常见函数的傅里叶级数

24FOURIERSERIESDefinitionofaFourierSeriesTheFourierseriescorrespondingtoafunctionf(x)definedintheintervalcxcL+2wherecandL0areconstants,isdefinedas24.1.aanxLbnxLnnn012++⎛⎝⎜⎞⎠⎟=∞∑cossinππwhere24.2.aLfxnxLdxbLfxnxLdxnccLnc==+∫112()cos()sinππccL+∫⎧⎨⎪⎩⎪2Iff(x)andf′(x)arepiecewisecontinuousandf(x)isdefinedbyperiodicextensionofperiod2L,i.e.,f(x2L)f(x),thentheseriesconvergestof(x)ifxisapointofcontinuityandto1200{()()}fxfx++−ifxisapointofdiscontinuity.ComplexFormofFourierSeriesAssumingthattheseries24.1convergestof(x),wehave24.3.fxceninxLn()/==−∞∞∑πwhere24.4.cLfxedxaibnaibninxLnnn==−+−−1201212()()(/π−−+=⎧⎨⎪⎩⎪∫nccLnan)001202Parseval’sIdentity24.5.122022212LfxdxaabnnnccL{()}()=++=∞+∑∫GeneralizedParsevalIdentity24.6.122001LfxgxdxacacbdccLnnnnn()()()+=∞∫∑=++wherean,bnandcn,dnaretheFouriercoefficientscorrespondingtof(x)andg(x),respectively.14414524.7.fxxx()=−−⎧⎨⎩1010ππFig.24-1413355πsinsinsinxxx+++⎛⎝⎜⎞⎠⎟24.8.fxxxxxx()||==−−⎧⎨⎩00ππFig.24-2ππ2413355222−+++⎛⎝⎜⎞⎠⎟coscoscosxxx24.9.fxxx(),=−ππFig.24-3212233sinsinsinxxx−+−⎛⎝⎜⎞⎠⎟24.10.fxxx(),=02πFig.24-4π−+++⎛⎝⎜⎞⎠⎟212233sinsinsinxxx24.11.fxxx()|sin|,=−ππFig.24-524213435657ππ−+++⎛⎝⎜⎞⎠⎟coscoscosxxxiiiSpecialFourierSeriesandTheirGraphsFOURIERSERIES14624.12.fxxxx()sin=⎧⎨⎩002πππFig.24-61122213435657ππ+−+++⎛⎝⎜sincoscoscosxxxxiii⎞⎞⎠⎟24.13.fxxxxx()coscos=−−⎧⎨⎩00ππFig.24-7821324353657πsinsinsinxxxiii+++⎛⎝⎜⎞⎠⎟24.14.fxxx(),=−2ππFig.24-8π22223412233−−+−⎛⎝⎜⎞⎠⎟coscoscosxxx24.15.fxxxx()(),=−ππ0Fig.24-9π22226214263−+++⎛⎝⎜⎞⎠⎟coscoscosxxx24.16.fxxxxx()()(),=−+−ππππFig.24-101212233333sinsinsinxxx−+−⎛⎝⎜⎞⎠⎟FOURIERSERIES147MiscellaneousFourierSeries24.17.fxxxx()=−−++⎧⎨⎪⎩⎪00102παπαπαπαπFig.24-11αππααα−−⎛⎝⎜+−21222333sincossincossincosxxx⎞⎞⎠⎟24.18.fxxxxxxx()()()=−−−−⎧⎨⎩ππππ00Fig.24-12813355333πsinsinsinxxx+++⎛⎝⎜⎞⎠⎟24.19.fxxxx()sin,,sinsin=−≠−μππμμππμinteger2122−−−+−−⎛⎝⎜⎞⎠⎟2223332222sinsinxxμμ24.20.fxxx()cos,,sincos=−≠+μππμμμππμinteger2122xxxx12233222222−−−+−−⎛⎝⎜⎞⎠⎟μμμcoscos24.21.fxaxaxxaa()tan[(sin)/(cos)],,||=−−−111ππssinsinsinxaxax+++23223324.22.fxaxaxaaxa()ln(cos),,||cos=−+−−+121222ππ222333coscosxax++⎛⎝⎜⎞⎠⎟24.23.fxaxaxaa()tan[(sin)/()],,||=−−−1221112ππssinsinsinxaxax+++35335524.24.fxaxaxaa()tan[(cos)/()],,||=−−−1221112ππccoscoscosxaxax−+−353355FOURIERSERIES14824.25.fxexnxnxn(),sinh()(cossi=−+−−μππμππμμ2121nn)nxnnμ221+⎛⎝⎜⎞⎠⎟=∞∑24.26.fxxxxx()sinh,sinhsinsin=−+−μππμππμ212222222222333+++−⎛⎝⎜⎞⎠⎟μμsinx24.27.fxxxx()cosh,sinhcosc=−−++μππμμππμμ2121222ooscos22332222xx+−++⎛⎝⎜⎞⎠⎟μμ24.28.fxxxxx()ln|sin|,lncoscoscos=−+++12021223πxx3+⎛⎝⎜⎞⎠⎟24.29.fxxxxx()ln|cos|,lncoscoscos=−−−+−122122ππ333x+⎛⎝⎜⎞⎠⎟24.30.fxxxxxx(),coscosco=−+++162121422202122πππss332x+24.31.fxxxxxxx()()(),sinsins=−−++11233202122πππiin333x+24.32.fxxxxxx(),cos=−+−19041122211231484021ππππ44442233+++coscosxxFOURIERSERIES