数学专业复试课程complex analysis elias stein

1、COMPLEX ANALYSISIbookrootOctober 20, 2007Princeton Lectures in AnalysisI Fourier Analysis: An IntroductionII Complex AnalysisIII Real Analysis: Measure Theory, Integration, andHilbert SpacesPrinceton Lectures in AnalysisIICOMPLEX ANALYSISElias M. Stein&Rami ShakarchiPRINCETON UNIVERSITY PRESSPRI

2、NCETON AND OXFORDCopyright © 2003 by Princeton University Press Published by Princeton University Press, 41 William Street,Princeton, New Jersey 08540In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TWLibrary of Congress Control Number 2005274996

3、ISBN 978-0-691-11385-2British Library Cataloging-in-Publication Data is availableThe publisher would like to acknowledge the authors of this volume forproviding the camera-y copy from which this book was printedPrinted on acid-paper. Printed in the United States of America5 7 9 10

4、 8 6To my grandchildrenCarolyn, Alison, JasonE.M.S.To my parentsmed & Mireilleand my brotherKarimR.S.ForewordBeginning in the spring of 2000, a series of four omester courseswere taught at Princeton University whose purpose was to present, inan integrated manner, the core areas of analysis. The

5、objective was tomake plain the organic uthat exists between the various parts of thesubject, and to illustrate the wide applicability of ideas of analysis to other elds of mathematics and science. The present series of books is an elaboration of the lectures that were given.While there are a number

6、of excellent texts dealing with individual parts of what we cover, our exposition aims at a dierent goal: pre- senting the various sub-areas of analysis not as separate disciplines, but rather as highly interconnected. It is our view that seeing these relationsand their resulting synergies will moti

7、vate theer to attain a betterunderstanding of the subjea whole. With this outcome in mind, wehave concentrated on the main ideas and theorems that have shaped theeld (sometimes sacricing a more systematic approach), and we have been sensitive to the historical order in which the logic of the subject

8、 developed.We have organized our exposition into four volumes, each reecting the material covered in a semester. Their contents may be broadly sum- marized as follows:I.II. III. IV.Fourier series and integrals. Complex analysis.Measure theory, Lebesgue integration, and Hilbert spaces.A selection of

9、further topics, including functional analysis, distri- butions, and elements of probability theory.However, this listing does not by itself give a complete picture of the many interconnections that are presented, nor of the applications to other branches that a ighlighted. To give a few examples: th

10、e ele- ments of (nite) Fourier series studied in Book I, which lead to Dirichlet characters, and from there to the innitude of primes in an arithmetic progression; the X-ray and Radon transforms, which arise in a number ofviiviiiFOREWORDproblems in Book I, and reappear in Book III to play an importa

11、nt role in understanding Besicovitch-like sets in two and three dimensions; Fatous theorem, which guarantees the existence of boundary values of bounded holomorphic functions in the disc, and whose proof relies on ideas devel- oped in each of the rst three books; and the theta function, which rst oc

12、curs in Book I in the solution of the heat equation, and is then used in Book II to nd the number of ways an integer can be represented as the sum of two or four squares, and in the analytic continuation of the zeta function.A few further words about the books and the courses on which they were base

13、d. These courses where given at a rather intensive pace, with 48lectuours a semester. The weekly problem sets played an indispens-able part, and as a result exercises and problems have a similarly im- portant role in our books. Each chapter has a series of “Exercises” that are tied directly to the t

14、ext, and while some are easy, others may requiremore eort. However, the substal number of hints that are givenshould enable theer to attack most exercises. There are also moreinvolved and challenging “Problems”; the ones that are most dicult, or go beyond the scope of the text, are marked with an as

15、terisk.Despite the substal connections that exist between the dierentvolumes, enough overlapmaterial has been provided so that each ofthe rst three books requires only minimal prerequisites: acquaintance with elementary topics in analysis such as limits, series, dierentiable functions, and Riemann i

16、ntegration, together with some exposure to lin-ear algebra. This makes these books accessible tos interestedin such diverse disciplines as mathematics, physics, engineering, and nance, at both the undergraduate and graduate level.It is with great pleasure that we express our appreciation to all who

17、have aided in this enterprise. We are particularly grateful to the stu- dents who participated in the four courses. Their continuing interest,enthusiasm, and dedication provided the encouragement thatthisproject possible. We also wish to thank Adrian Banner and Jose LuisRodrigo for their special hel

18、p in running the courses, and their eorts tosee that the Banner alsos got the most from each class. In addition, Adrian valuable suggestions that are incorporated in the text.ixFOREWORDWe wish also to record a note of special thanks for the following in- dividuals: Charles Feerman, who taught the rs

19、t week (successfullylaunching the whole project!); Paul Hagelstein, who in addition to-ing part of the manuscript taught several weeks of one of the courses, andhas since taken over the teaching of the second round of the series; andDaniel Levine, who gave valuable help in proof-ing. Last but notlea

