Table Of Content

245 Graduate Texts in Mathematics Editorial Board S. Axler K.A. Ribet Graduate Texts in Mathematics 1 Takeuti/Zaring. Introduction to Axiomatic 39 Arveson. An Invitation to C-Algebras. Set Theory. 2nd ed. 40 Kemeny/Snell/Knapp. Denumerable 2 Oxtoby. Measure and Category. 2nd ed. Markov Chains. 2nd ed. 3 Schaefer. Topological Vector Spaces. 41 Apostol. Modular Functions and Dirichlet 2nd ed. Series in Number Theory. 2nd ed. 4 Hilton/Stammbach. A Course in 42 J.-P. Serre. Linear Representations of Finite Homological Algebra. 2nd ed. Groups. 5 Mac Lane. Categories for the Working 43 Gillman/Jerison. Rings of Continuous Mathematician. 2nd ed. Functions. 6 Hughes/Piper. Projective Planes. 44 Kendig. Elementary Algebraic Geometry. 7 J.-P. Serre. A Course in Arithmetic. 45 Loève. Probability Theory I. 4th ed. 8 Takeuti/Zaring. Axiomatic Set Theory. 46 Loève. Probability Theory II. 4th ed. 9 Humphreys. Introduction to Lie Algebras 47 Moise. Geometric Topology in Dimensions and Representation Theory. 2 and 3. 10 Cohen. A Course in Simple Homotopy 48 Sachs/Wu. General Relativity for Theory. Mathematicians. 11 Conway. Functions of One Complex 49 Gruenberg/Weir. Linear Geometry. Variable I. 2nd ed. 2nd ed. 12 Beals. Advanced Mathematical Analysis. 50 Edwards. Fermat’s Last Theorem. 13 Anderson/Fuller. Rings and Categories 51 Klingenberg. A Course in Differential of Modules. 2nd ed. Geometry. 14 Golubitsky/Guillemin. Stable Mappings 52 Hartshorne. Algebraic Geometry. and Their Singularities. 53 Manin. A Course in Mathematical Logic. 15 Berberian. Lectures in Functional Analysis 54 Graver/Watkins. Combinatorics with and Operator Theory. Emphasis on the Theory of Graphs. 16 Winter. The Structure of Fields. 55 Brown/Pearcy. Introduction to Operator 17 Rosenblatt. Random Processes. 2nd ed. Theory I: Elements of Functional Analysis. 18 Halmos. Measure Theory. 56 Massey. Algebraic Topology: An 19 Halmos. A Hilbert Space Problem Book. Introduction. 2nd ed. 57 Crowell/Fox. Introduction to Knot Theory. 20 Husemoller. Fibre Bundles. 3rd ed. 58 Koblitz. p-adic Numbers, p-adic Analysis, 21 Humphreys. Linear Algebraic Groups. and Zeta-Functions. 2nd ed. 22 Barnes/Mack. An Algebraic Introduction 59 Lang. Cyclotomic Fields. to Mathematical Logic. 60 Arnold. Mathematical Methods in 23 Greub. Linear Algebra. 4th ed. Classical Mechanics. 2nd ed. 24 Holmes. Geometric Functional Analysis 61 Whitehead. Elements of Homotopy and Its Applications. Theory. 25 Hewitt/Stromberg. Real and Abstract 62 Kargapolov/Merizjakov. Fundamentals Analysis. of the Theory of Groups. 26 Manes. Algebraic Theories. 63 Bollobas. Graph Theory. 27 Kelley. General Topology. 64 Edwards. Fourier Series. Vol. I. 2nd ed. 28 Zariski/Samuel. Commutative Algebra. Vol. I. 65 Wells. Differential Analysis on Complex 29 Zariski/Samuel. Commutative Algebra. Vol. II. Manifolds. 2nd ed. 30 Jacobson. Lectures in Abstract Algebra I. 66 Waterhouse. Introduction to Affi ne Group Basic Concepts. Schemes. 31 Jacobson. Lectures in Abstract Algebra II. 67 Serre. Local Fields. Linear Algebra. 68 Weidmann. Linear Operators in Hilbert 32 Jacobson. Lectures in Abstract Algebra III. Spaces. Theory of Fields and Galois Theory. 69 Lang. Cyclotomic Fields II. 33 Hirsch. Differential Topology. 70 Massey. Singular Homology Theory. 34 Spitzer. Principles of Random Walk. 2nd ed. 71 Farkas/Kra. Riemann Surfaces. 2nd ed. 35 Alexander/Wermer. Several Complex 72 Stillwell. Classical Topology and Variables and Banach Algebras. 3rd ed. Combinatorial Group Theory. 2nd ed. 36 Kelley/Namioka et al. Linear Topological 73 Hungerford. Algebra. Spaces. 74 Davenport. Multiplicative Number Theory. 37 Monk. Mathematical Logic. 3rd ed. 38 Grauert/Fritzsche. Several Complex 75 Hochschild. Basic Theory of Algebraic Variables. Groups and Lie Algebras. (continued after index) Jane P. Gilman Irwin Kra Rubí E. Rodríguez Complex Analysis In the Spirit of Lipman Bers Jane P. Gilman Irwin Kra Rubí E. Rodríguez Department of Math for America Pontifi cia Universidad Mathematics and 50 Broadway, 23 Floor Católica de Chile Computer Science New York, NY 10004 Santiago Rutgers University and Chile Newark, NJ 07102 Department of Mathematics [email protected] USA State University of New York [email protected] at Stony Brook Stony Brook, NY 11794 USA [email protected] Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California at Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA [email protected] [email protected] ISBN: 978-0-387-74714-9 e-ISBN: 978-0-387-74715-6 Library of Congress Control Number: 2007940048 Mathematics Subject Classifi cation (2000): 30-xx 32-xx © 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identifi ed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com To the memory of Mary and Lipman Bers Preface 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. Complex