Complex Function Theory

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Second Edition

Donald Sarason

AMERICAN MATHEMATICAL SOCIETY http://dx.doi.org/10.1090/mbk/049

Complex Function Theory Second Edition

Donald Sarason

AMERICAN MATHEMATICAL SOCIETY 2000 Mathematics Subject Classiﬁcation. Primary 30–01.

Front Cover: The ﬁgure on the front cover is courtesy of Andrew D. Hwang.

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Library of Congress Cataloging-in-Publication Data Sarason, Donald. Complex function theory / Donald Sarason. — 2nd ed. p. cm. Includes index. ISBN-13: 978-0-8218-4428-1 (alk. paper) ISBN-10: 0-8218-4428-8 (alk. paper) 1. Functions of complex variables. I. Title. QA331.7.S27 2007 515.9—dc22 2007060552

Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2007 by the American Mathematical Society. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 121110090807 Contents

Preface to the Second Edition ix

Preface to the First Edition xi

Chapter I. Complex Numbers 1 §I.1. Deﬁnition of C 2 §I.2. Field Axioms 2 §I.3. Embedding of R in C. The Imaginary Unit 3 §I.4. Geometric Representation 3 §I.5. Triangle Inequality 4 §I.6. Parallelogram Equality 5 §I.7. Plane Geometry via Complex Numbers 5 §I.8. C as a Metric Space 6 §I.9. Polar Form 6 §I.10. De Moivre’s Formula 7 §I.11. Roots 8 §I.12. Stereographic Projection 9 §I.13. Spherical Metric 10 §I.14. Extended Complex Plane 11

Chapter II. Complex Diﬀerentiation 13 §II.1. Deﬁnition of the Derivative 13 §II.2. Restatement in Terms of Linear Approximation 14 §II.3. Immediate Consequences 14

iii iv Contents

§II.4. Polynomials and Rational Functions 15 §II.5. Comparison Between Differentiability in the Real and Complex Senses 15 §II.6. Cauchy-Riemann Equations 16 §II.7. Sufficient Condition for Differentiability 17 §II.8. Holomorphic Functions 17 §II.9. Complex Partial Differential Operators 18 §II.10. Picturing a Holomorphic Function 19 §II.11. Curves in C 20 §II.12. Conformality 21 §II.13. Conformal Implies Holomorphic 22 §II.14. Harmonic Functions 23 §II.15. Holomorphic Implies Harmonic 24 §II.16. Harmonic Conjugates 24 Chapter III. Linear-Fractional Transformations 27 §III.1. Complex projective space 27 §III.2. Linear-fractional transformations 28 §III.3. Conformality 29 §III.4. Fixed points 29 §III.5. Three-fold transitivity 29 §III.6. Factorization 30 §III.7. Clircles 31 §III.8. Preservation of clircles 31 §III.9. Analyzing a linear-fractional transformation—an example 32 Chapter IV. Elementary Functions 35 §IV.1. Definition of ez 35 §IV.2. Law of Exponents 36 §IV.3. ez is holomorphic 36 §IV.4. Periodicity 37 §IV.5. ez as a map 37 §IV.6. Hyperbolic functions 38 §IV.7. Zeros of coshz and sinhz.38 §IV.8. Trigonometric functions 39 §IV.9. Logarithms 40 §IV.10. Branches of arg z and log z.40 Contents v

