將下列各周期函數(shù)展開成傅里葉級數(shù)(下面給出函數(shù)在一個...

1、習(xí)題11?81?將下列各周期函數(shù)展開成傅里葉級數(shù)(下面給出函數(shù)在一個周期內(nèi)的表達式)?(1)?)2121(1)(2?????xxxf解因為f(x)?1?x2為偶函數(shù)?所以bn?0(n?1?2????)?而?611)1(4)1(212210221020???????dxxdxxa???210221cos)1(212dxxnxan?(n?1?2????)?2212102)1(2cos)1(4??nxdxnxn??????由于f(x)在(??

2、???)內(nèi)連續(xù)?所以?x?(?????)????????12122cos)1(11211)(nnxnnxf??(2)?????????????????1211210101)(xxxxxf解?21)(1212100111????????????dxdxxdxdxxfan??????????1212100111coscoscoscos)(xdxnxdxnxdxnxxdxnxfan????(n?1?2????)?2sin2])1(1[122?

3、??nnnn????dxxnxdxnxdxnxxdxnxfbn??????????1212100111sinsinsinsin)(????(n?1?2????)????nnn12cos2???故?x?[0?l]?????122sin2sin14)(nlxnnnlxf???余弦級數(shù)?對f(x)進行偶延拓?則函數(shù)的傅氏系數(shù)為?2])([2212100ldxxlxdxlal???????????lndxlxnxldxlxnxla21210]c

4、os)(cos[2??(n?1?2????)])1(12cos2[222nnnl??????bn?0(n?1?2????)?故?x?[0?l]?lxnnnllxfnn???cos])1(12cos2[124)(122????????(2)f(x)?x2(0?x?2)?解正弦級數(shù)?對f(x)進行奇延拓?則函數(shù)的傅氏系數(shù)為a0?0(n?0?1?2????)??]1)1[()(168)1(2sin2231202????????nnnnndxx