20、st, our thanks go to Gerree Pecht, for her consummate skill in type- setting and for the time and energy she spent in the preparation of allaspects of the lectures, such as transparencies, notes, and the manuscript.We are also happy to acknowledge our indebtedness for the supportwe received from the

21、 250thFund of Princeton University,and the National Science Foundations VIGRE program.Elias M. SteinRami ShakarchiPrinceton, New JerseyAugust 2002ContentsviixvForewordIntroductionChapter 1. Preliminaries to Complex Analysis111558881418241Complex numbers and the complex plane1.1 Basic properties1.2 C

22、onvergence1.3 Sets in the complex plane Functions on the complex planeContinuous functions Holomorphic functions Power series34Integration along curves Exercises3234Chapter 2. Cauchys Theorem and Its Applications12Goursats theoremLocal existence of primitives and Cauchys theorem in a discE

23、valuation of some integrals Cauchys integral formulas Further applications3741455353535557606467345.45.5Moreras theoremSequences of holomorphic functionsHolomorphic functions dened in terms of integrals Schwarz reection principleRunges approximation theorem67ExercisesProblemsChapter 3. Mer

24、omorphic Functions and the Logarithm71727677838912Zeros and polesThe residue formula2.1 ExamplesSingularities and meromorphic functions The argument principle and applications34xixiiCONTENTS5 Homotopies and simply connected domains6 The complex logarithm7 Fourier series and harmonic functions8 Exerc

25、ises9 Problems9397101103108Chapter 4. The Fourier Transform11111311412112713112345The class FAction of the Fourier transform on FPaley-Wiener theorem ExercisesProblems134135138140140142145147153156Chapter 5. Entire Functions123Jensens formula Functions of nite order Innite products3.1 Generalities3.

26、2 Example: the product formula for the sine function Weierstrass innite productsHadamards factorization theorem ExercisesProblems4567Chapter 6. The Gamma and Zeta Functions1591601611631681681741791The gamma function1.1 Analytic continuation1.2 Further properties of The zeta function2.1 Functional eq

27、uation and analytic continuation ExercisesProblems234Chapter 7. The Zeta Function and Pri orem1 Zeros of the zeta function1.1 Estimates for 1/(s)2 Reduction to the functions and 12.1 Proof of the asymptotics for 1Note on interchanging double sums3 Exercisesmber The-181182187188194197199xiiiCONTENTS4

28、Problemhapter 8. Conformal Maps1Conformal equivalence and examples1.1 The disc and upper half-plane1.2 Further examples1.3 The Dirichlet problem in a stripThe Schwarz lemma; automorphisms of the disc and upper half-plane2.1 Automorphisms of the disc2.2 Automorphisms of the upper

29、half-plane22182192212242242252282312312352382412452482543The Riemann maptheorem3.1 Necessary conditions and statement of the theorem3.2 Montels theorem3.3 Proof of the Riemann maptheorem4Conformal maps onto polygons4.44.5Some examplesThe Schwarz-Christoel integral Boundary behaviorThe mapfo

30、rmulaReturn to elliptic integrals56ExercisesProblemsChapter 9. An Introduction to Elliptic Functions2612622642661Elliptic functions1.1 Liouvilles theorems1.2 The Weierstrass functionThe modular character of elliptic functions and Eisenstein series2.1 Eisenstein series2.2 Eisenstein series and diviso

31、r functions ExercisesProblems227327327627828134Chapter 10. Applications of Theta Functions2832842892932962971Product formula for the Jacobi theta function1.1 Further transformation laws Generating functionsThe theorems about sums of squares3.1 The two-squares theorem23xivCONTENTS3.2 The four-squares

32、 theorem ExercisesProblems30430931431831932332833434145Appendix A: Asymptotics1 Bessel functions2 Laplaces method; Stirlings formula3 The Airy function4 The partition function5 ProblemsAppendix B: Simple Connectivity and Jordan Curve Theorem1 Equivalent formulations of simple connectivity2 The Jorda

33、n curve theorem2.1 Proof of a general form of Cauchys theorem344345351361Notes and References365Bibliography369373Symbol GlossaryIndex375Introduction. In eect, if one extends these functions by allowingcomplex values for the arguments, then there arises a harmony and regularity which without it woul

34、d re- main hidden.B. Riemann, 1851When we begin the study of complex analysis we enter a marvelous world, full of wonderful insights. We are tempted to use the adjectives magical, or even miraculous when describing the rst theorems we learn; and in pursuing the subject, we continue to be astonished

35、by the elegance and sweep of the results.The starting point of our study is the idea of extending a function initially given for real values of the argument to one that is dened when the argument is complex. Thus, here the central objects are functions from the complex plane to itselff : C C,or more

36、 generally, complex-valued functions dened on open subsets of C. At rst, one might object that nothing new is gained from this extension,since any complex number z can be written as z = x + iy where x, y Rand z is identied with the point (x, y) in R2.However, everything changes drastically if we mak

