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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 List of derivatives and integrals in alternative calculi ： ウィキペディア英語版 List of derivatives and integrals in alternative calculi There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi.〔M. Grossman and R. Katz, (''Non-Newtonian Calculus'' ), ISBN 0-912938-01-3, Lee Press, 1972.〕 Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea.〔Agamirza E. Bashirov, Emine Misirli Kurpinar, and Ali Ozyapici. ("Multiplicative calculus and its applications" ), Journal of Mathematical Analysis and Applications, 2008.〕〔Diana Andrada Filip and Cyrille Piatecki. ("A non-Newtonian examination of the theory of exogenous economic growth" ), (CNCSIS – UEFISCSU )(project number PNII IDEI 2366/2008) and (LEO ), 2010.〕〔Luc Florack and Hans van Assen.("Multiplicative calculus in biomedical image analysis" ), Journal of Mathematical Imaging and Vision, DOI: 10.1007/s10851-011-0275-1, 2011.〕 The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). Interested readers are encouraged to improve the table by inserting citations for verification, and by inserting more functions and more calculi. == Table== In the following table $\backslash psi(x)=\backslash frac\backslash ,$ is the digamma function, $K(x)=e^=e^+\backslash frac\; z2\; \backslash ln\; (2\backslash pi)-\backslash psi^(z)\}$ is the K-function, $(!x)=\backslash frac$ is subfactorial, $B\_a(x)=-a\backslash zeta(-a+1,x)\backslash ,$ are the generalized to real numbers Bernoulli polynomials. f(x)\, (constant term is omitted) ! Discrete multiplicative derivative〔(H. R. Khatami & M. Jahanshahi & N. Aliev (2004). "An analytical method for some nonlinear difference equations by discrete multiplicative differentiation"., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings, pp. 455—462 )〕 (multiplicative difference) ! Discrete multiplicative integral〔(M. Jahanshahi, N. Aliev and H. R. Khatami (2004). "An analytic-numerical method for solving difference equations with variable coefficients by discrete multiplicative integration"., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings, pp. 425—435 )〕 (indefinite product) $\backslash prod\; \_x\; f(x)\backslash ,$ (constant factor is omitted) |- | $a\backslash ,$ | $0\backslash ,$ | $ax\backslash ,$ | $1\backslash ,$ | $a^x\backslash ,$ | $0\backslash ,$ | $ax\backslash ,$ | $1\backslash ,$ | $a^x\backslash ,$ |- | $x\backslash ,$ | $1\backslash ,$ | $\backslash frac\backslash ,$ | $\backslash sqrt()\backslash ,$ | $\backslash frac\backslash ,$ | $1\backslash ,$ | $\backslash frac-\backslash frac\; x2\backslash ,$ | $1+\backslash frac\; 1x\backslash ,$ | $\backslash Gamma(x)\backslash ,$ |- | $ax+b\backslash ,$ | $a\backslash ,$ | $\backslash frac\backslash ,$ | $\backslash exp\backslash left(\backslash frac\backslash right)\backslash ,$ | $\backslash frac+x\}\}\backslash ,$ | $a\backslash ,$ | $\backslash frac\backslash ,$ | $1+\backslash frac\backslash ,$ | $\backslash frac)\})\}\backslash ,$ |- | $\backslash frac\; 1x\backslash ,$ | $-\backslash frac\backslash ,$ | $\backslash ln\; |x|\backslash ,$ | $\backslash frac\backslash ,$ | $-\backslash frac\backslash ,$ | $\backslash psi(x)\backslash ,$ | $\backslash frac\backslash ,$ | $\backslash frac\backslash ,$ |- | $x^a\backslash ,$ | $ax^\backslash ,$ | $\backslash frac\backslash ,$ | $e^\backslash ,$ | $e^\; x^\backslash ,$ | $(x+1)^a-x^a\backslash ,$ | $a\backslash notin\; \backslash mathbb^-\backslash ,;$$\backslash frac,\backslash ,$ $a\backslash in\backslash mathbb^-\backslash ,;$$\backslash frac(x)\},\backslash ,$ | $\backslash left(1+\backslash frac\; 1x\backslash right)^a\backslash ,$ | $\backslash Gamma(x)^a\backslash ,$ |- | $a^x\backslash ,$ | $a^x\backslash ln\; a\backslash ,$ | $\backslash frac\backslash ,$ | $a\backslash ,$ | $a^\}\backslash ,$ | $(a-1)a^x\backslash ,$ | $\backslash frac\backslash ,$ | $a\backslash ,$ | $a^\}\backslash ,$ |- | $\backslash sqrt()\backslash ,$ | $-\backslash frac\backslash ,$ | $x\backslash sqrt()-\backslash operatorname\backslash left(\backslash frac\backslash right)\backslash ln\; a\backslash ,$ | $a^\}\backslash ,$ | $a^\backslash ,$ | $a^\}-a^\}\backslash ,$ | $?\; \backslash ,$ | $a^\}\backslash ,$ | $a^\backslash ,$ |- | $\backslash log\_a\; x\backslash ,$ | $\backslash frac\backslash ,$ | $\backslash log\_a\; x^x-\backslash frac$ | $\backslash exp\; \backslash left(\backslash frac\backslash right)\; \backslash ,$ | $\backslash frac\}\backslash ,$ | $\backslash log\_a\backslash left(\backslash frac\; 1x\; -1\backslash right)\backslash ,$ | $\backslash log\_a\; \backslash Gamma(x)\backslash ,$ | $\backslash log\_x\; (x+1)\backslash ,$ | $?\backslash ,$ |- | $x^x\backslash ,$ | $x^x(1+\backslash ln\; x)\backslash ,$ | $?\backslash ,$ | $ex\backslash ,$ | $e^x^2(1-2\backslash ln\; x)\}\backslash ,$ | $(x+1)^-x^x\backslash ,$ | $?\; \backslash ,$ | $\backslash frac\backslash ,$ | $\backslash operatorname(x)\backslash ,$ |- | $\backslash Gamma(x)\backslash ,$ | $\backslash Gamma(x)\backslash psi(x)\backslash ,$ | $?\backslash ,$ | $e^\backslash ,$ | $e^\backslash ,$ | $(x-1)\backslash Gamma(x)\backslash ,$ | $(-1)^\backslash Gamma(x)(!(-x))\backslash ,$ | $x\backslash ,$ | $\backslash frac(x)\}\backslash ,$ |} 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「List of derivatives and integrals in alternative calculi」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. 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