mathematical methods for physicists PDF
Preview mathematical methods for physicists
MATHEMATICAL METHODS FOR PHYSICISTS SEVENTH EDITION MATHEMATICAL METHODS FOR PHYSICISTS A Comprehensive Guide SEVENTH EDITION George B. Arfken MiamiUniversity Oxford,OH Hans J. Weber UniversityofVirginia Charlottesville,VA Frank E. Harris UniversityofUtah,SaltLakeCity,UT and UniversityofFlorida,Gainesville,FL AMSTERDAM•BOSTON•HEIDELBERG•LONDON NEWYORK•OXFORD•PARIS•SANDIEGO SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO AcademicPressisanimprintofElsevier AcademicPressisanimprintofElsevier 225WymanStreet,Waltham,MA02451,USA TheBoulevard,LangfordLane,Kidlington,Oxford,OX51GB,UK ©2013ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicor mechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem,without permissioninwritingfromthepublisher.Detailsonhowtoseekpermissionandfurtherinformation aboutthePublisher’spermissionspoliciesandourarrangementswithorganizationssuchastheCopyright ClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions. ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher (otherthanasmaybenotedherein). Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperience broadenourunderstanding,changesinresearchmethods,professionalpractices,ormedicaltreatment maybecomenecessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingand usinganyinformation,methods,compounds,orexperimentsdescribedherein.Inusingsuchinformation ormethodstheyshouldbemindfuloftheirownsafetyandthesafetyofothers,includingpartiesfor whomtheyhaveaprofessionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeany liabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproductsliability,negligence orotherwise,orfromanyuseoroperationofanymethods,products,instructions,orideascontainedin thematerialherein. LibraryofCongressCataloging-in-PublicationData Applicationsubmitted. BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary. ISBN:978-0-12-384654-9 ForinformationonallAcademicPresspublications, visitourwebsite:www.elsevierdirect.com Typesetby:diacriTech,India PrintedintheUnitedStatesofAmerica 12 13 14 9 8 7 6 5 4 3 2 1 P REFACE This,theseventheditionofMathematicalMethodsforPhysicists,maintainsthetradition setbythesixpreviouseditionsandcontinuestohaveasitsobjectivethepresentationofall themathematicalmethodsthataspiringscientistsandengineersarelikelytoencounteras studentsandbeginningresearchers.Whiletheorganizationofthiseditiondiffersinsome respectsfromthatofitspredecessors,thepresentationstyleremainsthesame:Proofsare sketched for almost all the mathematical relations introduced in the book, and they are accompaniedbyexamplesthatillustratehowthemathematicsappliestoreal-worldphysics problems.Largenumbersofexercisesprovideopportunitiesforthestudenttodevelopskill intheuseofthemathematicalconceptsandalsoshowawidevarietyofcontextsinwhich themathematicsisofpracticaluseinphysics. Asinthepreviouseditions,themathematicalproofsarenotwhatamathematicianwould considerrigorous,buttheyneverthelessconveytheessenceoftheideasinvolved,andalso provide some understanding of the conditions and limitations associated with the rela- tionshipsunderstudy.Noattempthasbeenmadetomaximizegeneralityorminimizethe conditions necessary to establish the mathematical formulas, but in general the reader is warned of limitations that are likely to be relevant to use of the mathematics in physics contexts. T S O THE TUDENT Themathematicspresentedinthisbookisofnouseifitcannotbeappliedwithsomeskill, andthedevelopmentofthatskillcannotbeacquiredpassively,e.g.,bysimplyreadingthe text and understanding what is written, or even by listening attentively to presentations by your instructor. Your passive understanding needs to be supplemented by experience in using the concepts, in deciding how to convert expressions into useful forms, and in developingstrategiesforsolvingproblems.Aconsiderablebodyofbackgroundknowledge xi xii Preface needstobebuiltupsoastohaverelevantmathematicaltoolsathandandtogainexperi- enceintheiruse.Thiscanonlyhappenthroughthesolvingofproblems,anditisforthis