Complex Integration: A Compendium of Smart and Little-Known Techniques for Evaluating Integrals and Sums PDF
Preview Complex Integration: A Compendium of Smart and Little-Known Techniques for Evaluating Integrals and Sums
Undergraduate Lecture Notes in Physics Ron Gordon Complex Integration A Compendium of Smart and Little-Known Techniques for Evaluating Integrals and Sums Undergraduate Lecture Notes in Physics SeriesEditors NeilAshby,UniversityofColorado,Boulder,CO,USA WilliamBrantley,DepartmentofPhysics,FurmanUniversity,Greenville,SC,USA MatthewDeady,PhysicsProgram,BardCollege,Annandale-on-Hudson, NY,USA MichaelFowler,DepartmentofPhysics,UniversityofVirginia,Charlottesville, VA,USA MortenHjorth-Jensen,DepartmentofPhysics,UniversityofOslo,Oslo,Norway MichaelInglis,DepartmentofPhysicalSciences,SUNYSuffolkCounty CommunityCollege,Selden,NY,USA BarryLuokkala ,DepartmentofPhysics,CarnegieMellonUniversity,Pittsburgh, PA,USA Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems,workedexamples,chaptersummaries,andsuggestionsforfurtherreading. ULNPtitlesmustprovideatleastoneofthefollowing: • Anexceptionallyclearandconcisetreatmentofastandardundergraduatesubject. • Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standard subject. • Anovelperspectiveoranunusualapproachtoteachingasubject. ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysics teachingattheundergraduatelevel. ThepurposeofULNPistoprovideintriguing,absorbingbooksthatwillcontinueto bethereader’spreferredreferencethroughouttheiracademiccareer. Ron Gordon Complex Integration A Compendium of Smart and Little-Known Techniques for Evaluating Integrals and Sums RonGordon Northborough,MA,USA ISSN 2192-4791 ISSN 2192-4805 (electronic) UndergraduateLectureNotesinPhysics ISBN 978-3-031-24227-4 ISBN 978-3-031-24228-1 (eBook) https://doi.org/10.1007/978-3-031-24228-1 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2023 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 Thisbookisdedicatedtothememoryofmy eleventhgrademathteacher,Ms.EffieVellios. Preface Inreal-worldmaths,computersdoalmostallthecalculating; bycontrast,ineducationalmaths, peopledoalmostallthecalculating. —ConradWolfram,TheMath(s)Fix:AnEducationBlueprint fortheAIAge. Conrad Wolfram, in his book The Math(s) Fix: An Education Blueprint for the AI Age,laysoutnotonlyacaseforabandoningthetraditionalmathcurriculuminwhich we teach concepts involving hand calculations, but also a proposed all-computer computation curriculum. Wolfram is related to Stephen Wolfram who gifted the computationalsystemMathematicatousallsincethe1980s. You might think this Preface and this book are meant to be a broadside against Wolfram,andIamgoingtofurnishargumentsastowhyheisdeadwrongandwe shouldkeepteachingtraditionalhandcomputationtoeveryone.Butitisn’t,andIam not.Dr.Wolfram(theauthor)islargelycorrect:Weneedtoteachmathdifferentlyto readystudentsfordealingwithconceptssuchasmachinelearning,Kalmanfilters, graphs, and networks, which have become the backbone of the problems of our societytoday. SoamIsayingthatabookdedicatedtohandcalculationsisacompletewasteof time. No.Theopposite,actually. The problems requiring hand calculation have never really gone away. We still generatemodelsofthingslikeimagingdetailedobjectsinturbulentmedia,orreturn on European put options, or knocking in an automotive engine. These problems are solved, others will replace them. Mr. Wolfram will tell us simply to fire up Wolfram|AlphaorMathematicaandwhateverintegralorsumordifferentialequation