常见函数的傅里叶级数.pdf

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1、24FOURIERSERIESDefinitionofaFourierSeriesTheFourierseriescorrespondingtoafunctionf(x)definedintheintervalcxcL+2wherecandL>0areconstants,isdefinedas∞a0⎛nxππnx⎞24.1.2++∑⎜anncosLbsinL⎟⎝⎠n=1where⎧1cL+2nxπa=fx()cosdx⎪nL∫cL24.2.⎨1ccL+2nxπ⎪b=fx()sindxn∫⎩LcLIff(x)andf′(x)arepiecewisecontinuou

2、sandf(x)isdefinedbyperiodicextensionofperiod2L,i.e.,f(x2L)f(x),thentheseriesconvergestof(x)ifxisapointofcontinuityandto1{(fx++−00)fx()}if2xisapointofdiscontinuity.ComplexFormofFourierSeriesAssumingthattheseries24.1convergestof(x),wehave∞inxLπ/24.3.fx()=∑cenn=−∞where1()ai−>bn0⎧2nn1cL+2

3、⎪==fxe()−inxLπ/dx1(ai+b)0n<24.4.cn2L∫c⎨2−n−−n⎪1an=0⎩20Parseval’sIdentity2∞1cL+2a202224.5.∫{()}fxdx=+∑(abnn+)Lc2n=1GeneralizedParsevalIdentity∞1cL+2ac24.6.00fxgxdx()()=+(ac+bd)L∫c2∑nnnnn=1wherea,bandc,daretheFouriercoefficientscorrespondingtof(x)andg(x),respectively.nnnn144FOURIERSERIES1

4、45SpecialFourierSeriesandTheirGraphs⎧10<

5、

6、==x⎨⎩−−<

7、.11.fx()

8、sin

9、,=−xππ

10、⎞−−+−4⎜222⎟3⎝123⎠Fig.24-824.15.fxx()=−(ππx),0<

11、x<24.18.fx()=⎨⎩−−−<