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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 Product integral ： ウィキペディア英語版 Product integral The expression "product integral" is used informally for referring to any product-based counterpart of the usual sum-based integral of classical calculus. The first product integral was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations.〔V. Volterra, B. Hostinský, ''Opérations Infinitésimales Linéaires'', Gauthier-Villars, Paris (1938).〕〔A. Slavík, (''Product integration, its history and applications'' ), ISBN 80-7378-006-2, Matfyzpress, Prague, 2007.〕 (Please see "Type II" below.) Other examples of product integrals are the geometric integral ("Type I" below), the bigeometric integral, and some other integrals of non-Newtonian calculus.〔 Product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic population dynamics using multiplication integrals (multigrals), analysis and quantum mechanics. The geometric integral, together with the geometric derivative, is useful in biomedical image analysis.〔Luc Florack, Hans van Assen. ("Multiplicative Calculus in Biomedical Image Analysis" ), Journal of Math Imaging and Vision, , 2011.〕 This article adopts the "product" $\backslash prod$ notation for product integration instead of the "integral" $\backslash int$ (usually modified by a superimposed "times" symbol or letter P) favoured by Volterra and others. An arbitrary classification of types is also adopted to impose some order in the field. ==Basic definitions== The classical Riemann integral of a function $f:()\backslash to\backslash mathbb$ can be defined by the relation :$\backslash int\_a^b\; f(x)\backslash ,dx\; =\; \backslash lim\_\backslash sum\; f(x\_i)\backslash ,\backslash Delta\; x,$ where the limit is taken over all partitions of interval $()$ whose norm approach zero. Roughly speaking, product integrals are similar, but take the limit of a product instead of the limit of a sum. They can be thought of as "continuous" versions of "discrete" products. The most popular product integrals are the following: 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「Product integral」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.