《高数双语》课件section 7-5

Section6.5FourierSeries12TheFourierSeriesWheninvestigatingtheproblemofheatconductioninalongthininsulatedrod,JosephFourierneededtoexpressafunctionasatrigonometricseries[三角级数].coefficientsand()forwhichGenerally,ifisdefinedontheinterval,weneedtoknowthe(1)3TheFourierSeriesNoticethattheintervalissymmetricabouttheorigin.ThisequationcalledaFourierseries

forontheinterval.AssumethatisexpressibleasthetrigonometricseriesgivenbytheEquation(1).WewanttofindawaytocalculatethecoefficientsHowcanwedeterminethecoefficients?4OrthogonalityofthesystemoftrigonometricfunctionsItiseasytoprovethatdefiniteintegraloftheproductofanytwodifferentfunctionsofsquareofanyfunctionofthesystemoverisnotequalto

zero.overtheintervaliszero;andthedefiniteintegralofthe

on.Thekeytothecalculationsisthedefiniteintegral5Orthogonalityofthesystemoftrigonometricfunctions5.1.6.2.3.4.TrigonometricIntegrals6OrthogonalityofthesystemoftrigonometricfunctionsDefinition(Orthogonality[正交性])thenfandgaresaidtobeorthogonal

onIffunctionsfandgarebothintegrableovertheintervalandSupposethatisasequenceoffunctionsondifferentfunctionsofthesequenceareorthogonalonsystemoforthogonalfunctions

on

andthenthesequenceiscalledaIfanytwo7FourierSeriesCalculationofa0operationsforintegrationandsummationcanbeinterchangedtoobtainWeintegratebothsidesofEquation(1)fromto(2)Therefore,Foreverypositiveintegern,thelasttwointegralsontheright-handsideofSolvingfora0yieldsequation(2)arezero.andassumethatthe8FourierSeriesCalculationofamWemultiplybothsidesofEquation(1)bytheresultfromto(3)andintegrate9FourierSeriesTherefore,canbefurtherreducedtoThefirstintegralontheright-handsideofEquation(3)iszeroandtheequationCalculationofam10FourierSeriesCalculationofbmWemultiplybothsidesofEquation(1)bytotheresultfromandintegrate(4)11FourierSeriesThus,theequation(4)canbefurtherreducedtoTherefore,Calculationofbm12FourierSeriesThetrigonometricseriesTheconstantsa0,anandbnaretheFouriercoefficients

of

whosecoefficientsaredeterminedbyf.iscalledtheFourierseriesofthefunctionfovertheintervalDefinition(Fourierseries)13FindingaFourierExpansionExample

FindtheFourierseriesofthefunctionSolutionThepiecewisecontinuousfunctionFromthelastdefinitionwehave14FindingaFourierExpansionSolution(continued)Example

FindtheFourierseriesofthefunction15FindingaFourierExpansionSolution(continued)Therefore,theFourierseriesofthegivenfunctionisFinish.Example

FindtheFourierseriesofthefunction16FindingaFourierExpansiontermsisgiveninthefollowingfigure.AgraphoftheFourierseriesapproximationsasnvariesupto1,5,and2017ConvergenceofFourierSeriesTheorem(Dirichlet’stheorem)Assumethatthefunctionispiecewisemonotoneontheintervalandiscontinuousexceptforafinitenumberofdiscontinuouspointsofthefirsttype.ThentheFourierseriesofthefunctionmustconvergeontheintervalanditssumfunctionis18ConvergenceofFourierSeriestheFourierSeriesconvergesintheintervalForeveryconvergestotheaveragevaluethefunctionhasajumpAttoThefunctionsatisfiestheconditionsofTheoremofConvergenceofFourierSeries.discontinuityandtheFourierseries19FindtheFourierSeriesofaFunctionSupposethatfisaperiodicfunctionwithperiod2π,andasfollows:ExampleFinditsFourierexpansion.isdefinedintheintervalSawtoothwave20FindtheFourierSeriesofaFunctionObviously,fispiecewisecontinuousandmonotone.BytheDirichlettheoremitcanbeexpandedinaFourierseries.Solution21convergestofatallotherpointsinFindtheFourierSeriesofaFunctionHenceaccordingtotheDirichlettheorem,weknowthattheFourierseriesoff,thatistheseriesontherightsideSolution(continued)forandwhenThus,theofthelastequationconvergestoatFourierseriesoffconvergestoFinish.22FindtheFourierSeriesofaFunctionyxOπ3π2π-3π-π-2π1Supposethatfisaperiodicfunctionwithperiod2π,andasfollows:ExampleFinditsFourierexpansion.isdefinedintheinterval23FindtheFourierSeriesofaFunctionObviously,fsatisfiestheDirichletconditions.BytheDirichlettheoremitcanbeexpandedinaFourierseries.SolutionHence,theFourierseriesoffis24FindtheFourierSeriesofaFunctionSolution(continued)AccordingtotheDirichlettheoremweknowthatthesumfunctionoftheFourierseriesontheintervalisIntheintervalitisTheFourierexpansionofthefunctionfisFinish.25FindtheFourierSeriesofaFunctionyxO1π-πtermsisgiveninthefollowingfigure.Agrap