§IV.11. log z as a holomorphic function 41 §IV.12. Logarithms of holomorphic functions 42 §IV.13. Roots 43 §IV.14. Inverses of holomorphic functions 43 §IV.15. Inverse trigonometric functions 44 §IV.16. Powers 45 §IV.17. Analytic continuation and Riemann surfaces 45 Chapter V. Power Series 49 §V.1. Inﬁnite Series 49 §V.2. Necessary Condition for Convergence 49 §V.3. Geometric Series 50 §V.4. Triangle Inequality for Series 50 §V.5. Absolute Convergence 50 §V.6. Sequences of Functions 51 §V.7. Series of Functions 51 §V.8. Power Series 52 §V.9. Region of Convergence 53 §V.10. Radius of Convergence 54 §V.11. Limits Superior 54 §V.12. Cauchy-Hadamard Theorem 55 §V.13. Ratio Test 56 §V.14. Examples 57 §V.15. Diﬀerentiation of Power Series 58 §V.16. Examples 60 §V.17. Cauchy Product 61 §V.18. Division of Power Series 63 Chapter VI. Complex Integration 65 §VI.1. Riemann Integral for Complex-Valued Functions 65 §VI.2. Fundamental Theorem of Calculus 66 §VI.3. Triangle Inequality for Integration 66 §VI.4. Arc Length 67 §VI.5. The Complex Integral 67 §VI.6. Integral of a Derivative 68 §VI.7. An Example 68 vi Contents

§VI.8. Reparametrization 69 §VI.9. The Reverse of a Curve 70 §VI.10. Estimate of the Integral 71 §VI.11. Integral of a Limit 71 §VI.12. An Example 71 Chapter VII. Core Versions of Cauchy’s Theorem, and Consequences 75 §VII.1. Cauchy’s Theorem for a Triangle 75 §VII.2. Cauchy’s Theorem for a Convex Region 78 §VII.3. Existence of a Primitive 78 §VII.4. More Definite Integrals 79 §VII.5. Cauchy’s Formula for a Circle 79 §VII.6. Mean Value Property 81 §VII.7. Cauchy Integrals 82 §VII.8. Implications for Holomorphic Functions 83 §VII.9. Cauchy Product 84 §VII.10. Converse of Goursat’s Lemma 85 §VII.11. Liouville’s Theorem 86 §VII.12. Fundamental Theorem of Algebra 86 §VII.13. Zeros of Holomorphic Functions 87 §VII.14. The Identity Theorem 89 §VII.15. Weierstrass Convergence Theorem 89 §VII.16. Maximum Modulus Principle 90 §VII.17. Schwarz’s Lemma 91 §VII.18. Existence of Harmonic Conjugates 93 §VII.19. Infinite Differentiability of Harmonic Functions 94 §VII.20. Mean Value Property for Harmonic Functions 94 §VII.21. Identity Theorem for Harmonic Functions 94 §VII.22. Maximum Principle for Harmonic Functions 95 §VII.23. Harmonic Functions in Higher Dimensions 95 Chapter VIII. Laurent Series and Isolated Singularities 97 §VIII.1. Simple Examples 97 §VIII.2. Laurent Series 98 §VIII.3. Cauchy Integral Near ∞ 99 §VIII.4. Cauchy’s Theorem for Two Concentric Circles 100 Contents vii

§VIII.5. Cauchy’s Formula for an Annulus 101 §VIII.6. Existence of Laurent Series Representations 101 §VIII.7. Isolated Singularities 102 §VIII.8. Criterion for a Removable Singularity 105 §VIII.9. Criterion for a Pole 105 §VIII.10. Casorati-Weierstrass Theorem 106 §VIII.11. Picard’s Theorem 106 §VIII.12. Residues 106 Chapter IX. Cauchy’s Theorem 109 §IX.1. Continuous Logarithms 109 §IX.2. Piecewise C1 Case 110 §IX.3. Increments in the Logarithm and Argument Along a Curve 110 §IX.4. Winding Number 111 §IX.5. Case of a Piecewise-C1 Curve 111 §IX.6. Contours 113 §IX.7. Winding Numbers of Contours 114 §IX.8. Separation Lemma 115 §IX.9. Addendum to the Separation Lemma 117 §IX.10. Cauchy’s Theorem 118 §IX.11. Homotopy 119 §IX.12. Continuous Logarithms—2-D Version 119 §IX.13. Homotopy and Winding Numbers 120 §IX.14. Homotopy Version of Cauchy’s Theorem 121 §IX.15. Runge’s Approximation Theorem 121 §IX.16. Second Proof of Cauchy’s Theorem 122 §IX.17. Sharpened Form of Runge’s Theorem 123 Chapter X. Further Development of Basic Complex Function 125 §X.1. Simply Connected Domains 125 §X.2. Winding Number Criterion 126 §X.3. Cauchy’s Theorem for Simply Connected Domains 126 §X.4. Existence of Primitives 127 §X.5. Existence of Logarithms 127 §X.6. Existence of Harmonic Conjugates 128 §X.7. Simple Connectivity and Homotopy 128 viii Contents

