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Eberhard Freitag Rolf Busam Complex Analysis ABC ProfessorDr.EberhardFreitag Translator Dr.RolfBusam Dr.DanFulea FacultyofMathematics FacultyofMathematics InstituteofMathematics InstituteofMathematics UniversityofHeidelberg UniversityofHeidelberg ImNeuenheimerFeld288 ImNeuenheimerFeld288 69120Heidelberg 69120Heidelberg Germany Germany E-mail:[email protected] E-mail:[email protected] [email protected] MathematicsSubjectClassification(2000):30-01,11-01,11F11,11F66,11M45,11N05, 30B50,33E05 LibraryofCongressControlNumber:2005930226 ISBN-10 3-540-25724-1SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-25724-0SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springeronline.com (cid:1)c Springer-VerlagBerlinHeidelberg2005 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandTechBooksusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11396024 40/TechBooks 543210 In Memoriam Hans Maaß (1911–1992) Preface to the English Edition This book is a translation of the forthcoming fourth edition of our German book “Funktionentheorie I” (Springer 2005). The translation and the LATEX files have been produced by Dan Fulea. He also made a lot of suggestions for improvementwhichinfluencedtheEnglishversionofthebook.Itisapleasure for us to express to him our thanks. We also want to thank our colleagues DiarmuidCrowley,WinfriedKohnenandJo¨rgSixtforusefulsuggestionscon- cerning the translation. Over the years, a great number of students, friends, and colleagues have con- tributed many suggestions and have helped to detect errors and to clear the text. The many new applications and exercises were completed in the last decade to also allow a partial parallel approach using computer algebra systems and graphictools,whichmayhaveafruitful,powerfulimpactespeciallyincomplex analysis. Last but not least, we are indebted to Clemens Heine (Springer, Heidelberg), who revived our translation project initially started by Springer, New York, and brought it to its final stage. Heidelberg, Easter 2005 Eberhard Freitag Rolf Busam Contents I Differential Calculus in the Complex Plane C ............ 9 I.1 Complex Numbers ................................... 9 I.2 Convergent Sequences and Series....................... 24 I.3 Continuity .......................................... 36 I.4 Complex Derivatives ................................. 42 I.5 The Cauchy–Riemann Differential Equations .......... 48 II Integral Calculus in the Complex Plane C................ 71 II.1 Complex Line Integrals ............................... 72 II.2 The Cauchy Integral Theorem ........................ 79 II.3 The Cauchy Integral Formulas........................ 94 III Sequences and Series of Analytic Functions, the Residue Theorem.................................................105 III.1 Uniform Approximation ..............................106 III.2 Power Series.........................................111 III.3 Mapping Properties for Analytic Functions..............126 III.4 Singularities of Analytic Functions .....................136 III.5 Laurent Decomposition .............................145 A Appendix to III.4 and III.5............................158 III.6 The Residue Theorem ................................165 III.7 Applications of the Residue Theorem ...................174 IV Construction of Analytic Functions ......................195 IV.1 The Gamma Function ................................196 IV.2 The Weierstrass Product Formula ...................214 IV.3 The Mittag–Leffler Partial Fraction Decomposition ...223 IV.4 The Riemann Mapping Theorem ......................228 A Appendix : The Homotopical Version of the Cauchy Integral Theorem ....................................239 B Appendix : The Homological Version of the Cauchy Integral Theorem ....................................244 X Contents C Appendix : Characterizations of Elementary Domains ....249 V Elliptic Functions ........................................257 V.1 The Liouville Theorems.............................258 A Appendix to the Definition of the Periods Lattice ........265 V.2 The Weierstrass ℘-function .........................267 V.3 The Field of Elliptic Functions.........................274 A Appendix to Sect. V.3 : The Torus as an Algebraic Curve .279 V.4 The Addition Theorem ...............................287 V.5 Elliptic Integrals .....................................292 V.6 Abel’s Theorem.....................................299 V.7 The Elliptic Modular Group...........................310 V.8 The Modular Function j ..............................319 VI Elliptic Modular Forms ..................................327 VI.1 The Modular Group and Its Fundamental Region ........328 VI.2 The k/12-formula and the Injectivity of the j-function .....................................335 VI.3 The Algebra of Modular Forms ........................345 VI.4 Modular Forms and Theta Series.......................348 VI.5 Modular Forms for Congruence Groups .................362 A Appendix to VI.5 : The Theta Group...................374 VI.6 A Ring of Theta Functions ............................381 VII Analytic Number Theory ................................391 VII.1 Sums of Four and Eight Squares .......................392 VII.2 Dirichlet Series ....................................409 VII.3 Dirichlet Series with Functional Equations ............418 VII.4 The Riemann ζ-function and Prime Numbers ...........431 VII.5 The Analytic Continuation of the ζ-function.............439 VII.6 A Tauberian Theorem ...............................446 VIII Solutions to the Exercises................................459 VIII.1 Solutions to the Exercises of Chapter I..................459 VIII.2 Solutions to the Exercises of