reason thatthe text includesnearly 1400 exercises, manywith answers (butnot methods ofsolution).Ifyouareusingthisbookforself-study,orifyourinstructordoesnotassign aconsiderablenumberofproblems,youwouldbewelladvisedtoworkontheexercises untilyouareabletosolveareasonablefractionofthem. This book can help you to learn about mathematical methods that are important in physics, as well as serve as a reference throughout and beyond your time as a student. Ithasbeenupdatedtomakeitrelevantformanyyearstocome. W ’ N HAT S EW Thisseventheditionisasubstantialanddetailedrevisionofitspredecessor;everywordof thetexthasbeenexaminedanditsappropriacyandthatofitsplacementhasbeenconsid- ered.Themainfeaturesoftherevisionare:(1)Animprovedorderoftopicssoastoreduce theneedtouseconceptsbeforetheyhavebeenpresentedanddiscussed.(2)Anintroduc- tory chapter containing material that well-prepared students might be presumed to know and which will be relied on (without much comment) in later chapters, thereby reducing redundancyinthetext;thisorganizationalfeaturealsopermitsstudentswithweakerback- groundstogetthemselvesreadyfortherestofthebook.(3)Astrengthenedpresentationof topicswhoseimportanceandrelevancehasincreasedinrecentyears;inthiscategoryare the chapters on vector spaces, Green’s functions, and angular momentum, and the inclu- sionofthedilogarithmamongthespecialfunctionstreated.(4)Moredetaileddiscussion ofcomplexintegrationtoenablethedevelopmentofincreasedskillinusingthisextremely importanttool.(5)Improvementinthecorrelationofexerciseswiththeexpositioninthe text,andtheadditionof271newexerciseswheretheyweredeemedneeded.(6)Addition of a few steps to derivations that students found difficult to follow. We do not subscribe tothepreceptthat“advanced”means“compressed”or“difficult.”Wherevertheneedhas beenrecognized,materialhasbeenrewrittentoenhanceclarityandeaseofunderstanding. In order to accommodate new and expanded features, it was necessary to remove or reduce in emphasis some topics with significant constituencies. For the most part, the material thereby deleted remains available to instructors and their students by virtue of itsinclusionintheon-linesupplementarymaterialforthistext.On-lineonlyarechapters onMathieufunctions,onnonlinearmethodsandchaos,andanewchapteronperiodicsys- tems. These are complete and newly revised chapters, with examples and exercises, and are fully ready for use by students and their instuctors. Because there seems to be a sig- nificant population of instructors who wish to use material on infinite series in much the same organizational pattern as in the sixth edition, that material (largely the same as in theprintedition,butnotallinoneplace)hasbeencollectedintoanon-lineinfiniteseries chapterthatprovidesthismaterialinasingleunit.Theon-linematerialcanbeaccessedat www.elsevierdirect.com. Preface xiii P M ATHWAYS THROUGH THE ATERIAL This book contains more material than an instructor can expect to cover, even in a two-semestercourse.Thematerialnotusedforinstructionremainsavailableforreference purposes or when needed for specific projects. For use with less fully prepared students, a typical semester course might use Chapters 1 to 3, maybe part of Chapter 4, certainly Chapters5to7,andatleastpartofChapter11.Astandardgraduateone-semestercourse might have the material in Chapters 1 to 3 as prerequisite, would cover at least part of Chapter 4, all of Chapters 5 through 9, Chapter 11, and as much of Chapters 12 through 16 and/or 18 as time permits. A full-year course at the graduate level might supplement theforegoingwithseveraladditionalchapters,almostcertainlyincludingChapter20(and Chapter 19 if not already familiar to the students), with the actual choice dependent on the institution’s overall graduate curriculum. Once Chapters 1 to 3, 5 to 9, and 11 have beencoveredortheircontentsareknowntothestudents,mostselectionsfromtheremain- ing chapters should be reasonably accessible to students. It would be wise, however, to includeChapters15and16ifChapter17isselected. A CKNOWLEDGMENTS This