Complex Analysis This user-friendly textbook introduces complex analysis at the beginning graduate or advanced undergraduate level. Unlike other textbooks, it follows Weierstrass’s approach, stressing the importance of power series expansions instead of starting with the Cauchy integ- ral formula, an approach that illuminates many important concepts. This view allows readers to quickly obtain and understand many fundamental results of complex analysis, such as the maximum principle, Liouville’s theorem and Schwarz’s lemma. The book covers all the essential material on complex analysis, and includes several elegant proofs that were recently discovered. It includes the zipper algorithm for computing conformal maps, a constructive proof of the Riemann mapping theorem, and culminates in a complete proof of the uniformization theorem. Aimed at students with some undergraduate background in real analysis, though not Lebesgue integration, this classroom-tested textbook will teach the skills and intuition necessary to understand this important area of mathematics. Donald E. Marshallis Professor of Mathematics at the University of Washington. He received his PhD from UCLA in 1976. Professor Marshall is a leading complex analyst with a very strong research record that has been continuously funded throughout his career. He has given invited lectures in over a dozen countries. He is coauthor of the research-level monograph Harmonic Measure, published by Cambridge University Press. CAMBRIDGE MATHEMATICAL TEXTBOOKS Cambridge Mathematical Textbooks is a program of undergraduate and beginning graduate-level text- books for core courses, new courses, and interdisciplinary courses in pure and applied mathematics. These texts provide motivation with plenty of exercises of varying difficulty, interesting examples, modern applications, and unique approaches to the material. Advisory Board John B. Conway,George Washington University Gregory F. Lawler,University of Chicago John M. Lee,University of Washington John Meier,Lafayette College Lawrence C. Washington,University of Maryland, College Park A complete list of books in the series can be found at www.cambridge.org/mathematics Recent titles include the following: Chance, Strategy, and Choice: An Introduction to the Mathematics of Games and Elections, S. B. Smith Set Theory: A First Course, D. W. Cunningham Chaotic Dynamics: Fractals, Tilings, and Substitutions, G. R. Goodson A Second Course in Linear Algebra, S. R. Garcia & R. A. Horn Introduction to Experimental Mathematics, S. Eilers & R. Johansen Exploring Mathematics: An Engaging Introduction to Proof, J. Meier & D. Smith A First Course in Analysis, J. B. Conway Introduction to Probability, D. F. Anderson, T. Seppäläinen & B. Valkó Linear Algebra, E. S. Meckes & M. W. Meckes A Short Course in Differential Topology, B. I. Dundas Abstract Algebra with Applications, A. Terras Complex Analysis Donald E. Marshall University of Washington, Seattle, WA, USA University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title:www.cambridge.org/9781107134829 © Donald E. Marshall 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed in the United States of America by Sheridan Books, Inc. A catalog record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Marshall, Donald E. (Donald Eddy), 1947– author. Title: Complex analysis / Donald E. Marshall. Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2019. Identifiers: LCCN 2018029851 | ISBN 9781107134829 (Hardback) Subjects: LCSH: Functions of complex variables – Textbooks. | Mathematical analysis – Textbooks. Classification: LCC QA331.7 M365 2019 | DDC 515/.9–dc23 LC record available athttps://lccn.loc.gov/2018029851 ISBN 978-1-107-13482-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents List of Figures pageviii Preface xi Prerequisites xv PART I 1 Preliminaries 3 1.1 Complex Numbers 3 1.2 Estimates 6 1.3 Stereographic Projection 8 1.4 Exercises 10 2 Analytic Functions 13 2.1 Polynomials 13 2.2 Fundamental Theorem of Algebra and Partial Fractions 15 2.3 Power Series 17 2.4 Analytic Functions 20 2.5 Elementary Operations 23 2.6 Exercises 27 3 The Maximum Principle 31 3.1 The Maximum Principle 31 3.2 Local Behavior 33 3.3 Growth onCandD 35 3.4 Boundary Behavior 38 3.5 Exercises 41 4 Integration and Approximation 43 4.1 Integration on Curves 43 4.2 Equivalence of Analytic and Holomorphic 47 4.3 Approximation by Rational Functions 53 4.4 Exercises 60 5 Cauchy’s Theorem 63 5.1 Cauchy’s Theorem 63 5.2 Winding Number 64 5.3 Removable Singularities 70 5.4 Laurent Series 72 vi Contents 5.5 The Argument Principle 76 5.6 Exercises 78 6 Elementary Maps 81 6.1 Linear