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INTEGRATION: THE FEYNMAN WAYANONYMOUS

Abstract. In this paper we will learn a common technique not often described in collegiate calculus courses. After reviewing the necessary theory,wewillproceedtoworkthroughsometypicalexamples.Throughoutthisprocess,wewillseetrivialintegralsthatcanbeevaluatedusingbasictechniquesofintegration(suchasintegrationbyparts),howeverwewillalsoencounterintegralsthatwouldotherwiserequiremoreadvancedtechniquessuchascontourintegration.

1. IntroductionManyup-and-comingmathematicians,beforeeveryreachingtheuniversitylevel,

heard about a certain method for evaluating definite integrals from the followingpassage in [1]:

OnethingIneverdidlearnwascontourintegration. Ihadlearnedto do integrals by various methods show in a book that my highschoolphysicsteacherMr. Baderhadgivenme.

The book also showed how to differentiate parameters underthe integralsign- Itsacertainoperation. Itturnsoutthatsnottaughtverymuchintheuniversities;theydontemphasizeit. ButI caught on how to use that method, and I used that one damntoolagainandagain. SobecauseIwasself-taughtusingthatbook,Ihadpeculiarmethodsofdoingintegrals.

Theresultwasthat,whenguysatMITorPrincetonhadtroubledoing a certain integral, it was because they couldnt do it withthestandardmethodstheyhadlearnedinschool. Ifitwascontourintegration, they would have found it; if it was a simple seriesexpansion, they would have found it. Then I come alongand trydifferentiating under the integral sign, and often it worked. So Igotagreatreputationfordoingintegrals,onlybecausemyboxoftools was different from everybody elses, and they had tried alltheirtoolson itbeforegivingtheproblemtome.

ThemethodMr. Feynmanisreferringtooftengoesbythenameofdifferentiatingunder the integral sign, differentiationwith respect to a parameter, or sometimesevenFeynmanIntegration. Howeveronewishestonameit,theeleganceandappeallies in how this method can be employed to evaluate seemingly complex integralswithnothingmorethan1 elementarycalculus.

1OnceonegetspastthemeasuretheoryrequiredtoprovetheTheorem2.1

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2. SomeKeyTheoremsThe technique of Feynman Integration is a simple application of a theorem

attributedtoLeibniz. Inthissectionwestatethetheorem in itsmostbasic form,andendbystatingamoregeneralversionthatallowsforevenweakerhypotheses.Inbothcases,weaddresssituationswherethefollowingequation(whichwewouldlovetobetrue)holds:

d f(x, y)dy= f(x, y)dy.

dx Y Y xBeforestatingthesetheorems,recallthatdifferentiation issimplyaparticularexampleofalimitinsofaraswedefine

df(x):=:f(x):= lim f(x+h)f(x),

dx h 0 hwithatruedefinitiononthefarright. Thus,weseethat(2)willholdwheneverwemay

make

the

following

statement,

lim f(x, y)dy= limf(x, y)dy.x a x a Y Y

Theorem2.1(ElementaryCalculusVersion). Letf : [a, b] Y Rbeafunction,with [a, b] being a closed interval, and Y being a compact subset ofRn. Supposethat both f(x, y)and f(x, y)/xare continuous in the variables xand yjointly.Then f(x, y)dyexistsasacontinuouslydifferentiablefunctionofxon[a, b],with

Yderivative

ddx Y f(x,y)dy= Y

xf(x,y)dy.

Asmentionedabove,theveracityof(2)iscompletelydependentuponifwecanexchangetheoperationsoflimitingandintegration. Ifweweretoprovetheabovetheorem, our argument would make full use of the compactness of Y, which ofcourseimpliesuniform continuity. Fromthisfact,wecouldshowthatitisjustifiedtoswitchchangetheorderoflimitsand integration,thusproving(2).However, in many cases the restriction of compactness can be too severe. Oftentimes we would like Y to be (, a),(a,),(,),etc... In these situations,thefollowingmeasuretheoreticversionoftheabovecomestoourrescue:Theorem2.2(MeasureTheoryVersion). LetX beanopensubsetofR,andbeameasurespace. Supposef :XRsatisfies thefollowingconditions:

(1) f(x, ) isaLebesgue-integrablefunctionof foreach xX.(2) Foralmostall, thederivative f(x, )/xexistsforallxX.(3) There isan integrablefunction : Rsuch that f(x, )/x ()

forall xX. | |

ThenforallxX,d

f(x, )d= f(x, )d.dx x

A sketch of the proof of Theorem 2.2 would most likely make some form of afamous result from measure theory, the Dominated Convergence Theorem. Thiswill of course provide us with thejustification to switch the order of limit and

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INTEGRATION: THE FEYNMAN WAY 3

integration. For the interested reader, we state the theorem whose proof may befound in [5]:Theorem2.3(DominatedConvergenceTheorem).LetXbeameasurespace,andlet , f1, f2, . . . bemeasurablefunctions such that

1,

all

conditions

of

Theorem

2.1

are

satisfied

and

we

may

differentiateunderthe integralsign:

0

d x xI(b) =

1 b 1dx=

1 b 1dx

db 0 logx 0 b logx 1 b+11b x = x =

b+ 100

1=

b+ 1whereuponintegrationyields

I(b)=log(b+1)+C.Inordertofindoutourconstantofintegration,we letb=0sothatour integrandis0,implyingthatC=0. Lettingb=2willofcoursesolveouroriginalproblem: 1 2x

log

x

1dx=I(2)=log(3).

0Example3.2. Compute the improperdefinite integral,

sin(x)dx.

x

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Asbefore,wemuststrategically introduceaparametersothatwecanactuallyuseourtheorems. Inthisexample,wegeneralizebysolvingthefollowingintegral

sin(x)I(b) = ebxdx,

x

0whereupon setting b = 0 and doubling will give us the desired value. But beforewe proceed, how do we know that we can indeed differentiate under the integralas we would hope? As mentioned in the previous section, it is clear (why?) thatour integrand is Lebesgue integrable and differentiable a.e.; all that remains is toverifythatitisdominated. Thekeyhereistorealizethatsince|sin(x)| |x|,thisimpliesthat

= ebx ebxsin(x) sin(x)ebx .x x

Lastlysince 1

ebx

dx=