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Section6.6FourierSeriesofOtherForms1FourierExpansionsofPeriodicFunctionswithPeriod2L2Supposeaperiodicfunctionf(x)satisfiesDirichletconditionsontheinterval[?L,L]andhasperiod2L.SetThen,g(t)isaperiodicfunctionwithperiod2πandsatisfiesDirichletconditionsontheinterval[?π,π].Wehavewhere3FourierExpansionsofPeriodicFunctionswithPeriod2LThetrigonometricseriesTheconstantsa0,anandbnaretheFouriercoefficients

of

whosecoefficientsaredeterminedbyf.iscalledtheFourierseriesofthefunctionfovertheintervalDefinition(Fourierseriesofafunctionwithperiod2L)4ConvergenceTheoremTheorem(Dirichlet’stheorem)Assumethatthefunctionfispiecewisemonotoneontheintervalandiscontinuousexceptforafinitenumberofdiscontinuouspointsofthefirsttype.ThentheFourierseriesofthefunctionfmustconvergeontheintervalanditssumfunctionisInthiscasewealsosaytheFourierseriesistheFourierExpansionof

f.5FourierExpansionsofPeriodicFunctionswithPeriod2LSolutionExample

FindtheFourierseriesforthefunctionandthesumoftheseries.6FourierExpansionsofPeriodicFunctionswithPeriod2LSolution(continued)7FourierExpansionsofPeriodicFunctionswithPeriod2LSolution(continued)TheFourierseriesforthegivenfunctionis8FourierExpansionsofPeriodicFunctionswithPeriod2LSolution(continued)yxO264-6-2-41-29IntegralsofEvenandOddFunctionsEvenfunction:Oddfunction:Becauseofthesetworules,evenandoddextensionsofafunctionThefollowingresultsalsoholdforevenandoddfunctions.areconvenienttouse.1.

Theproductoftwoevenfunctionsiseven.2.

Theproductofanevenfunctionswithanoddfunctionisodd.3.

Theproductoftwooddfunctionsiseven.FourierExpansionsofPeriodicFunctionswithPeriod2L10Corollary

SupposethatthefunctionfsatisfiesDirichletconditionsontheinterval[-L,L]andhasperiod2L.Let.(1)Whilef(x)isanoddfunction,itsFourierexpansioniswhere(2)Whilef(x)isanevenfunction,itsFourierexpansioniswhere11FourierExpansionsofPeriodicFunctionswithPeriod2LExample

FindtheFourierExpansionforf,wherefisaperiodicfunctionwithperiod6anddefinedoninterval[-3,3]asfollowsSolution12FourierExpansionsofPeriodicFunctionswithPeriod2LSolution(continued)Sincef(x)iscontinuousexceptwehaveExample

FindtheFourierExpansionforf,wherefisaperiodicfunctionwithperiod6anddefinedoninterval[-3,3]asfollowsExample

FindtheFourierExpansionforf,wherefisaperiodicfunctionwithperiod4anddefinedoninterval[-2,2]asfollows13FourierExpansionsofPeriodicFunctionswithPeriod2LyxOδ4+δ4-δ-δ-4+δ-4-δ1/(2δ)-44-2214FourierExpansionsofPeriodicFunctionswithPeriod2LSolutionSo,Example

FindtheFourierExpansionforf,wherefisaperiodicfunctionwithperiod4anddefinedoninterval[-2,2]asfollows15FourierExpansionsofFunctionsDefinedon[0,L]Question:HowthancanwecalculateaFourierseriesexpansionforfon[0,L]?Todoso,weextendthefunctionsothatitisdefinedoverthesymmetricintervalHow,though,dowedefinetheextensionoffforTheansweristhatwedefinetheextensiontobeanyfunctionoverwechooseaslongastheextensionanditsderivativearepiecewisecontinuous(inordertosatisfytheDirichletconditions).LOx16Anyway,therearetwospecialextensionsthatareparticularlyusefulandwhoseFourierExpansionsofFunctionsDefinedon[0,L]Fouriercoefficientsareespeciallyeasytocalculate;extensions

off.LOxthesearetheevenandodd-LQuestion:HowthancanwecalculateaFourierseriesexpansionforfon[0,L]?17IntegralsofEvenandOddFunctionsThegraphofanoddfunctionissymmetricabouttheorigin.Thegraphofanevenfunctionissymmetricaboutthey-axis.Thisobservationcanmaketheintegralsofevenandoddfunctionsoverintervalssymmetricabouttheoriginrelativelyeasytocalculate.18EvenExtension:FourierCosineSeriesWedefinetheevenextension

offbyrequiringthatisspecifiedfortheintervalSupposethatthefunctionaboutGraphically,weobtaintheevenextensionbyreflectingTheevenextensionofafunctionisillustratedinthefollowing.they-axis.19Therefore,ifweusedtheevenextensionforafunctionf,weobtaintheEvenExtension:FourierCosineSeriesFouriercoefficientsTheFourierseriesoffis20FourierCosineSeriesFourierCosineSeriesisthecosineserieswhereofanevenfunctionFontheintervalSupposeweextendthefunctionffrominterval[0,L]to[-L,0)sothattheextendedfunctionFisanevenfunction,thatisThisextensioniscalledtheevenextension.Inthiscase,TheFourierexpansionTheFourierexpansionofthefunctionfontheinterval[0,L]is21FourierCosineSeriesExample

FindtheFouriercosineseriesforthefunctionSolution22FourierCosineSeriesFortheFouriercosineseries,weselecttheevenSolution(continued)TheFouriercoefficientsareextensionofthefunctionoverTherefore,wehavetheFouriercosineexpansionFinish.23FourierCosineSeriesNextfigureshowstheFouriercosineapproximationfasnvariesupto1,5and20terms.24OddExtension:FourierSineSeriesspecifiedfortheintervalConsideragainafunctionWedefinedtheoddextension

offbyrequiringthataboutGraphically,weobtaintheevenextensionbyreflectingTheoddextensionofafunctionisillustratedinthefollowing.theorigin.25Therefore,ifweusedtheoddextensionforafunctionf,weobtaintheOddExtension:FourierSineSeriesFouriercoefficientsTheFourierseriesoffis26FourierSineSeriesFourierSineSeriesisthesineseriesanoddfunctionFontheintervalwhereSupposeweex

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