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TheycanbeusedtoteachandillustratealargepartofmodernmathematicsTheyoccurassoonasonegoesbeyondtheleadingorderinperturbativequan-tum fieldtheory.WritetheResultreputationfordoingintegrals,onlybecausemyboxoftoolswasdierentfromeverybodyelse’s,andtheyhadtriedalltheirtoolsonit beforegivingtheproblemapproach,usingFeynman’strickyAsbefore,wesimplifymattersbycomputingtherighthalfoftheintegral,Zex2dx:SetI(t):=Zex+ (x=t)2dx:Thisisdefinedforallt>0,andwe’rehopingtodetermineI(1)HomeResultChapterIntroductionStep–Orient&SetaGoalIdentifysomeofthe hardestormostimportantconceptsyouwanttolearnWritethenameoftheconceptatthetopofablankpieceofpaperLet'schoosethemethodyoumightbe inclinedtouseonthisintegral:integrationbypartsIfyouchoosetodoitthiswaythenyou'llhaveZdr=eIntheselecturenoteswegiveanintroductiontothe verywideandactiveeldofFeynmanintegralsandthetechniquesusedtoevaluatethemAComprehensiveTreatmentforStudentsandResearchersWeassume familiaritywiththebasicideasofperturbativequantumeldtheoryandFeynmandiagrams,butintroducealloftheconceptsthatareusedintheexamplecalculations below.Step–Orient&SetaGoalIdentifysomeofthehardestormostimportantconceptsyouwanttolearn.Theyoccurassoonasonegoesbeyondtheleading orderinperturbativequantumfieldtheoryInlargepartsthesetheideabehindtheFeynmanTechnique(Sincexisthevariableofintegration,xisnotaparameter) Ingeneral,wemightwritesuchanintegralasTherearetwowaystoevaluatethisintegral,bothofwhichareeasy(atthisstage)butoneisslightlyfasterLet's choosethemethodyoumightbeinclinedtouseonResultFeynmanintegralsareindispensableforprecisioncalculationsinquantumfieldthe-oryrInbothcases, weaddresssituationswherethefollowingequation(whichwewouldlovetobetruetomeRichardFeynman[5,pp{72]IntroductionThemethodofdierentiation undertheintegralsign,duetoLeibnizin[4],concernsintegralsdependingonaparameter,suchasRx2etxdxInthissectionwestatethetheoreminitsmostbasic form,andendbystatingamoregeneralversionthatallowsforevenweakerhypotheses.re.Feynmanintegralsareindispensableforprecisioncalculationsinquantum fieldtheoryFeynmanintegralsarealsofascinatingfromamathematicalpointofviewFeynmanintegralsarealsofascinatingfromamathematicalpointofview Feynmanintegralsareindispensableforprecisioncalculationsinquantumfieldthe-oryTheyoccurassoonasonegoesbeyondResultFeynman’sFavoriteTrick Leibniz’sFormulaThestartingpointforFeynman’strickof‘differentiatingundertheintegralsign,’mentionedattheendofChap,ResultEvaluateFeynmanintegrals TherearetwostrategiestosolveaFeynmanintegral:ReductionExpresstheintegralwiththeaidofrecurrencerelationsbyother,Resulttheideabehindthe FeynmanTechniqueWesimplifythisintegralbymakingthenaturalsubstitution:I(t)=tZet2x+x2dx:I’mgratefultoBThetechniqueof“FeynmanIntegration”is asimpleapplicationofatheoremattributedtoLeibnizResultTherearetwowaystoevaluatethisintegral,bothofwhichareeasy(atthisstage)butoneisslightly faster.Heretistheextraparameter.Step–PracticeRecallWritedownanexplanationoftheconceptinyourownwords;beasthoroughaspossibleandFeynman integralsareindispensableforprecisioncalculationsinquantumfieldtheoryTheyoccurassoonasonegoesbeyondtheleadingorderinperturbativeResultFeynman Integrals:AComprehensiveTreatmentforStudentsandResearchersSpringerLink