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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 integrals of Gaussian functions ： ウィキペディア英語版 List of integrals of Gaussian functions In these expressions :$\backslash phi(x)\; =\; \backslash frac\}\}^x\; \backslash phi(t)dt\; =\; \backslash frac12\backslash left(1\; +\; \backslash operatorname\backslash left(\backslash frac\; \backslash ,\; dx$ which is known as the Owen's T function. Owen has an extensive list of Gaussian-type integrals; only a subset is given below. == Indefinite integrals == :$\backslash int\; \backslash phi(x)\; \backslash ,\; dx\; =\; \backslash Phi(x)\; +\; C$ :$\backslash int\; x\; \backslash phi(x)\; \backslash ,\; dx\; =\; -\backslash phi(x)\; +\; C$ :$\backslash int\; x^2\; \backslash phi(x)\; \backslash ,\; dx\; =\; \backslash Phi(x)\; -\; x\backslash phi(x)\; +\; C$ :$\backslash int\; x^\; \backslash phi(x)\; \backslash ,\; dx\; =\; -\backslash phi(x)\; \backslash sum\_^k\; \backslash fracx^\; +\; C$〔 lists this integral above without the minus sign, which is an error. See calculation by (WolframAlpha )〕 :$\backslash int\; x^\; \backslash phi(x)\; \backslash ,\; dx\; =\; -\backslash phi(x)\backslash sum\_^k\backslash fracx^\; +\; (2k+1)!!\backslash ,\backslash Phi(x)\; +\; C$ In these integrals, ''n''!! is the double factorial: for even ''n''’s it is equal to the product of all even numbers from 2 to ''n'', and for odd ''n''’s it is the product of all odd numbers from 1 to ''n'', additionally it is assumed that . : $\backslash int\; \backslash phi(x)^2\; \backslash ,\; dx\; =\; \backslash tfrac)\; +\; C$ : $\backslash int\; \backslash phi(x)\backslash phi(a\; +\; bx)\; \backslash ,\; dx\; =\; \backslash tfrac\backslash phi(\backslash tfrac)\backslash Phi(tx\; +\; \backslash tfrac)\; +\; C,\; \backslash qquad\; t\; =\; \backslash sqrt$〔 report this integral with error, see (WolframAlpha )〕 : $\backslash int\; x\backslash phi(a+bx)\; \backslash ,\; dx\; =\; -b^\backslash left\; (\backslash phi(a+bx)\; +\; a\backslash Phi(a+bx)\backslash right)\; +\; C$ : $\backslash int\; x^2\backslash phi(a+bx)\; \backslash ,\; dx\; =\; b^\; \backslash left\; ((a^2+1)\backslash Phi(a+bx)\; +\; (a-bx)\backslash phi(a+bx)\; \backslash right\; )\; +\; C$ : $\backslash int\; \backslash phi(a+bx)^n\; \backslash ,\; dx\; =\; \backslash frac\; \backslash Phi\backslash left(\backslash sqrt(a+bx)\backslash right)\; +\; C$ : $\backslash int\; \backslash Phi(a+bx)\; \backslash ,\; dx\; =\; b^\; \backslash left\; ((a+bx)\backslash Phi(a+bx)\; +\; \backslash phi(a+bx)\backslash right)\; +\; C$ : $\backslash int\; x\backslash Phi(a+bx)\; \backslash ,\; dx\; =\; \backslash tfrac\backslash left((b^2x^2\; -\; a^2\; -\; 1)\backslash Phi(a+bx)\; +\; (bx-a)\backslash phi(a+bx)\backslash right)\; +\; C$ : $\backslash int\; x^2\backslash Phi(a+bx)\; \backslash ,\; dx\; =\; \backslash tfrac\backslash left((b^3x^3\; +\; a^3\; +\; 3a)\backslash Phi(a+bx)\; +\; (b^2x^2-abx+a^2+2)\backslash phi(a+bx)\backslash right)\; +\; C$ : $\backslash int\; x^n\; \backslash Phi(x)\; \backslash ,\; dx\; =\; \backslash frac\backslash left(\; \backslash left\; (x^-nx^\; \backslash right\; )\backslash Phi(x)\; +\; x^n\backslash phi(x)\; +\; n(n-1)\backslash int\; x^\backslash Phi(x)\backslash ,dx\; \backslash right)\; +\; C$ : $\backslash int\; x\backslash phi(x)\backslash Phi(a+bx)\; \backslash ,\; dx\; =\; \backslash tfrac\backslash phi(\backslash tfrac)\backslash Phi(xt\; +\; \backslash tfrac)\; -\; \backslash phi(x)\backslash Phi(a+bx)\; +\; C,\; \backslash qquad\; t\; =\; \backslash sqrt$ : $\backslash int\; \backslash Phi(x)^2\; \backslash ,\; dx\; =\; x\; \backslash Phi(x)^2\; +\; 2\backslash Phi(x)\backslash phi(x)\; -\; \backslash tfrac)\; +\; C$ : $\backslash int\; e^\backslash phi(bx)^n\; \backslash ,\; dx\; =\; \backslash frac\}\}\backslash Phi\; \backslash left\; (\backslash frac\{b\backslash sqrt\{n\}\}\; \backslash right\; )\; +\; C,\; \backslash qquad\; b\backslash ne\; 0,\; n>0$ 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 integrals of Gaussian functions」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.