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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 Hermite's identity ： ウィキペディア英語版 Hermite's identity In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function. It states that for every real number ''x'' and for every positive integer ''n'' the following identity holds:〔.〕〔.〕 : $\backslash sum\_^\backslash left\backslash lfloor\; x+\backslash frac\backslash right\backslash rfloor=\backslash lfloor\; nx\backslash rfloor\; .$ ==Proof== Split $x$ into its integer part and fractional part, $x=\backslash lfloor\; x\backslash rfloor+\backslash$. There is exactly one $k\text{'}\backslash in\backslash$ with : By subtracting the same integer $\backslash lfloor\; x\backslash rfloor$ from inside the floor operations on the left and right sides of this inequality, it may be rewritten as : Therefore, : and multiplying both sides by $n$ gives : Now if the summation from Hermite's identity is split into two parts at index $k\text{'}$, it becomes : $\backslash sum\_^\backslash left\backslash lfloor\; x+\backslash frac\backslash right\backslash rfloor$ =\sum_^ \lfloor x\rfloor+\sum_^ (\lfloor x\rfloor+1)=n\, \lfloor x\rfloor+n-k' =n\, \lfloor x\rfloor+\lfloor n\,\\rfloor=\left\lfloor n\, \lfloor x\rfloor+n\, \ \right\rfloor=\lfloor nx\rfloor. 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「Hermite's identity」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.