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求下列周期函数的傅里叶级数展开式：（1) f（x)=|cosx|（周期π)；（2) f（x)=x－[x]（周期1)；（3) f（x)=sin4x（

大学 2021-02-04 23:32:57 周期,周期函数,级数已帮助人

求下列周期函数的傅里叶级数展开式：

(1) f(x)=|cosx|(周期π)；

(2) f(x)=x-[x](周期1)；

(3) f(x)=sin⁴x(周期π)；

(4) f(x)=sgn(cosx)(周期2π)．

热搜：傅里叶级数展开公式a0怎么求

答案解析

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由于f(x)在[-π，π]上是偶函数，f(x)及其周期延拓后如图15-6所示，它是按段光滑，即可展成傅里叶级数，且这个级数为余弦级数，其中：
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(2) f(x)及其周期延拓后如图15-7所示，显然f(x)是按段光滑，即可展开为傅里叶级数，其中

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而当x=0或x=1时，上式右边级数收敛于 $latex.action?latex=XGZyYWN7MX17Mn0%3D$ ，即左、右两边相等
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(4)由于f(x)是以2π为周期的偶，且按段光滑，即可展开为傅里叶级数，并有bn=0，而
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