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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 Grünwald–Letnikov derivative ： ウィキペディア英語版 Grünwald–Letnikov derivative In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868. ==Constructing the Grünwald–Letnikov derivative== The formula :$f\text{'}(x)\; =\; \backslash lim\_\; \backslash frac$ for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be: :$f\text{'}\text{'}(x)\; =\; \backslash lim\_\; \backslash frac$ :$=\; \backslash lim\_\; \backslash frac-\backslash lim\_\; \backslash frac\}$ Assuming that the ''h'' 's converge synchronously, this simplifies to: :$=\; \backslash lim\_\; \backslash frac,$ which can be justified rigorously by the mean value theorem. In general, we have (see binomial coefficient): :$f^(x)\; =\; \backslash lim\_\; \backslash fracf(x+(n-m)h)\}.$ Removing the restriction that ''n'' be a positive integer, it is reasonable to define: :$\backslash mathbb^q\; f(x)\; =\; \backslash lim\_\; \backslash frac\backslash sum\_(-1)^m\; f(x+(q-m)h).$ This defines the Grünwald–Letnikov derivative. To simplify notation, we set: :$\backslash Delta^q\_h\; f(x)\; =\; \backslash sum\_(-1)^m\; f(x+(q-m)h).$ So the Grünwald–Letnikov derivative may be succinctly written as: :$\backslash mathbb^q\; f(x)\; =\; \backslash lim\_\backslash frac.$ 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「Grünwald–Letnikov derivative」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.