thatneedstobesolvedwillbesolvedbythecomputer.And,again,heisn’twrong aboutit. Muchofthetime,anyway. vii viii Preface The truth is the systems behind the computers we use as substitutes for hand calculationsrelyonalgorithmswrittenbyhumanbeings.Humanbeingsthatmake mistakes,alot. Quiscustodietipsoscustodes? Sowewillalwaysneedpeoplewhoknowhowtodothehandcalculations.Idoubt Dr.Wolframwoulddisagree—otherwise,whywouldtheotherDr.Wolframneedto hireallofthosepeopletokeepMathematicarunning? Isupposethishasalwaysbeenmycalling.WhenIworkedforIBMMicroelec- tronics,Iwasinagroupthatdidlithographicprocessdesignanddidsobyrunning interminable simulations of the lithography process on vendor-provided software. (Previously, I worked for such a vendor.) One of my projects was to design test cases that we could model by hand, so we could evaluate the simulation results. Thiswasmyfavoritepartofmyjob,andwewereabletofindnumerousissuesto taketothesoftwarevendors.Myenthusiasmspilledouttomymanagement,andI wasallowedtoteachashortcoursetotheengineersrunningthesimulationsabout how to do simple hand calculations of the lithography models—a sort of Conrad Wolfram-in-reverse,ifyouwill. EvenMathematicaitself—andbyextension,Wolfram|Alpha—arenotperfect.I myselfcanpointtoaspecificexampleMathematicagotwrong.Theproblemwasto computethefollowingsum(whichwillbeconsideredinthebook): (cid:2)∞ 1 13π3 (cid:3)√ (cid:4) =− √ n=1 n3sin 2πn 360 2 WhenIcheckedtheresult,Mathematicareturned− 13π√2 .π squared,notcubed. 360 2 WhenIconvincinglyestablishedthatthecubewascorrect,thegoodstaffatWolfram Researcheventuallyfixedtheerror.Butittooksomeonewhodidhandcalculations tofixapresumedbugintheircode. Again, this is not to disparage the Dr. Wolfram’s point about math education. In fact, I think he would agree with me that there is a fundamental need to have peoplewhoknowthebasicsverywelltobemakingsurethecomputermodelswe arerunningareaccurate. Butthisisallaverylongwayofansweringthequestion,whydowewanttowork outthesecrazyintegralsandsumswhenwehavegreatcomputersoftwarethatcando itforus?Buteveniftherewerenoneedforcustodes,evenifthelikesofMathematica wereperfectandneedednochecking,thereisamoreimportant,overrridingreason toworkouttheseintegralsandsumsbyhand: Itisfun! Ifyouareperusingthroughthisbook,itisnotlikelybecauseyouhavecalculus homeworkduetomorrow.Itisbecauseyouhaveacuriosityastohowreallydifficult problemsincalculuscanbesolvedusingcomplexintegration.Itisbecauseyouwant tolearnthesemethodsforyourself.Thereisapowerbehindthemthatisaddictive. It is like the classic line, “amuse your friends, confound your enemies” taken in a wholenewdirection. Preface ix WhyComplexIntegration? Ithinkwehaveansweredthequestionofwhyevaluatetheseintegralsandsumsby handingeneral.Butunansweredisthequestion,whyarewefocusingoncomplex integrationtechniques? TheoriginofthecomplexintegrationtechniquesisaresultofmyworkontheMath StackExchangewebsite.Onthiswebsite,therearelotsofpeopleaskingotherpeople likemehowtoevaluatevariousintegralsandsums.Infact,therewouldbeseveral ofusworkingontheseproblemsatonce,whichwouldresultinverysimilar,good answers being offered at roughly the same time. Most of the time, these answers would use fundamentally real techniques—“Feynman’s trick” of differentiating a parameterundertheintegralsign,Taylorexpansion,orthogonalfunctiondecompo- sition,forexample.IstartedevaluatingintegralsusingtheResidueTheoremjustas awaytoputdownadifferentanswer. Itoccurredtome,however,thattheResidueTheoremasIhadlearneditinschool was not