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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Shift theorem ： ウィキペディア英語版 Shift theorem In mathematics, the (exponential) shift theorem is a theorem about polynomial differential operators (''D''-operators) and exponential functions. It permits one to eliminate, in certain cases, the exponential from under the ''D''-operators. The theorem states that, if ''P''(''D'') is a polynomial ''D''-operator, then, for any sufficiently differentiable function ''y'', :$P(D)(e^y)\backslash equiv\; e^P(D+a)y.\backslash ,$ To prove the result, proceed by induction. Note that only the special case :$P(D)=D^n\backslash ,$ needs to be proved, since the general result then follows by linearity of ''D''-operators. The result is clearly true for ''n'' = 1 since :$D(e^y)=e^(D+a)y.\backslash ,$ Now suppose the result true for ''n'' = ''k'', that is, :$D^k(e^y)=e^(D+a)^k\; y.\backslash ,$ Then, : &\+ae^\\\ &+a\right)(D+a)^ky\right\}\\ &y.\end This completes the proof. The shift theorem applied equally well to inverse operators: :$\backslash frac(e^y)=e^\backslash fracy.\backslash ,$ There is a similar version of the shift theorem for Laplace transforms ( :$\backslash mathcal(e^\; f(t))=\backslash mathcal(f(t-a)).\backslash ,$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Shift theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.