37、e a natural, but misleadingly simple-looking assumption on f : that it is dierentiable in the complex sense. This condition is called holomorphicity, and it shapes most of the theory discussed in this book.A function f : C C is holomorphic at the point z C if the limitf (z + h) f (z)limh0(h C)hexist

38、s. This is similar to the denition of dierentiability in the case of a real argument, except that we allow h to take complex values. The reason this assumption is so far-reaching is that, in fact, it encompasses a multiplicity of conditions: so to speak, one for each angle that h can approach zero.x

39、vxviINTRODUCTIONAlthough one might now be tempted to prove theorems about holo- morphic functions in terms of real variables, theer will soon discoverthat complex analysis is a new subject, one whichs proofs to thetheorems that are proper to its own nature. In fact, the proofs of themain properties

40、of holomorphic functions which we discuss in the next chapters are generally very short and quite illuminating.The study of complex analysis proceeds along two paths that often intersect. In following the rst way, we seek to understand the univer- sal characteristics of holomorphic functions, withou

41、t special regard for specic examples. The second approach is the analysis of some partic- ular functions that have proved to be of great interest in other areas of mathematics. Of course, we cannot go too far along either path without having traveled some way along the other. We shall start our stud

42、y with some general characteristic properties of holomorphic functions, which are subsumed by three rather miraculous facts:1.Contour integration: If f is holomorphic in , then for appro-priated paths in f (z)dz = 0.Regularity: If f is holomorphic, then f is indenitely dieren- tiable.2.3.Analytic co

43、ntinuation: If f and g aolomorphic functionsin which are equal in an arbitrarily small disc in , then f = geverywhere in .These three phenomena and other general properties of holomorphic functions are treated in the beginning chapters of this book. Instead of trying to summarize the contents of the

44、 rest of this volume, we men- tion briey several other highlights of the subject. The zeta function, which is expressed as an innite series 1ns(s) = ,n=1and is initially dened and holomorphic in the half-plane Re(s) > 1, where the convergence of the sum is guaranteed. This function and its varian

45、ts (the L-series) are central in the theory of prime numbers, and have al y appeared in Chapter 8 of Book I, wherexviiINTRODUCTIONwe proved Dirichlets theorem. Here we will prove that extends to a meromorphic function with a pole at s = 1. We shall see that thebehavior of (s) for Re(s) = 1 (and in p

46、articular that does notvanish on that line) leads to a proof of the priThe theta functionmber theorem. 2ein e2inz ,(z| ) = n=which in fact is a function of the two complex variables z and , holomorphic for all z, but only for in the half-plane Im( ) > 0. On the one hand, when we x , and think of

47、as a function ofz, it isly related to the theory of elliptic (doubly-periodic)functions. On the other hand, when z is xed, displays featuresof a modular function in the upper half-plane. The function (z| )arose in Book I as a fundamental solution of the heat equation on the circle. It will be used a

48、gain in the study of the zeta function, as well as in the proof of certain results in combinatorics and number theory given in Chapters 6 and 10.Two additional noteworthy topics that we treat are: the Fourier trans- form with its elegant connection to complex analysis via contour integra- tion, and

49、the resulting applications of the Poisson summation formula;also conformal maps, with the maps of polygons whose inversesare realized by the Schwarz-Christoel formula, and the particular case of the rectangle, which leads to elliptic integrals and elliptic functions.1Preliminaries to ComplexAnalysis

50、The sweedevelopment of mathematics during thelast two centuries is due in large part to the introduc-tion of complex numbers; paradoxically, this is basedon the seely absurd notion that there are num-bers whose squares are negative.E. Borel, 1952This chapter is devoted to the exposition of basic pre

51、liminary material which we use extensively throughout of this book.We begin with a quick review of the algebraic and analytic properties of complex numbers followed by some topological notions of sets in the complex plane. (See also the exercises at the end of Chapter 1 in Book I.) Then, we dene pre

52、cisely the key notion of holomorphicity, which is the complex analytic version of dierentiability. This allows us to discussthe Cauchy-Riemann equations, and power series.Finally, we dene the notion of a curve and the integral of a function along it. In particular, we shall prove an important result

53、, which we state loosely as follows: if a function f has a primitive, in the sense that there exists a function F that is holomorphic and whose derivative is preciselyf , then for anyd curve f (z) dz = 0.This is the rst step towards Cauchys theorem, which plays a central role in complex function the

54、ory.1 Complex numbers and the complex planeMany of the facts covered in this section were aly used in Book I.1.1 Basic propertiesA complex number takes the form z = x + iy where x and y are real, and i is an imaginary number that satises i2 = 1. We call x and y the2Chapter 1. PRELIMINARIES TO COMPLE

55、X ANALYSISreal part and the imaginary part of z, respectively, and we writex = Re(z)andy = Im(z).The real numbers are precisely those complex numbers with zero imagi- nary parts. A complex number with zero real part is said to be purely imaginary.Throughout our presentation, the set of all complex numbers is de- noted by C. The complex numbers can be visualized as the usual Eu- clidean plane