seventh edition has benefited from the advice and help of many people; valuable advicewasprovidedbothbyanonymousreviewersandfrominteractionwithstudentsat the University of Utah. At Elsevier, we received substantial assistance from our Acqui- sitions Editor Patricia Osborn and from Editorial Project Manager Kathryn Morrissey; production was overseen skillfully by Publishing Services Manager Jeff Freeland. FEH gratefullyacknowledgesthesupportandencouragementofhisfriendandpartnerSharon Carlson. Without her, he might not have had the energy and sense of purpose needed to helpbringthisprojecttoatimelyfruition. CHAPTER 1 M ATHEMATICAL P RELIMINARIES This introductory chapter surveys a number of mathematical techniques that are needed throughoutthebook.Someofthetopics(e.g.,complexvariables)aretreatedinmoredetail inlaterchapters,andtheshortsurveyofspecialfunctionsinthischapterissupplemented byextensivelaterdiscussionofthoseofparticularimportanceinphysics(e.g.,Besselfunc- tions).Alaterchapteronmiscellaneousmathematicaltopicsdealswithmaterialrequiring more background than is assumed at this point. The reader may note that the Additional Readingsattheendofthischapterincludeanumberofgeneralreferencesonmathemati- calmethods,someofwhicharemoreadvancedorcomprehensivethanthematerialtobe foundinthisbook. 1.1 INFINITE SERIES Perhaps the most widely used technique in the physicist’s toolbox is the use of infinite series(i.e.,sumsconsistingformallyofaninfinitenumberofterms)torepresentfunctions, tobringthemtoformsfacilitatingfurtheranalysis,orevenasapreludetonumericaleval- uation.Theacquisitionofskillincreatingandmanipulatingseriesexpansionsistherefore anabsolutelyessentialpartofthetrainingofonewhoseekscompetenceinthemathemat- icalmethodsofphysics,anditisthereforethefirsttopicinthistext.Animportantpartof thisskillsetistheabilitytorecognizethefunctionsrepresentedbycommonlyencountered expansions,anditisalsoofimportancetounderstandissuesrelatedtotheconvergenceof infiniteseries. 1 MathematicalMethodsforPhysicists.DOI:10.1016/B978-0-12-384654-9.00001-3 ©2013ElsevierInc.Allrightsreserved. 2 Chapter1 MathematicalPreliminaries FundamentalConcepts The usual way of assigning a meaning to the sum of an infinite number of terms is by introducingthenotionofpartialsums.Ifwehaveaninfinitesequenceoftermsu ,u ,u , 1 2 3 u ,u ,...,wedefinetheithpartialsumas 4 5 i s u . (1.1) i n = n 1 X= This is a finite summation and offers no difficulties. If the partial sums s converge to a i finitelimitasi , →∞ lim s S, (1.2) i i = →∞ the infinite series ∞n 1un is said to be convergent and to have the value S. Note that wedefinetheinfinite=seriesasequalto S andthatanecessaryconditionforconvergence P to a limit is that lim u 0. This condition, however, is not sufficient to guarantee n n convergence. →∞ = Sometimes it is convenient to apply the condition in Eq. (1.2) in a form called the Cauchy criterion, namely that for each ε > 0 there is a fixed number N such that s s <ε foralli and j greaterthan N.Thismeansthatthepartialsumsmustcluster j i | − | togetheraswemovefaroutinthesequence. Someseriesdiverge,meaningthatthesequenceofpartialsumsapproaches ;others ±∞ mayhavepartialsumsthatoscillatebetweentwovalues,asforexample, ∞ u 1 1 1 1 1 ( 1)n . n = − + − + −···− − +··· n 1 X= This series does not converge to a limit, and can be called oscillatory. Often the term divergent is extended to include oscillatory series as well. It is important to be able to determinewhether,orunderwhatconditions,aserieswewouldliketouseisconvergent. Example 1.1.1 THEGEOMETRICSERIES The geometric series, starting with u 1 and with a ratio of successive terms r 0 = = u /u ,hastheform n 1 n + 1 r r2 r3 rn 1 . − + + + +···+ +··· Itsnthpartialsums (thatofthefirstn terms)is1 n 1 rn sn − . (1.3) = 1 r − Restricting attention to r <1, so that for large n,rn approaches zero, and s possesses n | | thelimit 1 lim s , (1.4) n n = 1 r →∞ − 1Multiplyanddividesn= nm−10rmby1−r. = P 1.1 InfiniteSeries 3 showingthatfor r <1,thegeometricseriesconverges.Itclearlydiverges(orisoscilla- tory)for