§X.8. The Residue Theorem 129 §X.9. Cauchy’s Formula 130 §X.10. More Definite Integrals 130 §X.11. The Argument Principle 137 §X.12. Rouché’s Theorem 138 §X.13. The Local Mapping Theorem 140 §X.14. Consequences of the Local Mapping Theorem 140 §X.15. Inverses 141 §X.16. Conformal Equivalence 141 §X.17. The Riemann Mapping Theorem 142 §X.18. An Extremal Property of Riemann Maps 143 §X.19. Stieltjes-Osgood Theorem 144 §X.20. Proof of the Riemann Mapping Theorem 146 §X.21. Simple Connectivity Again 148 Appendix 1. Sufficient condition for differentiability 151 Appendix 2. Two instances of the chain rule 153 Appendix 3. Groups, and linear-fractional transformations 155 Appendix 4. Differentiation under the integral sign 157 References 159 Index 161 Preface to the Second Edition

As is usual in a second edition, various minor flaws that had crept into the first edition have been corrected. Certain topics are now presented more clearly, it is hoped, after being rewritten and/or reorganized. The treatment has been expanded only slightly: there is now a section on division of power series, and a brief discussion of homotopy. (The latter topic was relegated to a couple of exercises in the first edition.) Four appendices have been added; they contain needed background which, experience has shown, is not possessed nowadays by all students taking introductory complex analysis. In this edition, the numbers of certain exercises are preceded by an asterisk. The asterisk indicates that the exercise will be referred to later in the text. In many cases the result established in the exercise will be needed as part of a proof. I am indebted to a number of students for detecting minor errors in the first edition, and to Robert Burckel and Bjorn Poonen for their valuable comments. Special thanks go to George Bergman and his eagle eye. George, while teaching from the first edition, read it carefully and provided a long list of suggested improvements, both in exposition and in typography. I owe my colleague Henry Helson, the publisher of the first edition, thanks for encouraging me to publish these Notes in the first place, and for his many kindnesses during our forty-plus years together at Berkeley.

ix x Preface to the Second Edition

The ﬁgures from the ﬁrst edition have been redrawn by Andrew D. Hwang, whose generous help is greatly appreciated. I am indebted to Ed- ward Dunne and his AMS colleagues for the patient and professional way they shepherded my manuscript into print. Finally, as always, I am deeply grateful to my wife, Mary Jennings, for her constant support, in particular, for applying her TEXnical skills to this volume.

Berkeley, California January 26, 2007 Preface to the First Edition1

These are the notes for a one-semester introductory course in the theory of functions of a complex variable. The aim of the notes is to help students of mathematics and related sciences acquire a basic understanding of the subject, as a preparation for pursuing it at a higher level or for employing it in other areas. The approach is standard and somewhat old-fashioned. The user of the notes is assumed to have a thorough grounding in basic real analysis, as one can obtain, for example, from the book of W. Rudin cited in the list of references. Notions like metric, open set, closed set, inte- rior, boundary, limit point, and uniform convergence are employed without explanation. Especially important are the notions of a connected set and of the connected components of a set. Basic notions from abstract algebra also occur now and then. These are all concepts that ordinarily are familiar to students by the time they reach complex function theory. As these notes are a rather bare-bones introduction to a vast subject, the student or instructor who uses them may well wish to supplement them with other references. The notes owe a great deal to the book by L. V. Ahlfors and to the book by S. Saks and A. Zygmund, which, together with the teaching of George Piranian, were largely responsible for my own love aﬀair with the subject. Several other excellent books are mentioned in the list of references.

