6, 10, 13, 171, 174, 190, 199 root test, 46 Rouch´e’s Theorem, 152, 165 Rouch´e, E., 139, 152 Roy, R., 6 Runge’s Theorem, 188 rational approximation, 188 Runge, C., 4, 171, 188, 190

S S 1 , 204 S 2 , 77 schlicht mapping, 174 Schwarz’s lemma, 133 invariant form, 135 Schwarz, H. A., 13, 14, 133, 135, 136, 206, 211, 219, 236, 237 Schwarz-Pick Lemma, 135, 224 sequence Cauchy, 23 convergent, 23 series absolutely convergent, 40 convergent at a point, 40 divergent, 40 geometric, 42 infinite, 40 Laurent, 139, 140 Maclaurin, 124 normally convergent, 40

306 series (cont.) pointwise convergent, 40 power, 41 Taylor, 122 uniformly convergent, 40 simple closed curve, 83 simple closed path, 129 simply connected domain, 99 singularity, 70 essential, 144 isolated, 70, 143 at 1, 144 pole, 70, 144 principal part, 144 removable, 143 SL.2; C/, 201 Sokhotskii, Y., 13 stereographic projection, 77 Stirling’s formula, 284, 287 strongly bounded, 183 subharmonic, 245 sup, 43 superharmonic, 246 supremum, 43 symmetric w.r.t. the real axis, 244

Index T Taylor, B., 4, 11, 12, 122, 124, 125 total differential of a function, 87 transcendental function, 54 triangular inequality, 19 U U.c; / D Uc ./, 24 V Vitali’s theorem, 187 W Weierstrass elementary functions, 272 Weierstrass, K., 4, 7, 10, 12, 14, 25, 41, 145, 146, 172, 272 winding number, 100 of a closed curve with respect to a point, 96 Z zero simple, 69