1、245Editorial BoardS. AxlerK.A. RibetGraduate Texts in MathematicsGraduate Texts in Mathematics1Takeuti/Zaring. Introduction to Axiomatic Set Theory. 2nd ed.Oxtoby. Measure and Category. 2nd ed. Schaefer. Topological Vector Spaces. 2nd ed.Hilton/Stammbach. A Course in Homological Algebra. 2nd ed.Mac

2、Lane. Categories for the Working Mathematician. 2nd ed.Hughes/Piper. Projective Planes. J.-P. Serre. A Course in Arithmetic.Takeuti/Zaring. Axiomatic Set Theory. Humphreys. Introduction to Lie Algebras and Representation Theory.Cohen. A Course in Simple Homotopy Theory.Conway. Functions of One Compl

3、ex Variable I. 2nd ed.Beals. Advanced Mathematical Analysis. Anderson/Fuller. Rings and Categories of Modules. 2nd ed.3940Arveson. An Invitation to C-Algebras. Kemeny/Snell/Knapp. Denumerable Markov Chains. 2nd ed.Apostol. Modular Functions and Dirichlet Series in Number Theory. 2nd ed.J.-P. Serre.

4、Linear Representations of Finite Groups.Gillman/Jerison. Rings of Continuous Functions.Kendig. Elementary Algebraic Geometry. Love. Probability Theory I. 4th ed.Love. Probability Theory II. 4th ed. Moise. Geometric Topology in Dimensions 2 and 3.Sachs/Wu. General Relativity for Mathematicians.Gruenb

5、erg/Weir. Linear Geometry. 2nd ed.Edwards. Fermats Last Theorem. Klingenberg. A Course in Differential Geometry.Hartshorne. Algebraic Geometry. Manin. A Course in Mathematical Logic. Graver/Watkins. Combinatorics with Emphasis on the Theory of Graphs.Brown/Pearcy. Introduction to Operator Theory I:

6、Elements of Functional Analysis. Massey. Algebraic Topology: An Introduction.Crowell/Fox. Introduction to Knot Theory. Koblitz. p-adic Numbers, p-adic Analysis, and Zeta-Functions. 2nd ed.Lang. Cyclotomic Fields. Arnold. Mathematical Methods in Classical Mechanics. 2nd ed.Whitehead. Elements of Homo

7、topy Theory.Kargapolov/Merizjakov. Fundamentals of the Theory of Groups.Bollobas. Graph Theory.Edwards. Fourier Series. Vol. I. 2nd ed. Wells. Differential Analysis on Complex Manifolds. 2nd ed.Waterhouse. Introduction to Affine Group Schemes.Serre. Local Fields.Weidmann. Linear Operators in Hilbert

8、 Spaces.Lang. Cyclotomic Fields II. Massey. Singular Homology Theory.Farkas/Kra. Riemann Surfaces. 2nd ed. Stillwell. Classical Topology and Combinatorial Group Theory. 2nd ed.Hungerford. Algebra.Davenport. Multiplicative Number Theory. 3rd ed.Hochschild. Basic Theory of Algebraic Groups and Lie Alg

9、ebras.(continued after index)2341442543678944454647104811491213505114Golubitsky/Guillemin. Stable Map and Their Singularities.s52535415Berberian. Lectures in Functional Analysisand Operator Theory.Winter. The Structure of Fields. Rosenblatt. Random Processes. 2nd ed. Halmos. Measure Theory.Halmos. A

10、 Hilbert Space Problem Book. 2nd ed.Husemoller. Fibre Bundles. 3rd ed. Humphreys. Linear Algebraic Groups. Barnes/Mack. An Algebraic Introduction to Mathematical Logic.Greub. Linear Algebra. 4th ed. Holmes. Geometric Functional Analysis and Its Applications.Hewitt/Stromberg. Real and Abstract Analys

11、is.Manes. Algebraic Theories. Kelley. General Topology.161718195556575820212259602324612562262763646528 Zariski/Samuel. Commutative Algebra. Vol. I. 29 Zariski/Samuel. Commutative Algebra. Vol. II.30 Jacobson. Lectures in Abstract Algebra I. Basic Concepts.31 Jacobson. Lectures in Abstract Algebra I

12、I. Linear Algebra.32 Jacobson. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory.33 Hirsch. Differential Topology.34 Spitzer. Principles of Random Walk. 2nd ed.35 Alexander/Wermer. Several Complex Variables and Banach Algebras. 3rd ed.6667686970717236 Kelley/Namioka et al. Linear

13、Topological Spaces.37 Monk. Mathematical Logic.38 Grauert/Fritzsche. Several Complex Variables.737475Jane P. Gilman Irwin KraRub E. RodrguezComplex AnalysisIn the Spirit of Lipman BersJane P. Gilman Department ofMathematics and Computer ScienceRutgers University Newark, NJ 07102 USA