1 The ﬁrst edition was published by Henry Helson under the title Notes on Complex Function Theory.

xi xii Preface to the First Edition

The notes contain only a handful of pictures, not enough to do justice to the strong geometric component of complex function theory. The user is advised to make his or her own sketches as an aid to visualization. Thanks go to Andrew Hwang for drawing the pictures. The approach in these notes to Cauchy’s theorem, the central theorem of the subject, follows the one used by Ahlfors, attributed by him to A. F. Beardon. An alternative approach based on Runge’s approximation theorem, adapted from Saks and Zygmund, is also presented. The terminology used in the notes is for the most part standard. Two exceptions need mention. Some authors use the term “region” specifically to refer to an open connected subset of the plane. Here the term is used, from time to time, in a less formal way. On the other hand, the term “contour” is used in the notes in a specific way not employed by other authors. I wish to thank my Berkeley Math H185 class in the Spring Semester, 1994, for pointing out a number of corrections to the prepublication version of the notes, and my wife, Mary Jennings, who read the first draft of the notes and helped me to anticipate some of the questions students might raise as they work through this material. She also typed the manuscript. I deeply appreciate her assistance and support. The notes are dedicated to her.

Berkeley, California June 8, 1994 References

1. Ahlfors, L. V., Complex Analysis, McGraw-Hill, 1979. 2. Axler, S., Bourdon, P., and Ramey, W., Harmonic Function Theory, Springer-Verlag, 1992. 3. Burckel, R. B., An Introduction to Classical Complex Analysis, Vol. 1,Birkh¨auser, 1979. 4. Conway, J. B., Functions of One Complex Variable, Springer-Verlag, 1978. 5. Marsden, J. E., and Hoﬀman, M. J., Basic Complex Analysis, W. H. Freeman, 1987. 6. Newman, M. H. A., Elements of the Topology of Plane Sets of Points, Cambridge, 1961. 7. Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, 1976. 8. Saks, S., and Zygmund, A., Analytic Functions, Elsevier, 1971.

159

Index

absolute convergence, 50 complex conjugate, 4 absolute value, 3 complex integral, 67 analytic, 17 complex number system, 1 analytic continuation, 45 complex partial differential operators, 18 arc length, 67 complex plane, 3 argument, 6 complex projective space, 27 argument principle, 137, 138 conformal automorphism, 142 conformal equivalence, 141 conformal map, 22 Bernoulli numbers, 104 contour, 113 bijection, 28 simple, 114 branch convergence function, 44 local uniform, 51 of arg z,40 of an infinite series, 49 of log z,40 uniform, 51 of a holomorphic function, 42 cross ratio, 30, 32 of an inverse, 44 curve, 20 of the logarithm, 42 differentiable, 20 Burckel, R. B., 85 regular, 20

Carathéodory, C., 143 De Moivre’s formula, 7 Casorati-Weierstrass theorem, 106 differentation of power series, 58 Cauchy integral, 83, 99 differentiable Cauchy product, 61, 84 in the complex sense, 13 Cauchy’s formula, 130 in the real sense, 15, 151 for a circle, 79 differentiation under the integral sign, 157 Cauchy’s formula for an annulus, 101 division of power series, 63 Cauchy’s theorem, 75, 109, 118, 121, 122 domain, 125 for a convex region, 78 for a triangle, 75 for simply connected domains, 126 elementary function, 35 for two concentric circles, 100 entire function, 86 Cauchy-Hadamard theorem, 55 Euler, L., 35 z Cauchy-Riemann equations, 16 exponential function (e ), 35 in polar coordinates, 16 extended complex plane, 11 chain rule, 14, 153 clircle, 31 Fejér, L., 143

