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Complex Function Theory Complex Function Theory 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. The Hindustan Book Agency has the rights to distribute this book in India, Bangladesh, Bhutan, Nepal, Pakistan, Sri Lanka, and the Maldives. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-49 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. Rouch´e’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.
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