analysis is also known as function theory. In this text we address the theory of complex-valued functions of a single complex variable. This is a prerequisite for the study of many current and rapidlydeveloping areasofmathematics, includingthetheoryofseveral andinfinitely many complex variables, the theory of groups, hyperbolic geometry and three-manifolds, and number theory. Complex analysis has connections and applications to many other many other subjects in mathematics, and also to other sciences as an area where the classic and the modern techniques meet and benefit from each other. We will try to illustrate this in the applications we give. Because function theory 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 subject and explain and interpret standard definitions and results using the most convenient tools from analysis, geometry, and algebra. The key first step in the theory is to extend the concept of differentiability from real-valued functions of a real variable to complex- valued functions of a complex variable. Although the definition of complex differentiability resembles the definition of real differentiability, its consequences are profoundly different. A complex-valued function of a complex variable that is differentiable iscalled holomorphic oranalytic, and the first part of this book is a study of the many equivalent ways of understanding the concept of analyticity. Many of 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. Chapter 1 begins with two motivating examples, followed by the statement of the Fundamental Theorem, an outline of the plan for vii viii PREFACE proving it, and a description of the text contents: the plan for the rest of the book. In devoting the first part of this book to the precise goal of stating andproving the Fundamental Theorem, we follow a path charted for us by Lipman Bers, from whom we learned the subject. In his teaching, expository, andresearchwritingheoftenstartedbyintroducingamain, often technical, result and then proceeded to derive its important and seemingly surprising consequences. Some of the grace and elegance of this subject will not emerge until a more technical framework has been established. In the second part of the text, we proceed to the leisurely exploration of interesting ramifications 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 with them helpedshapethisbook. Wetriedtofollowalluseful adviceandcorrect, of course, any mistakes or shortcomings they identified. Those that remain are entirely our responsibility. Jane Gilman Irwin Kra Rub´ı E. Rodr´ıguez June 2007 Standard Notation and Commonly Used Symbols A LIST OF SYMBOLS TERM MEANING Z integers Q rationals R reals C complex numbers (cid:2) C C∪{∞} ı a square root of −1 ıR the imaginary axis in C (cid:4)z real part of z (cid:5)z imaginary part of z z = x+ıy x = (cid:4)z and y = (cid:5)z z¯ conjugate of z r = |z| absolute value of z θ = arg z an argument of z z = reıθ r = |z| and θ = arg z u , u real partial derivatives x y u , u complex partial derivatives z z¯ ∂f partial derivative ∂x ∂R boundary of set R |R| cardinality of set R clR closure of set R intR interior of set R X the set of x ∈ X that satisfy condition condition ν (f) order of the function f at the point ζ ζ i(γ) interior of the Jordan curve γ e(γ) exterior of the Jordan curve γ f| restriction of the function f to the subset B B of its domain U(z,r) = U (r) {ζ ∈ C; |ζ −z| < r} z D U(0,1) H2 {z ∈ C; (cid:5)z > 0} ix x STANDARD NOTATION AND COMMONLY USED SYMBOLS STANDARD TERMINOLOGY TERM MEANING LHS left-hand side RHS right-hand side deleted neighborhood of z neighborhood with z removed CR Cauchy Riemann equations ⊂ proper subset ⊆ subset, may not be proper d Euclidean distance on C ρ hyperbolic distance on D D MMP Maximum Modulus Property MVP Mean Value Property Contents Preface vii Standard Notation and Commonly Used Symbols ix Chapter 1. The Fundamental Theorem in Complex Function Theory 1 1.1. Some motivation 1 1.2. The Fundamental Theorem 3 1.3. The plan for the proof 4 1.4. Outline of text 5 Chapter 2. Foundations 7 2.1. Introduction and preliminaries 7 2.2. Differentiability and holomorphic mappings 14 Exercises 18 Chapter 3. Power Series 23 3.1. Complex power series 24 3.2. More on power series 32 3.3. The exponential function, the logarithm function, and some complex trigonometric functions 36 3.4. An identity principle 42 3.5. Zeros and poles 47 Exercises 52 Chapter 4. The Cauchy Theory–A Fundamental Theorem 59 4.1. Line integrals and differential forms 60 4.2. The precise difference between closed and exact forms 65 4.3. Integration of closed forms and the winding number 70 4.4. Homotopy and simple connectivity 72 4.5. Winding number 75 4.6. Cauchy Theory: initial version 78 Exercises 80 Chapter 5. The Cauchy Theory–Key Consequences 83 5.1. Consequences of the Cauchy Theory 83 xi