数学外文翻译

PowerSeriesExpansionandItsApplicationsIntheprevioussection,wediscusstheconvergenceofpowerseries,initsconvergenceregion,thepowerseriesalwaysconvergestoafunction.Forthesimplepowerseries,butalsowithitemizedderivative,orquadraturemethods,findthisandfunction.Thissectionwilldiscussanotherissue,foranarbitraryfunction()fx,canbeexpandedinapowerseries,andlaunchedinto.Whetherthepowerseries()fxasandfunction?Thefollowingdiscussionwilladdressthisissue.1Maclaurin(Maclaurin)formulaPolynomialpowerseriescanbeseenasanextensionofreality,soconsiderthefunction()fxcanexpandintopowerseries,youcanfromthefunction()fxandpolynomialsstarttosolvethisproblem.Tothisend,togiveherewithoutproofthefollowingformula.Taylor(Taylor)formula,ifthefunction()fxat0xxinaneighborhoodthatuntilthederivativeoforder1n,thenintheneighborhoodofthefollowingformula:20000()()()()()()nnfxfxxxxxxxrx…(9-5-1)Among10()()nnrxxxThat()nrxfortheLagrangianremainder.That(9-5-1)-typeformulafortheTaylor.Ifso00x,get2()(0)()nnfxfxxxrx…,(9-5-2)Atthispoint,(1)(1)111()()()(1)!(1)!nnnnnffxrxxxnn(01).That(9-5-2)typeformulafortheMaclaurin.Formulashowsthatanyfunction()fxaslongasuntilthe1nderivative,ncanbeequaltoapolynomialandaremainder.Wecallthefollowingpowerseries()2(0)(0)()(0)(0)2!!nnfffxffxxxn……(9-5-3)FortheMaclaurinseries.So,isitto()fxfortheSumfunctions?IftheorderMaclaurinseries(9-5-3)thefirst1nitemsandfor1()nSx,which()21(0)(0)()(0)(0)2!!nnnffSxffxxxn…Then,theseries(9-5-3)convergestothefunction()fxtheconditions1lim()()nnsxfx.NotingMaclaurinformula(9-5-2)andtheMaclaurinseries(9-5-3)therelationshipbetweentheknown1()()()nnfxSxrxThus,when()0nrxThere,1()()nfxSxViceversa.Thatif1lim()()nnsxfx,Unitsmust()0nrx.ThisindicatesthattheMaclaurinseries(9-5-3)to()fxandfunctionastheMaclaurinformula(9-5-2)oftheremainderterm()0nrx(whenn).Inthisway,wegetafunction()fxthepowerseriesexpansion:()()0(0)(0)()(0)(0)!!nnnnnfffxxffxxnn…….(9-5-4)Itisthefunction()fxthepowerseriesexpression,if,thefunctionofthepowerseriesexpansionisunique.Infact,assumingthefunctionf(x)canbeexpressedaspowerseries20120()nnnnnfxaxaaxaxax……,(9-5-5)Well,accordingtotheconvergenceofpowerseriescanbeitemizedwithinthenatureofderivation,andthenmake0x(powerseriesapparentlyconvergesinthe0xpoint),itiseasytoget()2012(0)(0)(0),(0),,,,,2!!nnnffafafxaxaxn…….Substitutingtheminto(9-5-5)type,incomeand()fxtheMaclaurinexpansionof(9-5-4)identical.Insummary,ifthefunctionf(x)containszeroinarangeofarbitraryorderderivative,andinthisrangeofMaclaurinformulaintheremaindertozeroasthelimit(whenn→∞,),then,thefunctionf(x)canstartformingas(9-5-4)typeofpowerseries.PowerSeries()20000000()()()()()()()()1!2!!nnfxfxfxfxfxxxxxxxn……,KnownastheTaylorseries.Second,primaryfunctionofpowerseriesexpansionMaclaurinformulausingthefunction()fxexpandedinpowerseriesmethod,calledthedirectexpansionmethod.Example1Testthefunction()xfxeexpandedinpowerseriesofx.Solutionbecause()()nxfxe,(1,2,3,)n…Therefore()(0)(0)(0)(0)1nffff…,Sowegetthepowerseries21112!!nxxxn……,(9-5-6)Obviously,(9-5-6)typeconvergenceinterval(,),As(9-5-6)whethertype()xfxeisSumfunction,thatis,whetheritconvergesto()xfxe,butalsoexamineremainder()nrx.Because1e()(1)!xnnrxxn(01)，且xxx,Therefore11ee()(1)!(1)!xxnnnrxxxnn,Notingthevalueofanysetx,xeisafixedconstant,whiletheseries(9-5-6)isabsolutelyconvergent,sothegeneralwhentheitemwhenn,10(1)!nxn,sowhenn→∞,there10(1)!nxxen,Fromthislim()0nnrxThisindicatesthattheseries(9-5-6)doesconvergeto()xfxe,therefore21112!!xnexxxn……(x).SuchuseofMaclaurinformulaareexpandedinpowerseriesmethod,althoughtheprocedureisclear,butoperatorsareoftentooCumbersome,soitisgenerallymoreconvenienttousethefollowingpowerseriesexpansionmethod.Priortothis,wehavebeenafunctionx11,xeandsinxpowerseriesexpansion,theuseoftheseknownexpansionbypowerseriesofoperations,wecanachievemanyfunctionsofpowerseriesexpansion.Thisdemandfunctionofpowerseriesexpansionmethodiscalledindirectexpansion.Example2Findthefunction()cosfxx,0x,Departmentinthepowerseriesexpansion.Solutionbecause(sin)cosxx,And3521111sin(1)3!5!(21)!nnxxxxxn……，（x）Therefore,thepowerseriescanbeitemizedaccordingtotherulesofderivationcanbe342111cos1(1)2!4!(2)!nnxxxxn……,（x）Third,thefunctionpowerseriesexpansionoftheapplicationexampleTheapplicationofpowerseriesexpansionisextensive,forexample,canuseittosetsomenumericalorotherapproximatecalculationofintegralvalue.Example3Usingtheexpansiontoestimatearctanxthevalueof.Solutionbecauseπarctan14Becauseof357arctan357xxxxx…,(11x),Sothere1114arctan14(1)357…Availablerightendofthefirstnitemsoftheseriesandasanapproximationof.However,theconvergenceisveryslowprogressiontogetenoughitemstogetmoreaccurateestimatesofvalue.此外文文献选自于：Walter.Rudin.数学分析原理(英文版)[M].北京：机械工业出版社.幂级数的展开及其应用在上一节中，我们讨论了幂级数的收敛性，在其收敛域内，幂级数总是收敛于一个和函数．对于一些简单的幂级数，还可以借助逐项求导或求积分的方法，求出这个和函数．本节将要讨论另外一个问题，对于任意一个函数()fx，能否将其展开成一个幂级数，以及展开成的幂级数是否以()fx为和函数?下面的讨论将解决这一问