Fractional Transformations 82 6.2 Exp and Log 86 6.3 Power Maps 87 6.4 The Joukovski Map 89 6.5 Trigonometric Functions 91 6.6 Constructing Conformal Maps 93 6.7 Exercises 98 PART II 7 Harmonic Functions 105 7.1 The Mean-Value Property and the Maximum Principle 105 7.2 Cauchy–Riemann and Laplace Equations 111 7.3 Hadamard, Lindelöf and Harnack 114 7.4 Exercises 118 8 Conformal Maps and Harmonic Functions 123 8.1 The Geodesic Zipper Algorithm 123 8.2 The Riemann Mapping Theorem 129 8.3 Symmetry and Conformal Maps 132 8.4 Conformal Maps to Polygonal Regions 135 8.5 Exercises 137 9 Calculus of Residues 141 9.1 Contour Integration and Residues 141 9.2 Some Examples 142 9.3 Fourier and Mellin Transforms 145 9.4 Series via Residues 149 9.5 Laplace and Inverse Laplace Transforms 150 9.6 Exercises 154 10 Normal Families 156 10.1 Normality and Equicontinuity 156 10.2 Riemann Mapping Theorem Revisited 162 10.3 Zalcman, Montel and Picard 164 10.4 Exercises 168 11 Series and Products 170 11.1 Mittag-Leffler’s Theorem 170 11.2 Weierstrass Products 175 11.3 Blaschke Products 181 Contents vii 11.4 The Gamma and Zeta Functions 184 11.5 Exercises 189 PART III 12 Conformal Maps to Jordan Regions 195 12.1 Some Badly Behaved Regions 195 12.2 Janiszewski’s Lemma 197 12.3 Jordan Curve Theorem 199 12.4 Carathéodory’s Theorem 201 12.5 Exercises 205 13 The Dirichlet Problem 207 13.1 Perron Process 207 13.2 Local Barriers 209 13.3 Riemann Mapping Theorem Again 212 13.4 Exercises 214 14 Riemann Surfaces 216 14.1 Analytic Continuation and Monodromy 216 14.2 Riemann Surfaces and Universal Covers 220 14.3 Deck Transformations 226 14.4 Exercises 227 15 The Uniformization Theorem 230 15.1 The Modular Function 230 15.2 Green’s Function 232 15.3 Simply-Connected Riemann Surfaces 237 15.4 Classification of All Riemann Surfaces 243 15.5 Exercises 243 16 Meromorphic Functions on a Riemann Surface 246 16.1 Existence of Meromorphic Functions 246 16.2 Properly Discontinuous Groups onC∗ andC 246 16.3 Elliptic Functions 248 16.4 Fuchsian Groups 251 16.5 Blaschke Products and Convergence Type 253 16.6 Exercises 257 Appendix 260 A.1 Fifteen Conditions Equivalent to Analytic 260 A.2 Program for Color Pictures 261 Bibliography 267 Index 269 Figures 1.1 Cartesian and polar representations of complex numbers. page3 1.2 Addition. 4 1.3 Multiplication. 4 1.4 Triangle inequality. 6 1.5 Stereographic projection. 9 2.1 The power map. 14 2.2 p(z0+ζ) lies in a small disk of radiuss=C|ζ|k+1<r = |bk||ζ|k. 14 2.3 Convergence of a power series. 19 2.4 Proof of Theorem2.7. 21 3.1 Conformality. 34 3.2 Local behavior of an analytic function. 34 3.3 Stolz angle,±α(ζ). 38 4.1 A curve,γ. 43 4.2 A closed curve,γ =γ1+γ2. 45 4.3 Integrals around squares. 45 4.4 Proof of Morera’s theorem. 52 4.5 Ifais outsideS. 53 4.6 The squareSand its circumscribed circleC. 53 4.7 A union of closed squares coveringK and contained inU. 55 5.1 Integration on a cycle. 64 5.2 Calculating the winding number. 66 5.3 Parity: which red point is not in the same component as the other two? 67 5.4 n(σ, 0)= n(σ, 1) =0. 68 5.5 Proof of Rouché’s theorem. 77 6.1 A rational function. 81 6.2 The Cayley transform. 85 6.3 Mapsez and logz. 86 6.4 Mapsz4andz1/4. 87 6.5 The mapz1+iand its inversez(1−i)/2. 88 6.6 The map (z+1/z)/2. 89 6.7 Some inverses of (z+1/z)/2. 91 6.8 The map cos(z). 92 6.9 Four intersections of disks meeting at angleπ/4. 93 6.10 Mapping a circularly slit half-plane. 95 6.11 Below a parabola. 95 6.12 Exterior of an ellipse. 96 List of Figures ix 6.13 Region between the branches of a hyperbola. 96 6.14 “World’s greatest function,” exp((z+1)/(z−1)). 98 6.15 Tangent circles. 101 7.1 The angleθ(z) as a harmonic function. 115 8.1 Construction ofγ4andϕ4. 124 8.2 Proof of Jørgensen’s theorem. 126 8.3 A disk-chain. 127 8.4 A smaller disk²⊂Dj. 128 8.5 Conformal mapping using the geodesic zipper algorithm. 129 8.6 Turn angles at the vertices of a polygon. 135 9.1 Half-disk contour. 144 9.2 Rectangle contour. 145 9.3 Contour avoiding poles onR. 147 9.4 Keyhole contour. 148 9.5 Enlarged half-disk. 153 11.1 ζ(1 +iy),−40 ≤y≤ 40. 187 2 12.1 E2 and connecting segments. 196 12.2 Constructing a dense set of spirals. 196 12.3 Constructing the Lakes of Wada. 197 12.4 Janiszewksi’s lemma. 199 12.5 Boundary of a component. 200 12.6 A modification ofJ. 200 12.7 Crosscutsγ andϕ(γ ). 202 δn δn 12.8 Proof thatϕis one-to-one. 203 13.1 Proof of Bouligand’s lemma. 211 14.1 Homotopy ofγ0toγ1. 218 14.2 Parking-lot ramp surface. 220 15.1 Reflections of a circular triangleT. 231 15.2 Harmonic measure of∂rUinW\rU. 234 15.3 The surfaceWt. 241 16.1 OnCσ,ρ(z, 0)=ρ(z,σ(0)). 252 16.2 Normal fundamental domain. 253