really a great arrow in my quiver of integration techniques I could bring tothetable—yet.TheexamplesIhadlearnedinschool—andthesameexamplesI wasworkingoutintheStackExchangeproblems—wereextremelylimitedtoafew, scientificallyselectedexamples.Forexample,wearetaughttoevaluatethefollowing integralusinganalysisofbranchcutsinamultivaluedintegrand: (cid:5)1 dxxp(1−x)q 0 It turns out, however, that the techniques we are taught only work when p + q is an (cid:6)integer. So many times we would be taught to perform inte- grals such as 1dxx−1/3(1 − x)−2/3, but not something more interesting like (cid:6) 0 1dxx−1/3(1−x)−2/3logx.(Notethatthefactoroflogmaybetreatedasthelimit 0 ofthederivativeof xa asa → 0.)Thereasonwhythisissoandtheevaluationof themoredifficultintegralusingcomplexintegrationwillbeprovidedinthisbook. Howwouldwegoaboutattackingthesenewproblems?Bydoingthemcorrectly! For example, inthe branch cut integrals above, the usual way to approach them is toconsidersomethingcalleda“dog-bone”contourformedbythebranchcutsabove and below the integration interval. We would then need to consider a “residue at infinity” to handle the effect of integrating out to infinity—the residue at infinity wassimplyaformuladerivedfromanintegralaboutalargecircle.Butwhatifthe integralaboutthelargecircledidn’tconvergeastheradiusbecameinfinite?Whatif, instead,therewasasingularpiecethathadtocancelanothersimilarpieceelsewhere on the contour? By “the contour,” I no longer mean the dog-bone contour but the logicalextensionofittothelargecircle. The cancelations of divergences formed the impetus for attacking whole new classes of problems using complex integration. This is one of the main themes throughoutthisbook. x Preface Anotherthemeistheexpansionofourquiverofproblem-solvingtechniques.Once we know a more complete form of problems that may be attacked using complex integration techniques, we may attempt to transform difficult integrals into one of thoseforms.Awell-knownexampleofthisisthefollowingintegral. (cid:5)1 1(cid:7)1+x (cid:8)1+2x +2x2(cid:9) (cid:10) dx log =4πCot−1 φ x 1−x 1−2x +2x2 −1 whereφisthegoldenratio.Iattackedthisproblembytransformingtheintegralinto thefollowingform. (cid:5)1 (cid:7) (cid:8) (cid:9) (cid:5)∞ 1 1+x 1+2x +2x2 dx log = dy p(y)logy x 1−x 1−2x +2x2 −1 0 where p is a rational function of y. I did this because I knew that integrals of this form, where the poles of p are determined exactly, could be evaluated using the ResidueTheorem.Inthiscase,eventhoughthedenominatorof pwasdegreeeight, Iwasabletofindthepolesbecauseofthehighdegreeofsymmetryof p. Goals What are the goals one should expect to achieve by reading this book? Generally, to appreciate whole new types of integrals that may be evaluated using complex integrationtechniques.Specifically,however,thereareseveralgoals. 1. Appreciatethatpolesandbranchpointsmaybetreatedinaunifiedmannerwhen itcomestoevaluatingrealintegralsusingcomplexintegration. 2. Gainperspectiveonthetypesofintegralsthatmaybeevaluatedusingcomplex integration. 3. Learnhowtoconstructevenreasonablysophisticatedcontoursinevaluatingreal integralsusingcomplexintegration. 4. BecomecomfortablewithCauchyprincipalvaluesandcancelingdivergencesas partofevaluatingintegralswithcombinedpole/branchpoints. 5. AppreciatetransformtechniquessuchasParseval’sTheoremandtheConvolution Theoremforevaluatingintegrals. 6. Gainfacilitywithasymptotictechniquesandhowtheymaybeusedtoevaluate realintegralsandsumsexactly.
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