r 1,a|s|theindividualtermsdonotthenapproachzeroatlargen. (cid:4) | |≥ Example 1.1.2 THEHARMONICSERIES Asasecondandmoreinvolvedexample,weconsidertheharmonicseries ∞ 1 1 1 1 1 1 . (1.5) n = + 2 + 3 + 4 +···+ n +··· n 1 X= The terms approach zero for large n, i.e., lim 1/n 0, but this is not sufficient to n guaranteeconvergence.Ifwegrouptheterms(w→it∞houtch=angingtheirorder)as 1 1 1 1 1 1 1 1 1 1 , + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +···+ 16 +··· (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) eachpairofparenthesesencloses ptermsoftheform 1 1 1 p 1 > . p 1 + p 2 +···+ p p 2p = 2 + + + Formingpartialsumsbyaddingtheparentheticalgroupsonebyone,weobtain 3 4 5 n 1 s1 1, s2 , s3> , s4> ,..., sn> + , = = 2 2 2 2 andweareforcedtotheconclusionthattheharmonicseriesdiverges. Although the harmonic series diverges, its partial sums have relevance among other placesinnumbertheory,where H n m 1 aresometimesreferredtoasharmonic numbers. n = m=1 − (cid:4) P We now turn to a more detailed study of the convergence and divergence of series, consideringhereseriesofpositiveterms.Serieswithtermsofbothsignsaretreatedlater. ComparisonTest Iftermbytermaseriesoftermsu satisfies0 u a ,wherethea formaconvergent n n n n ≤ ≤ series, then the series u is also convergent. Letting s and s be partial sums of the n n i j j u series, with j >i, the difference s s is u , and this is smaller than the P j − i n i 1 n correspondingquantityforthea series,therebyprov=in+gconvergence.Asimilarargument P showsthatiftermbytermaseriesoftermsv satisfies0 b v ,wheretheb forma n n n n ≤ ≤ divergentseries,then v isalsodivergent. n n Fortheconvergentseriesa wealreadyhavethegeometricseries,whereastheharmonic n P series will serve as the divergent comparison series b . As other series are identified as n eitherconvergentordivergent,theymayalsobeusedastheknownseriesforcomparison tests. 4 Chapter1 MathematicalPreliminaries Example 1.1.3 ADIVERGENTSERIES Ttheestdive∞nr=g1ennt−hpa,rpm=oni0c.9s9e9ri,efso,rthceoncvoemrgpeanricseo.nStiensctesnh−ow0.9s99th>atn−1 nand0.9b9n9=isnd−iv1efrogremnts. n − GenePralizing, n p isseentobedivergentforall p 1. (cid:4) n − ≤ P P CauchyRootTest If (a )1/n r <1 for all sufficiently large n, with r independent of n, then a is convnergent≤.If(a )1/n 1forallsufficientlylargen,then a isdivergent. n n n ≥ n n P Thelanguageofthistestemphasizesanimportantpoint:Theconvergenceordivergence P ofaseriesdependsentirelyonwhathappensforlargen.Relativetoconvergence,itisthe behaviorinthelarge-n limitthatmatters. Thefirstpartofthistestisverifiedeasilybyraising(a )1/n tothenthpower.Weget n a rn<1. n ≤ Sincern isjustthenthterminaconvergentgeometricseries, a isconvergentbythe n n comparisontest.Conversely,if(a )1/n 1,thena 1andtheseriesmustdiverge.This n n ≥ ≥ P roottestisparticularlyusefulinestablishingthepropertiesofpowerseries(Section1.2). D’Alembert(orCauchy)RatioTest If a /a r <1 for all sufficiently large n and r is independent of n, then a is convn+er1gennt.≤Ifa /a 1forallsufficientlylargen,then a isdivergent. n n Thistestisestna+b1lishned≥bydirectcomparisonwiththegeometnricnseries(1 r rP2 ). P + + +··· Inthesecondpart,a a anddivergenceshouldbereasonablyobvious.Althoughnot n 1 n quiteassensitiveasth+eC≥auchyroottest,thisD’Alembertratiotestisoneoftheeasiestto applyandiswidelyused.Analternatestatementoftheratiotestisintheformofalimit:If <1, convergence, a n 1 lim + >1, divergence, (1.6) n→∞ an 1, indeterminate. = Becauseofthisfinalindeterminatepossibility,theratiotestislikelytofailatcrucialpoints, andmoredelicate,sensitiveteststhenbecomenecessary.Thealertreadermaywonderhow thisindeterminacyarose.Actuallyitwasconcealedinthefirststatement,a /a r < n 1 n 1. We might encounter a /a <1 for all finite n but be unable to choo+se an≤r <1 n 1 n and independent of n suc+h that a /a r for all sufficiently large n. An example is n 1 n providedbytheharmonicseries,for+which≤ a n n 1 + <1. a = n 1 n + Since a n 1 lim + 1, n an = →∞ nofixedratior <1existsandthetestfails.
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