161 162 Index

field axioms, 2 locally uniformly bounded, 144 fundamental theorem of algebra, 1, 86 logarithm, 40 fundamental theorem of calculus, 66 of a continuous function, 109, 119 of a holomorphic function, 42, 127 Gauss, C. F., 1 logarithmic derivative, 42 general linear group for dimension two, 29 geometric function theory, 148 maximum modulus principle, 90 geometric series, 50 maximum principle for harmonic functions, Goursat’s lemma, 75, 85 95 Goursat, E., 75 mean value property group, 155 for harmonic functions, 94 for holomorphic functions, 81 harmonic conjugate, 24, 93, 128 meromorphic function, 138 harmonic function, 23 modulus, 3 Herglotz’s formula, 137 Morera’s theorem, 85 holomorphic, 17 Morera, G., 85 homothetic map, 19 multiplicity of a zero, 87 homotopy, 119 multiply connected domain, 128 Hurwitz’s theorem, 139 hyperbolic functions, 38 normal family, 144 identity theorem, 89 order identity theorem for harmonic functions, 94 of a pole, 103 imaginary axis, 3 of a zero, 87 imaginary part, 4 imaginary unit, 3 parallelogram equality, 5 increment in the argument, 110 partial fraction decomposition, 104 increment in the logarithm, 110 partial sum, 49 index (of a point with respect to a curve), Picard’s theorem, 106 111 Picard, E., 106 inverse of a holomorphic function, 43, 141 Pick’s lemma, 92 inverse trigonometric functions, 44 piecewise C1,66 inversion, 30 piecewise continuous, 65 isolated singularity, 102 point at infinity, 11 essential, 102 Poisson’s formula, 137 pole, 102 polar form, 6 removable, 102 pole, 102 isomorphism, 155 order, 103 simple, 103 Koebe, P., 143 polynomials, 15 potential theory, 143 Laplace’s equation, 23, 25 power series, 52 in polar coordinates, 24 center, 52 Laurent series, 98 coefficient, 52 annulus of convergence, 98 primitive, 78, 127 convergence, 98 principle value of arg, 6 principal part, 102 product rule, 14 law of exponents, 35, 36, 62 purely imaginary number, 3 limit superior, 54 linear-fractional transformation, 27, 155 quotient rule, 14 conjugate, 31 factorization, 30 radius of convergence, 54 fixed points, 29 ratio test, 56 preservation of clircles, 31 rational functions, 15 three-fold transitivity, 29 real axis, 3 Liouville’s theorem, 86 real part, 4 local mapping theorem, 140 reparametrization, 69 Index 163

residue, 106 residue theorem, 113, 129 reverse of a curve, 70 Riemann, 47 Riemann integral for complex-valued functions, 65 Riemann map, 143 Riemann mapping theorem, 142, 146 Riemann sphere, 11 Riemann surface, 45 Riemann, G. F. B., 11 Riesz, F., 143 roots, 43 Rouch´e’s theorem, 138 Runge’s approximation theorem, 121, 123

Schmiegungsverfahren, 147 Schwarz’s lemma, 91 separation lemma, 115 simple connectivity and homotopy, 128, 148 simply connected domain, 125 spherical metric, 10 stereographic projection, 9 Stieltjes-Osgood theorem, 144 sum rule, 14 tangent direction, 21 Toeplitz matrix, 63 triangle inequality, 4, 50 triangle inequality for integration, 66 trigonometric functions, 39 uniformly Cauchy, 51 univalent, 44

Vitali’s theorem, 146

Weierstrass convergence theorem, 89 Weierstrass, K., 121 winding number, 111, 114 zero (of a holomorphic function) order, 87 simple, 88

Complex Function Theory is a concise and rigorous introduction to the theory of functions of a complex variable. Written in a classical style, it is in the spirit of the books by Ahlfors and by Saks and Zygmund. Being designed for a one-semester course, it is much shorter than many of the standard texts. Sarason covers the basic material through Cauchy’s theorem and applications, plus the Riemann mapping theorem. It is suitable for either an introductory graduate course or an undergraduate course for students with adequate preparation. The ﬁ rst edition was published with the title Notes on Complex Function Theory.

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