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1、MASSACHUSETTSINSTITUTEOFTECHNOLOGYPhysicsDepartment8.323:RelativisticQuantumFieldTheoryINOTESONTHEEULER-MACLAURINSUMMATIONFORMULAThesenotesareintendedtosupplementtheCasimireffectproblemofProblemSet4.ThatcalculationdependedcruciallyontheEuler-Maclaurinsummation
2、formula,whichwasstatedwithoutderivation.HereIwillgiveaself-containedderivationoftheEuler-Maclaurinformula.ForpedagogicalreasonsIwillfirstderivetheformulawithoutanyreferencetoBernoullinumbers,andafterwardIwillshowthattheanswercanbeex-pressedintermsofthesenumber
3、s.Anexplicitexpressionwillbeobtainedforthere-mainderthatsurvivesafterafinitenumberoftermsintheseriesaresummed,andinanoptionalappendixIwillshowhowtosimplifythisremaindertoobtaintheformgivenbyAbramowitzandStegun.TheEuler-Maclaurinformularelatesthesumofafunctione
4、valuatedatevenlyspacedpointstothecorrespondingintegralapproximation,providingasystematicmethodofcal-culatingcorrectionsintermsofthederivativesofthefunctionevaluatedattheendpoints.Considerfirstafunctiondefinedontheinterval−1≤x≤1,forwhichwecanimagineapproximating
5、thesumoff(−1)+f(1)bytheintegralofthefunctionovertheinterval:1f(−1)+f(1)=dxf(x)+R1,(1)−1whereR1representsacorrectiontermthatwewanttounderstand.OnecanfindanexactexpressionforR1byapplyinganintegrationbypartstotheintegral:11dxf(x)=f(−1)+f(1)−dxxf(x),(2)−1−1so
6、1R=dxxf(x),(3)1−1whereaprimedenotesaderivativewithrespecttox.EULER-MACLAURINSUMFORMULA,8.323,SPRING2003p.21.Expansionbysuccessiveintegrationsbyparts:Wewantanapproximationthatisusefulforsmoothfunctionsf(x),andasmoothfunctionisoneforwhichthehigherderivativeste
7、ndtobesmall.Therefore,ifwecanextractmoretermsinawaythatleavesaremaindertermthatdependsonlyonhighderivativesofthefunction,thenwehavemadeprogress.Thiscanbeaccomplishedbysuccessivelyintegratingbyparts,eachtimedifferentiatingf(x)andintegratingthefunctionthatmultip
8、liesit.WecandefineasetoffunctionsV0(x)≡1,V1(x)≡x,(4)andVn(x)≡dxVn−1(x).(5)Eq.(5)isnotquitewell-defined,however,becauseeachindefiniteintegralisdefinedonlyuptoanarbitraryconstantofintegration.
