求下列周期函数的傅里叶级数展开式:[tex=6.0x1.286]bQVUWNAQtxnnx3MDKSPt5Q==[/tex] (周期1).
解: [tex=0.643x1.286]+RQz+inOZSqc5WvKyEpD0Q==[/tex] 是以1为周期的周期函数, [tex=0.643x1.286]+RQz+inOZSqc5WvKyEpD0Q==[/tex] 的图形如图所示.由于 [tex=0.643x1.286]+RQz+inOZSqc5WvKyEpD0Q==[/tex] 是按段光滑的, 因此,可以展开成傅里叶级数.[img=354x180]176fb7d395fc876.png[/img][tex=22.857x6.071]qeiYnKXLEhyhuGRg8yLtrzP3UAUMKWODWhPt5nQXkKDFFZUxt0bnunWX8qIbUjX3ajJrTxfiHVMJLJCWWBLqOkhNt+aAE5xAclD14m0MN3daffUJJlyjJFC2FVjMHccRJnMOuOr5M271MwBbP2Cahc1oS+PPdTjOdz+Rq37+iEh4m1C153JZQBLHU7WWPtQf5rBHSjR9V+9TVN8L9JVnTCyIsW0mBJbxbItFgzZoR7lMVNswT7It4gVFRxRR22lI[/tex][tex=20.143x6.071]qeiYnKXLEhyhuGRg8yLtr7ttxDPQdFcmMGkx76tc2fiRAZNl0jHq5lwk6P/QWV16HJx5lVPRD8/BPpBQk3QGAvuf5TT8iSNFFiV3WEaWVO0DQ3/4XqAytF4gJTjBRnhtTzE3XCqVZw1xgn9MsVnTQGPJ/jEdn59Qlw/p459/2Dp5JLfH2E5K9OZX850ieWlqVNhPTRtLRbP+RKYQAQy/gyD+/HZZoGvwM4N3FE6uPmQ=[/tex][tex=19.071x5.786]vymAz+o1JsDktP+/I4nZymQahR/tdL+HNrSkvSPrGDvD093Zy+WTLiazvH+jMyCc/x3a0u5ut4w27YS4xnDNPq6CIfUSYV7MCDVo4J1lvhDlohPoiPlzGp3ZXuaJtHwPgN14tjFfW2TYpRRGi3KeBSdCzcmLVooAR3+ebJIKx4OiDqDsYCqrdjA7Pj2Cqj7w[/tex] [tex=8.857x3.643]J0pccxWeE3vvuiIpY8fACp0B+9nW26JPN4qaY+MpC1Jc0C5SVXuzueuPyBvzaYe322TQFA+bkekb0yBU+gUNmlKmD/IoqHUpPJbuJMeeVXE=[/tex]因此, 由收敛定理,当 [tex=2.357x1.286]Fi2NJRx2RFwIfP6VHB/3aQ==[/tex],[tex=1.214x1.286]gYTgRQ9fU02e0d0EGUXE2A==[/tex],[tex=1.286x1.286]oUpg4SAdO2BfRE9c0fmmtg==[/tex], [tex=1.143x1.286]PZ3wc82RrbgX5KwVcyJcmA==[/tex] 时[tex=3.286x1.357]0tUiOB4E2pYUXWrxoTN79g==[/tex][tex=8.286x2.714]lKmDV2sfQHOiX8xNu1sgDhJWCcMvJptV1G06QlvwuFyzopX3LpE6o4dYh0skBsupipJp9kbL4fSEKNPrXEIkZd6J/827qUyyNudjDACbTYc=[/tex]当 [tex=7.286x1.286]JlmfjnMaiMYEAztnh0j59w1u12d2U2Bp5RcTggfpPxs=[/tex]时,上式右端收敛于 [tex=0.714x2.0]4HxptsXXGVzE18Uu2hj3h6C5Sxg2DM0D87ElHtd7URI=[/tex].
举一反三
- 求下列周期函数的傅里叶级数展开式:[tex=5.857x1.286]qMOLsARs2Ww+rZ87Gf3UqkRFv4VMGpbOjRzRZ3JBlsY=[/tex] (周期 [tex=0.643x1.286]USGVpa36zb6HMu8k0moHJA==[/tex]) .
- 求下列周期函数的傅里叶级数展开式:[tex=5.571x1.286]poXx9kzw7gxFgsEa9Qp7sEY4aEblMoGWnnSty8zDy2w=[/tex]( 周期 [tex=0.643x1.286]USGVpa36zb6HMu8k0moHJA==[/tex])
- 求下列周期函数的傅里叶级数展开式：[tex=7.143x1.357]qAU07a/UMA9qmDdYIFZHiuDbQciYplFbD010frnC35s=[/tex]（周期[tex=1.071x1.0]cWYnFY7tUlCT6WhMhv7goA==[/tex]）
- 求下列函数的傅里叶级数展开式：[tex=8.214x1.357]kEPI5WwnwC/4osrhuwcJn1bEYDNSfFo9tjF3JhtLWkk=[/tex]
- 求下列函数的傅里叶级数展开式：[tex=10.143x1.357]qAU07a/UMA9qmDdYIFZHilEJmiBRkoK3A8RHEzZ4P4W8f1Zgl8fZyxMJ2QQ6azCB[/tex]
内容
- 0
求下列函数的傅里叶级数展开式：[tex=10.429x2.286]i41q0BRTWZJQmNRPJQhH+gumwgQDzGqtl/61T6v1cRw=[/tex]
- 1
求下列函数的傅里叶级数展开式：[tex=10.071x1.357]5cM/LvJqoCikO7A5c+WCIFvkMrcH8wM/sSdNFcbb5l7CiIY7PXE8owHe7h0e7UQ9[/tex]
- 2
将下列周期函数展开成傅里叶级数：[tex=5.143x1.357]/hc/F/pZCirS/Rf1yGjQyw==[/tex]。提示：注意函数[tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex]的周期为1。
- 3
求函数系的傅里叶级数展开式:[tex=7.786x1.286]lKoao7AHJRlCMQaCikocwd9FkYgBIUtHum5SptVMYS0=[/tex],[tex=5.071x1.286]Y9nIBZA1Le7w9q8FIC0XtoPFAMFg+TzqgyUq4oFDpNDYqB7143ZGAm8mHpbxFBSP[/tex].
- 4
求下列函数的傅里叶级数展开式：[tex=8.143x1.357]sg3p+2f20Op/P/0bHR/IRv9zhJM8b8m1rAS/tNhp098=[/tex]