the theory of riemann intergration

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1、September26,2000TheTheoryofRiemannIntegration11TheIntegralThroughtheworkoncalculus,particularlyintegration,anditsapplica-tionthroughoutthe18thcenturywasformidable,therewasnoactual“theory”forit.Theapplicationsofcalculustoproblemsofphysics,i.e.partialdifferentialequations,andthefl

2、edglingideasoffunctionrepresentationbytrigonometricseriesrequiredclarificationofjustwhatafunctionwas.Correspondingly,thischallengedthenotionthatanintegralisjustanantiderivative.Let’stracethisdevelopmentoftheintegralasaroughandreadywaytosolveproblemsofphysicstoafull-fledgedtheory

3、.Webeginthestorywithsequenceofevents..........1.LeonhardEuler(1707-1783)andJeand’Alembert(1717-1783)arguein1730-1750’soverthe“type”ofsolutionsthatshouldbeadmit-tedassolutionstothewaveequationuxy=0D’Alembertshowedthatasolutionmusthavetheform1F(x;t)=[f(x+t)+f(x¡t)]:2Fort=0wehaveth

4、einitialshapef(x).Note:Hereafunctionisjustthat.Thenewnotationanddesigna-tionarefixed.Butjustwhatkindsoffunctionsfcanbeadmitted?1°c2000,G.DonaldAllenTheRiemannIntegral22.D’Alembertarguedfmustbe“continuous”,i.e.givenbyasingleequation.Eulerarguedtherestrictiontobeunnecessaryandthat

5、fcouldbe“discontinuous”,i.e.itcouldbeformedofmanycurves.Inthemodernsensethoughbotharecontinuous.3.DanielBernoulli(1700-1782)enteredthefraybyannouncingthatsolutionsmustbeexpressibleinaseriesoftheformf(x)=a1sin(¼x=L)+a2sin(2¼x=L)+¢¢¢;whereListhelengthofthestring.Euler,d’Alembertan

6、dJosephLagrange(1736-1813)stronglyre-jectthis.4.Inthe19thcenturythenotionofarbitraryfunctionagaintookcenterstagewhenJosephFourier(1768-1830)presentedhiscelebratedpaper2onheatconductiontotheParisAcademy(1807).Initsmostgeneralform,Fourier’spropositionstates:Any(bounded)functionfde

7、finedon(¡a;a)canbeexpressedas1X1f(x)=a0+ancosn¼x=a+bnsinn¼x=a;2n=1where2Zaa0=f(x)dxa¡aZ1aan=f(x)cosn¼x=adx;a¡aZ1abn=f(x)sinn¼x=adx:a¡a5.ForFourierthenotionoffunctionwasrootedinthe18thcentury.Inspiteofthegeneralityofhisstatementsa“general”functionforhimwasstillcontinuousinthemode

8、rnsense.Forexample,hewouldcall8>:e