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Springer Undergraduate Mathematics Series Charles H. C. Little Kee L. Teo Bruce van Brunt An Introduction to Infinite Products Springer Undergraduate Mathematics Series SUMS Readings AdvisoryEditors MarkA.J.Chaplain,St.Andrews,UK AngusMacintyre,Edinburgh,UK SimonScott,London,UK NicoleSnashall,Leicester,UK EndreSu¨li,Oxford,UK MichaelR.Tehranchi,Cambridge,UK JohnF.Toland,Bath,UK SUMS Readings isacollection of books thatprovides students withopportunities to deepen understanding and broaden horizons. Aimed mainly at undergraduates, the series is intended for books that do not fit the classical textbook format, from leisurely-yet-rigorous introductions to topics of wide interest, to presentations of specialised topics that are not commonly taught. Its books may be read in parallelwithundergraduatestudies,assupplementaryreadingforspecificcourses, backgroundreadingforundergraduateprojects,oroutofsheerintellectualcuriosity. Theemphasisoftheseriesisonnovelty,accessibilityandclarityofexposition,as wellasself-studywitheasy-to-followexamplesandsolvedexercises. Moreinformationaboutthisseriesathttp://www.springer.com/series/16607 Charles H. C. Little • Kee L. Teo • Bruce van Brunt An Introduction to Infinite Products CharlesH.C.Little KeeL.Teo ResearchFellowandformerProfessor ResearchFellowandformerProfessor ofMathematicsInstituteof ofMathematicsInstituteof FundamentalSciences FundamentalSciences MasseyUniversity MasseyUniversity PalmerstonNorth,NewZealand PalmerstonNorth,NewZealand BrucevanBrunt AssociateProfessorofMathematics InstituteofFundamentalSciences MasseyUniversity PalmerstonNorth,NewZealand ISSN1615-2085 ISSN2197-4144 (electronic) SpringerUndergraduateMathematicsSeries ISSN2730-5813 ISSN2730-5821 (electronic) SUMSReadings ISBN978-3-030-90645-0 ISBN978-3-030-90646-7 (eBook) https://doi.org/10.1007/978-3-030-90646-7 MathematicsSubjectClassification:40A20,26-01,30-01 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Infinite series and products share a common mathematical heritage. A student usually gets a glimpse of series in a first course on the calculus followed by a substantial dose of the theory underlying infinite series in a course on elementary realanalysis.Incontrast,infiniteproductsareusuallyrelegatedtoafewcomments and perhaps examples at this stage. If a student encounters infinite products, it is ofteninasecondcourseoncomplexanalysisinthecontextofrepresentingfunctions given their distribution of zeros (e.g. Weierstrass factorization). This situation is understandable because series arguably play a larger rôle in analysis, and much of the theory for infinite products can be constructed from that for infinite series. Students might thus be forgiven if their exposure to products affirms that these creatures live in the shadow of infinite series. Indeed, this is often the case, yet a closerstudyshowsthattheyhavetheirownlivesandmanifestthemselvesthrough beautifulandunexpectedrelations. The earliest algorithm for determining π dates back to Viète (c. 1590), who claimedthat (cid:3) √ (cid:2) √ (cid:2) √ 2 2 2+ 2 2+ 2+ 2 = · · ··· . π 2 2 2 This is also the earliest published infinite product. Wallis (c. 1650) gave another expressionforπ: π 22 44 4n2 = · ··· ··· . 2 13 35 4n2−1 The theory underlying infinite processes (analysis) was young atthis time,and no formalproofwasavailabletosupporttheseclaims. The“father”ofinfiniteproductsmightbeEuler(c.1750).Amongthecornucopia ofresultsfromhispenaretheformulæ v vi Preface (cid:4) (cid:5)(cid:4) (cid:5)(cid:4) (cid:5) sinx x x x = cos cos cos ··· , x 2 22 23 and (cid:6) (cid:4) (cid:5) (cid:7)(cid:6) (cid:4) (cid:5) (cid:7)(cid:6) (cid:4) (cid:5) (cid:7) sinx x 2 x 2 x 2 = 1− 1− 1− ··· . x π 2π 3π Particular choices of x in these expressions lead to Viète’s and Wallis’s products. The alert student can immediately see relations between the Maclaurin series for sinx andEuler’sproducts.Isitobviousthat (cid:6)(cid:6) (cid:4) (cid:5) (cid:7)(cid:6) (cid:4) (cid:5) (cid:7)(cid:6) (cid:4) (cid:5) (cid:7) (cid:7) x 2 x 2 x 2 sinx =x 1− 1− 1− ··· π 2π 3π x3 x5 =x− + −···? 3! 5! The interplay between infinite series and products is certainly interesting and Eulerdidnotstophere.Hewentontoconnectprimenumberswithacertainseries latertobecalledtheRiemannzetafunction: (cid:6) 1 (cid:7)−1(cid:6) 1 (cid:7)−1(cid:6) 1 (cid:7)−1 (cid:8) 1 (cid:9)−1 1− 1− 1− ··· 1− ··· 2s 3s 5s ps j 1 1 1 1 =1+ + + + +··· , 2s 3s 4s 5s where p is the jth prime number and s > 1. Riemann (c. 1860) took this result j for a real test drive by considering s to be a complex variable. The Riemann zeta functionprovedpivotalinthetheoryofthedistributionofprimenumbers.Itisstill thestartingpointforalotofmathematicalresearch.Euleralsoderivedanumberof relationsbetweenproductsandseriesthatformabasisforthestudyofpartitionsof integers—yetanothercurrentfieldofresearch. Post Riemann, and independent of the aforementioned research directions, we notethatvonSeidel(c.1871)showedthat logx 2 2 2 = · · ··· , x−1 1+x1/2 1+x1/22 1+x1/23 fromwhichweget 2 2 2 log2= √ · (cid:2)√ · (cid:3)(cid:2)√ ··· . 