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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 Euler–MacLaurin summation ： ウィキペディア英語版 Euler–Maclaurin formula In mathematics, the Euler–Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence. The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735 (and later generalized as Darboux's formula). Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. ==The formula== If and are natural numbers and is a complex or real valued continuous function for real numbers in the interval $()$, then the integral :$I\; =\; \backslash int\_m^n\; f(x)\backslash ,dx$ can be approximated by the sum (or vice versa) :$S\; =\; f\backslash left(m\; +\; 1\backslash right)\; +\; \backslash cdots\; +\; f\backslash left(n\; -\; 1\backslash right)\; +\; f(n)$ (see trapezoidal rule). The Euler–Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives evaluated at the end points of the interval, that is to say when and . Explicitly, for a natural number $p$ and a function $f(x)$ that is $p$ times continuously differentiable in the interval $(n)$, we have :$S\; -\; I\; =\; \backslash sum\_^p\; \backslash left(f^(n)\; -\; f^(m)\backslash right)\}\; +\; R$ where is the th Bernoulli number, with , , , , , , , , ..., and is an error term which is normally small for suitable values of and depends on and . The formula is often written with the subscript taking only even values, since the odd Bernoulli numbers are zero except for ''B''1, in which case we have :$\backslash sum\_^n\; f(i)\; =$ \int^n_m f(x)\,dx + B_1 \left(f(n) - f(m)\right) + \sum_^p\frac\left(f^(n) - f^(m)\right) + R. 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「Euler–Maclaurin formula」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.