Chapter II.................471 VIII.3 Solutions to the Exercises of Chapter III ................476 VIII.4 Solutions to the Exercises of Chapter IV ................488 VIII.5 Solutions to the Exercises of Chapter V.................496 VIII.6 Solutions to the Exercises of Chapter VI ................505 VIII.7 Solutions to the Exercises of Chapter VII ...............513 References.....................................................523 Symbolic Notations............................................533 Index..........................................................535 Introduction The complex numbers have their historical origin in the 16th century when theywerecreatedduringattemptstosolvealgebraic equations.G. C√ardano (1545)hasalreadyintroducedformalexpressionsasforinstance5± −15,in order to express solutions of quadratic and cubic equations. Around 1560 R. Bombelli computed systematically using such expressions and found 4 as a solution of the equation x3 =15x+4 in the disguised form (cid:1) (cid:1) √ √ 4= 3 2+ −121+ 3 2− −121 . AlsointheworkofG.W. Leibniz(1675)onecanfindequationsofthiskind, e.g. (cid:1) (cid:1) √ √ √ 1+ −3+ 1− −3= 6 . √ Intheyear1777L.Eulerintroducedthenotationi= −1fortheimaginary unit. The terminology “complex number” is due to C.F. Gauss (1831). The rig- orous introduction of complex numbers as pairs of real numbers goes back to W.R. Hamilton (1837). Sometimes it is already advantageous to introduce and make use of complex numbers in real analysis. One should for example think of the integration of rational functions, which is based on the partial fraction decomposition, und therefore on the Fundamental Theorem of Algebra: Over the field of complex numbers any polynomial decomposes as a product of linear factors. AnotherexampleforthefruitfuluseofcomplexnumbersisrelatedtoFourier series.FollowingEuler(1748)onecancombinetherealangularfunctionssine and cosine, and obtain the “exponential function” eix :=cosx+isinx . Then the addition theorems for sine and cosine reduce to the simple formula 2 Introduction ei(x+y) =eixeiy . In particular, (cid:2) (cid:3) eix n =einx holds for all integers n . The Fourier series of a sufficiently smooth function f, defined on the real line, with period 1, can be written in terms of such expressions as (cid:4)∞ f(x)= a e2πinx . n n=−∞ Here it is irrelevant whether f is real or complex valued. In these examples the complex numbers serve as useful, but ultimatively dis- pensabletools.Newaspectscomeintoplaywhenweconsidercomplexvalued functions depending on a complex variable, that is when we start to study functionsf :D →Cwithtwo-dimensionaldomainsD systematically.Thedi- mensiontwoisensuredwhenwerestricttoopendomainsofdefinition D ⊂C. Analogouslytothesituationinrealanalysisoneintroducesthenotionofcom- plex differentiability by requiring the existence of the limit f(z)−f(a) f(cid:3)(a):= lim z→a z−a z(cid:5)=a foralla∈D.Itturnsoutthatthisnotionbehavesmuchmoredrasticallythen realdifferentiability.Wewillshowforinstancethata(firstorder)complexdif- ferentiable function is automatically arbitrarily often complex differentiable. We will see more, namely that complex differentiable functions can always be developed locally as power series. For this reason, complex differentiable functions (defined on open domains) are also called analytic functions. “Complex analysis” is the theory of such analytic functions. Many classical functions from real analysis can be analytically extended to complexanalysis.Itturnsoutthattheseextensionsareunique,asforinstance in the case ex+iy :=exeiy . From the relation e2πi =1 it follows that the complex exponential function is periodic with the purely imaginaryperiod2πi.Thisobservationisfundamentalforthecomplexanaly- sis. As a consequence one can observe further phenomena: 1. The complex logarithm cannot be introduced as the unique inverse functionoftheexponentialfunctioninanaturalway.Itisa priorideter- mined only up to a multiple of 2πi. Introduction 3 2. The function 1/z (z (cid:5)= 0) does not have any primitive in the punc- tured complex plane. A related fact is the following: the path integral of 1/z with respect to a circle line centered in the origin and oriented anticlockwise yields the non-zero value (cid:5) 1 dz = 2πi (r >0) . z |z|=r Centralresultsofcomplexanalysis,likee.g.theResidueTheorem,arenothing but a highly generalized version of these statements. Realfunctionsoftenshowtheirtruenaturefirstafterconsideringtheiranalytic extensions. For instance, in the real theory it is not directly transparent why the power series representation 1 =1−x2+x4−x6±··· 1+x2 is valid only for |x| < 1. In the complex theory this phenomenon becomes moreunderstandable,simplybecausetheconsideredfunctionhassingularities in ±i. Then its power series representation is valid in the biggest open disk excluding the singularities, namely the unit disk. IntherealtheoryitisalsohardtounderstandwhytheTaylorseriesaround 0 of the C∞ function (cid:6) e−1/x2 , x(cid:5)=0 , f(x)= 0 , x=0 , convergesforallx∈R,butdoesnotrepresentthefunctioninanypointother than zero. In the complex theory this phenomenon becomes understandable, because the function e−1/z2 has an essential singularity in zero. Lesstrivialexamplesaremoreimpressive.Here,oneshouldmentiontheRie- mann ζ-function (cid:4)∞ ζ(s)= n−s , n=1 whichwillbeextensivelystudiedinthelastchapterofthebookasafunctionof thecomplex variabletraditionallydenotedbysusingthemethodsofcomplex analysis, which will be presented throughout the preceeding chapters. From the analytical properties of the ζ-function we will deduce the Prime Number Theorem. Riemann’s celebrated work on the ζ function [Ri2] is a brilliant example for thethesishealreadypresentedeightyearsinadvanceinhisdissertation[Ri1]