1+ 2 1+ 2 1+ 2 Preface vii Catalan(c.1873)derivedaninfiniteproductexpressionforthenumbere,andsome hundredyearslaterPippenger[55]derivedtheexpression (cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7) e 2 1/2 24 1/22 4668 1/23 = ··· . 2 1 33 5577 Infiniteproductscontinuetofascinatemathematicians. This book is about the theory of infinite products and applications. The target readership is a student familiar with the basics of real analysis of a single variable and a first course in complex analysis up to and including the calculus of residues. The first chapter is largely a summary of results on infinite series, with which the reader is assumed to be familiar. This chapter serves mostly to make the book moreself-containedanditprovidesaplatformtointroducenotationanddefinitions. Except in the final section, there are no proofs or problems, and few examples. Thischaptercanbereadasneededbythestudent.Thefinalsectioncoversdouble sequencesandseries.Thismaterialisseldomencounteredinfirstcoursesonrealor complexanalysis.Hereweprovideproofs,examplesandproblems. Thesecondchaptercontainsthegeneraltheoryofinfiniteproducts.Muchofthe material is standard in at least older books used for a second course on analysis, but we also include topics such as conditional convergence and Abel’s theorem. The chapter finishes with Weierstrass and Blaschke products and factorization for analyticfunctions. The remaining two chapters deal with applications of infinite products. The gamma and related functions (digamma, beta, etc.) are of great interest in their own right, but they also provide a nice application of the use of infinite products. We use infinite products to define the gamma function in Chap.3, recover a number of well-known results and then consider functions related to the gamma function along with applications to determine the limit of certain types of infinite products. Certainly, infinite products leave a sizable footprint in the distribution of prime numbersandalsopartitionproblems.Theapplicationofproductstothesetopicsin numbertheoryisthesubjectofthefinalchapter.Thisisahugefieldsprinkledwith nameslikeEuler,Jacobi,Riemann,HardyandRamanujan.Wemakenoattemptto giveacompleteorevenlimitedaccountofthistopic.Nonetheless,wehopetowhet thereader’sappetiteforfurtherstudy. Thegoalofthisbookistoprovidethereaderwithashortintroductiontoinfinite products and motivation for their study. Our aim is neither to provide a complete treatiseonthesubjectnoranexhaustivecompilationofresults.Thechoiceofresults andapplicationsisalwaysdebatable.Thereareclearlyotheravenueswemighthave pursued that are of equal importance. We hope to arm readers so that they might understandtheseproductsinthecourseofmoreadvancedstudies,and,onadifferent level,toinspirethemwiththesheerbeautyofmathematics. viii Preface The support of Massey University is gratefully acknowledged. The book has certainly benefited from the reviewing process as the reviewers made a number of suggestions that improved the text. Finally, the authors appreciate the encourage- mentandsupportoftheirwives. PalmerstonNorth,NewZealand CharlesH.C.Little KeeL.Teo BrucevanBrunt Contents 1 Introduction .................................................................. 1 1.1 Series................................................................... 1 1.2 SerieswithNon-NegativeTerms...................................... 3 1.3 SerieswithGeneralTerms ............................................ 7 1.4 UniformConvergenceofSequencesandSeriesofFunctions....... 11 1.5 AnalyticFunctionsandPowerSeries................................. 16 1.6 DoubleSeries.......................................................... 22 2 InfiniteProducts ............................................................. 39 2.1 Introduction............................................................ 40 2.2 ConvergenceofProductsandSeries.................................. 44 2.3 ConditionallyConvergentProducts................................... 59 2.4 UniformConvergenceofProductsofFunctions..................... 76 2.5 InfiniteProductsofRealFunctions................................... 79 2.6 InfiniteProductExpansionsforsinx andcosx...................... 88 2.7 Abel’sLimitTheoremforInfiniteProducts.......................... 99 2.8 WeierstrassProducts................................................... 109 2.9 TheWeierstrassFactorizationTheorem.............................. 115 2.10 BlaschkeProducts ..................................................... 120 2.11 DoubleInfiniteProducts............................................... 123 3 TheGammaFunction ...................................................... 131 3.1 RepresentationsoftheGammaFunction............................. 132 3.2 SomeIdentitiesInvolvingtheGammaFunction..................... 153 3.3 AnalyticFunctionsRelatedto(cid:3)...................................... 157 3.4 Stirling’sFormula...................................................... 161 3.5 ApplicationstoProductsandSeries .................................. 173 3.6 TheBetaFunction..................................................... 186 ix