14、Irwin KraMath for America50 Broadway, 23 Floor New York, NY 10004 andDepartment of Mathematics State University of New Yorkat Stony BrookStony Brook, NY 11794 USARub E. Rodrguez Pontificia UniversidadCatlica de Chile SagoChile rubimat.puc.clEditorial BoardS. AxlerMathematics Depa

15、rtmentSan Frsco State University San Frsco, CA 94132 USAK.A. RibetMathematics Department University of California at Berkeley Berkeley, CA 94720-3840USAISBN: 978-0-387-74714-9e-ISBN: 978-0-387-74715-6Library of Congress Control Number: 2007940048Mathematics Subject

16、 Classification (2000): 30-xx 32-xx 2007 Springer Science+Business Media, LLC. 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 i

17、n 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, trademark

18、s, service marks, and similar terms, even if they are not identified 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-paper.9 8 7 6 5 4 3 2 1To the memory ofMary and Lipman BersPrefaceThis book presents fundamental ma

19、terial that should be part of theeducation 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

20、 variable. This is a prerequisite for the study of many current andrapidly develoareas of mathematics, including the theory of severaland innitely many complex variables, the theory of groups, hyperbolicgeometry and three-manifolds, and number theory. Complex analysis has connections and application

21、s 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 benet from each other. We will try to illustrate this in the applications we give.Because function theory has been used by generations of practicing mathemat

22、icians working in a number of dierent elds, the basic re- sults have been developed and redeveloped from a number of dierent 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 denitions and results using

23、the most convenient tools from analysis, geometry, and algebra.The key rst step in the theory is to extend the concept of dif-ferentiability from real-valued functions of a real variable to complex- valued functions of a complex variable. Although the denition of com- plex dierentiability resembles

24、the denition of real dierentiability, its consequences are profoundly dierent. A complex-valued function of a complex variable that is dierentiable is called holomorphic or analytic, and the rst part of this book is a study of the many equivalent ways of understanding the concept of analyticity. Man

25、y 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 forviivi

26、iiPREFACEproving it, and a description of the text contents: the plan for the restof the book.In devoting the rst part of this book to the precise goal of stating and proving the Fundamental Theorem, we follow a path charted for us by Lipman Bers, from whom we learned the subject. In his teaching, e

27、xpository, and research writing he often started by introducing a main, often technical, result and then proceeded to derive its important andseely surprising consequences.Some of the grace and eleganceof this subject will not emerge until a more technical framework hasbeen established. In the secon

28、d part of the text, we proceed to the leisurely exploration of interesting ramications 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 Berss original lectures. Versions of these notes h

29、ave been used by us at our respective home institutions, some for more than 20 years, as well as by others atvarious universities. We are grateful to many colleagues andswhoand commented on these notes. Our interaction with themhelped shape this book. We tried to follow all useful advice and correct

30、,of course, any mistakes or shortcoremain are entirely our responsibility.s they identied. Those thatJane GilmanIrwin Kra Rub E. RodrguezJune 2007StandardNotation and CommonlyUsedSymbolsA LIST OF SYMBOLSixTERMMEANINGZ Q R CC R z zz = x + y zr = |z| = arg z z = reux, uy uz, uzfxR|R| cl R int RXcondit

31、ion (f )i()e() f |BU (z, r) = Uz(r)D H2integers rationals realscomplex numbersC a square root of 1 the imaginary axis in Creal part of z imaginary part of z x = z and y = z conjugate of z absolute value of z an argument of zr = |z| and = arg zreal partial derivativescomplex partial derivatives parti

32、al derivative boundary of set R cardinality of set Rclosure of set Rinterior of set Rthe set of x X that satisfy condition order of the function f at the point interior of the Jordan curve exterior of the Jordan curve restriction of the function f to the subset Bof its domain C; | z| 0xSTANDARD NOTA

33、TION AND COMMONLY USED SYMBOLSSTANDARD TERMINOLOGYTERMMEANINGLHS RHSdeleted neighborhood of zCRdD MMP MVPleft-hand side right-hand sideneighborhood with z removed Cauchy Riemann equations proper subsetsubset, may not be proper Euclidean distance on C hyperbolic distance on Dum Modulus Property Mean

34、Value PropertyContentsPrefaceviiStandard Notation and Commonly Used SymbolsixChapter 1.The Fundamental Theorem in Complex FunctionTheory1134..Some motivationThe Fundamental Theorem The plan for the proof Outline of textChapter 2.Foundations2.1. Introduction and preliminaries2.2. Diere

35、ntiability and holomorphic map Exercises771418sChapter 3.Power Series23243.3.3.Complex power seriesMore on power seriesThe exponential function,the logarithmfunction,and some complex trigonometric functionsAn identity principle Zeros and poles3642475.ExercisesChapter 4.The Cauchy The

