第8章2拉普拉斯變換存在定理性質(zhì)-資料下載頁
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【正文】 ] ( 1 c o s ) tte ??0??? ds21ln( 1 ) 1ss????0?? 1 ln 22?計算廣義積分 9. 卷積性質(zhì) 函數(shù) 1()ft與 2()ft的 卷積 1 ()ft? 2()ft 1()f ? 2 ()ft ??d????? ?卷積 滿足交換律 1 ()ft? 2()ft 2 ()ft??1()ft卷積 也滿足 1 ()ft? 2[ ( )ft 3 ( )]ft? 1 ()ft??2()ft 1 ()ft??3()ft如果當 0t? 時 1 ( ) 0 ,ft ? 2 ( ) 0ft ?則 1 ()ft? 2()ft 1()f ? 2 ()ft ?? d0t??1? ()ft ( ) 1ft??0t?? ()f ? d?11? 0t?? 1d? t?1? te? 0t?? e?? d? e ???? 0t1 te ???對加法的分配律 在 Laplace變換中 用該公式計算卷積 L[ ] []L[ ] L[ ] 1()ft 1 ()Fs?若 卷積定理 L[ ] 2()ft 2 ()Fs?則 1 ()f t ?2()f t 1 ()Fs? 2()Fs證明 1 ()f t ? 2()f tL[ ] 1()f ? 2 ()ft ?? d?0t?0 [???? ] ste? dt0??? ?0t?1()f ? 2 ()ft ?? d? dtse ??()ste ???0??? ?1()f ? se ?? 2 ()ft ?? ()ste ???dt????2()Fsd?0??? ?1()f ? se ?? d? 1 ()Fs? 2()FsL[ ] 例 1 用卷積定理證明 ()ft0t? dt 1s? ()Fs證明 L[1] 1s? L[ ] ()ft ()Fs?1? ()ft()ft0t? dt =L[ ] 1s? ()FsL[ ] 例 2 求函數(shù) 的拉氏 逆 變換 ()Fs 221( 1 )s? ?解 L[ ] sint 2 1 1s? ?s i n s i ntt? 2 1 1s? ? 2 1 1s? ? ()Fs?()ft s i n s i ntt?? in? sin( )t ??d?0t??1 [ c os ( 2 )2 t? ?cos ]t?0t?? d?1 si n ( 2 )4 t???0t 1c os2 tt?1 si n2 t?1 c os2 tt?L[ ] 例 2 求函數(shù) 的拉氏 逆 變換 ()Fs 22( 1 )ss? ?解 L[ ] sint 2 1 1s? ?si n c ostt? 2 1 1s? ? 2 1ss? ? ()Fs?()ft si n c ostt?? sin? c o s( )t ??d?0t??1 [sin2 t sin( 2 ) ]t???0t?? d?1 si n2 tt? 0t1c os( 2 )4 t???1 si n2 tt?L[ ] cost 2 1ss? ?
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