函数幂级数毕业论文中英文资料外文翻译文献.docx

PowerSeriesExpansionandItsApp1.icationsIntheprevioussection,wediscusstheconvergenceofpowerseries,initsconvergenceregion,thepowerseriesa1.waysconvergestoafunction.Foi-thesimp1.epowerseries,buta1.sowithitemizedderivative,orquadraturemethods,findthisandfunction.Thissectionwi1.1.discussanotherissue,foranarbitraryfunction/(x),canbeexpandedinapowerseries,and1.aunchedinto.Whetherthepowerseries/(x)asandfunction?Thefo1.1.owingdiscussionwi1.1.addressthisissue.1Mac1.aurin(Mac1.aurin)formu1.aPo1.ynomia1.powerseriescanbeseenasanextensionofrea1.ity,soconsiderthefunction/（八）canexpandintopowerseries,youcanfromthefunction/(x)andpo1.ynomia1.sstarttoso1.vethisprob1.em.Tothisend,togiveherewithoutproofthefo1.1.owingformu1.a.Tay1.or(Tay1.or)formu1.a,ifthefunctionf(x)at.x=xvinaneighborhoodthatunti1thederivativeofordern+1,thenintheneighborhoodofthefo1.1.owingformu1.a:f(x)=f(-vn)+(axn)+(a-x0)'+-+(.r-x0)11+(x)(9-5-1)Among(x)=(-.vn,That()forthe1.agrangianremainder.That(9-5-1)-typeformu1.afortheTay1.or.Ifso=0,get/(.v)=(0)+.r+1+.r"+r,(),(9-5-2)tthispoint,4.(x)="厂V"*)I(m+1)!'(M+1.)!'Tha1.(9-5-2)typefor11u1.afortheMac1.aurin.Formu1.ashowsthatanyfundion/(.v)as1.ongasunti1the11+1.derivative,ncanbeequa1.toapo1.ynomia1.andaremainder.Weca1.1.thefo1.1.owingpowerseries/(-)=/(0)+,(0)x+.+1.(°).ro+2!j!(9-5-3)FortheMac1.aurinseries.So,isittof(x)fortheSumfunctions?IftheorderMac1.aurinseries(9-5-3)thefirst+1.iternsandforS11.(x),which5,i(x)=/(0)+,()x+零+-+Z2!！Then,theseries(9-5-3)convergestothefunctionf(x)theconditionsIimSnTa)=(x).NotingMac1.aurinformu1.a(9-5-2)andtheMac1.aurinseries(9-5-3)there1.ationshipbetweentheknown/(K)=SII“(x)U)Thus,when<,(x)=0There,/（八）=5,1()Viceversa.ThatifIimS“(x)=(x),Unitsmust,(x)=O.ThisindicatesthattheMac1.aurinseries(9-5-3)tof(x)andfunctionastheMac1.aurinformu1.a(9-5-2)oftheremainderterm,(x)0(whenjoo).Inthisway,wegetafunctionf(x)thePOWerseriesexpansion:/()=Y.rn=/(0)+/'(0)x+x"+.(9-5-4)仁！Itisthefunctionf()thepowerseriesexpression,if,thefunctionofthepowerseriesexpansionisunique.Infact,assumingthefunction/（八）canbeexpressedaspowerseries/()=an.r"=«0+aix+aix2+axn+,(9-55)We11,accordingtotheconvergenceofpowerseriescanbeitemizedwithinthenatureofderivation,andthenmakeK=O(powerseriesapparent1.yconvergesintheX=Opoint),itiseasytoget%=(0)M=f(0)x,a,=1.n,2!”!Substitutingtheminto(9-5-5)type,incomeandf(x)theMac1.aurinexpansionof(9-5-4)identica1.Insummary,ifthefunctionf(x)containszeroinarangeofarbitraryorderderivative,andinthisrangeofMac1.aurinformu1.aintheremaindertozeroasthe1imit(whenn-*,),then,thefunctionf(x)canstartformingas(9-5-4)typeofpowerseries.PowerSeries/(x)=()+(x-)+(x-)2+-7(x-)"sI!2!！KnownastheTay1.orseries.Second,primaryfundiono1.'powerseriesexpansionMac1.aurinformu1.ausingthefunctionf(x)expandedinpowerseriesmethod,ca1.1.edthec1.irectexpansionmethod.Example 1Testthefunction/()=eAexpandedinpowerseriesofx.So1.utionbecause%t)=e(rt=1.,2,3,)Therefore/(0)=f,(0)=*(0)=,"'(0)=I,Sowegetthepowerseries1.+.r+-.v1+-j,+,(9-5-6)2!n!Obvious1.y,(95-6)typeconvergenceinterva1.(-0>,+),As(9-5-6)whethertypef(x=e'isSumfunction,thatis,whetheritconvergestof(x)=eA,buta1.soexamineremainderrn(x).Becauseb(o<e<),且e,4同IM1.M，U?十1.)Thereforek=-1.r,<-1."1(M+1.)!11(n+y.'1Notingtheva1.ueofanyset,Kisafixedconstant,whiIetheseries(9-5-6)isabso1.ute1.yconvergent>sothegenera1.whentheitcmwhennoo,sowhenn-*,there(11+1)!>0,Fromthis1.imq(x)=0Thisindicatesthattheseries(9-5-6)doesconvergeto/(x)=er',thereforee1.=1+-x2+,+,+,(o<.r<+oo).2!nSuchuseofMac1aurinformu1.aareexpandedinpowerseriesmethod,a1.thoughtheprocedureisc1.ear,butoperatorsareoftentooCumbersome,soitisgenera1.1.ymoreconvenienttousethefo1.1.owingpowerseriesexpansionmethod.Priortothis,Wehavebeenafunctione,andsin.rpowerseries-xexpansion,theuseoftheseknownexpansionbypowerseriesofoperations,wecanachievemanyfunctionsofpowerseriesexpansion.Thisdemandfunctionofpowerseriesexpansionmethodisca1.1.edindirectexpansion.Example 2becauseFindthefunction/(x)=cos,x=0,Departmentinthepowerseriesexpansion.(SinX)'=cos.t,So1.utionsin.v=And3!5!'(2h+I)!Therefore,thepowerseriescanbeitemizedaccordingtotheru1.esofderivationcanbecos=1-+-A4+(-1.)"!x1.11+,(-co<.v<+<x)2!4!(2n)!Third,thefunctionpowerseriesexpansionoftheapp1.icationexamp1.eTheapp1.icationofpowerseriesexpansionisextensive,forexamp1.e,canuseittosetsomenumerica1.orotherapproximateca1.cu1.ationofintegra1.va1ue.Example 3 Usingtheexpansiontoestimatearciaii.vtheva1.ueof11.So1.utionbecausearctan1=-4Becauseofarctana,=A-+-+,(-1x1.),357SothereAvaiIab1.erightendofthefirstnitemsoftheseriesandasanapproximationof11.However,theconve