36、oryA Fundamental Theorem5960657072757880...Line integrals and dierential formsThe precise dierence betweend and exact formsIntegration ofd forms and the winding numberHomotopy and simple connectivityWinding numberCauchy Theory: initial versionExercisesChapter 5.The Cauchy Theory

37、Key Consequences5.1.Consequences of the Cauchy Theoryxi8383xiiCONTENTS..5.6.Cycles and homologyJordan curvesThe Mean Value Property On elegance and concisenessAppendix: Cauchys integral formula for smooth functions899093969697ExercisesChapter 6.Cauchy Theory: Local Behavior and Singula

38、rities of Holomorphic Functions1011011031061101131..6.5.Functions holomorphic on an annulusIsolated singularitiesZeros and poles of meromorphic functions Local properties of holomorphic maps Evaluation of denite integralsExercisesChapter 7.Sequences and Series of Holomorphic Function

39、s12312312613013413814614714815215415816716917317317617918618618819119119519..7.5.Consequences of uniform convergence on compact setsA metric on C(D)The cotangent function Compact sets in H(D)Approximation theorems and Runges theoremExercisesChapter 8.Conformal Equivalence.8.3.8

40、.4.8.5.Fractional linear (Mobius) transformations Aut(D) for D = C, C, D, and H2The Riemann MapTheoremHyperbolic geometryFinite Blaschke productsExercisesChapter 9.Harmonic Functions..9.5.Harmonic functions and the LaplacianIntegral representation of harmonic functions The Dirichlet pr

41、oblemThe Mean Value Property:The reection principlea characterizationExercisesChapter 10.Zeros of Holomorphic Functions.10.3.Innite productsHolomorphic functions with prescribed zeros Eulers -functionCONTENTSxiii10.4. The eld of meromorphic functions10.5. Innite Blaschke products ExercisesB

42、IBLIOGRAPHICAL NOTESBibliography Index207209209213215217CHAPTER 1The Fundamental Theorem in Complex FunctionTheoryThis introductory chapter is meant to convey the need for and theintrinsic beauty found in passing from a real variable x to a complex variable z. In the rst section we “solve” two natur

43、al problems using complex analysis. In the second, we state the most important result in the theory of functions of one complex variable that we call the Fun- damental Theorem of complex variables; its proof will occupy most of this volume. The next-to-last section of this chapter is an outline of o

44、ur plan for the proof; in subsequent chapters, we will dene all theconcepcountered in the statement of the theorem. Therefore, theer may not be able at this point to understand all (or any) of thestatements in the theorem, to fully appreciate the two motivating ex- amples, or to appreciate the depth

45、 of the various claims of the theoremand might choose initially to skim this material. Allers shouldperiodically, throughout the journey through this book, return to thischapter. Finally we end this chapter with a section that gives a more conventional outline of the text.1.1. Some motivation1.1.1.

46、Where do series converge? In the calculus of a real vari-able one encounters two series that converge for |x| 0 into nitely many (more than 1) innite arith- metic progressions with distinct dierences? The answer is “NO”. The assumption on distinct dierences is clearly necessary. So assume to the con

47、trary thatZ0 = S1 S2 . Sn,where n Z1, and for 1 i n, Si is an arithmetic progression withinitial term ai and dierence di, 1 d1 d2 . dn. Thenfor 1 i j n, Si Sj = , and iiiiz =z +z + . +z ,i=1iS1iS2iSnand each series converges for |z| 1. Sumthe above geometricseries, we see thatza1za2zanz+ . +for all

48、z with |z| 1. (1.1) 1 zdn+1 z1 zd11 zd2=2Choose a sequence of complex numbers z of absolute value lessk2than 1 with lim zk = e dn . Thenk2 zke dnlim=2k 1 zk1 e dnand2aiai e dnzklimk(all these qu=for i = 1, 2,.,n 1;d2di1 zik1 ednzan kdties are nite), whereas limdoes not exist.1 zknkThis is an obvious

49、 contradiction to (1.1).1In the sequel we usually use z, w, and , among others, but not x to denote a complex variable.2Notation for the polar form of a complex number is established in Chapter 2.1.2. THE FUNDAMENTAL THEOREM31.2. The Fundamental TheoremTheorem 1.1. Let D C denote a domain (an open c

50、onnectedset), and let f = u + v : D C be a complex-valued function denedon D.The following conditions are equivalent:(1)The complex derivativef (z) exists for all z D;(Riemann)that is, the function f is holomorphic on D.(2)The functions u and v are continuously dierentiable and sat-isfyuvuv=and= x.(CauchyRiemann: CR)xyyAlternatively, the function f is continuously dierentiableand satisesfz= 0.(CRcomplex form) (3)For each simply connected subdomain D of D there exists aholomorphic function F : D C such that F (z) = f (z) for all z D.(4)The functio