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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 Landen's transformation ： ウィキペディア英語版 Landen's transformation Landen's transformation is a mapping of the parameters of an elliptic integral, which shows how the value of the integral, changes when its parameters:amplitude and modular angle changes following some dependency. As a special case, we can see when the transformation does not change the value of the integral. :It was originally due to John Landen, although independently rediscovered by Carl Friedrich Gauss. For example,the incomplete elliptic integral of the first kind is : $F(\backslash varphi,k)\; =\; F(\backslash varphi\; \backslash ,|\backslash ,\; k^2)\; =\; F(\backslash sin\; \backslash varphi\; ;\; k)\; =\; \backslash int\_0^\backslash varphi\; \backslash frac\; \backslash frac\; K(\backslash frac)$ :(aa) Where :$k\text{'}=\}\; \backslash frac\; \}\backslash frac$ and $\backslash scriptstyle$ are replaced by their arithmetic and geometric means respectively, that is :$a\_1\; =\; \backslash frac,\backslash qquad\; b\_1\; =\; \backslash sqrt.\backslash ,$ $I\_1\; =\; \backslash int\; \_0^\}\backslash fracK(\backslash frac)$ :$I\_1=\backslash fracK(\backslash frac)$ :Accordingly formula (aa) :$K(\backslash frac)=\backslash fracK(\backslash frac)$ :(aaa) : As follows from the formula (aaa) :$I\_1=I$ :The same equation can be proved using a simple mathematical analysis. The transformation, may be achieved purely by integration by substitution. It is convenient to first cast the integral in an algebraic form by a substitution of $\backslash scriptstyle\backslash right)\}$, $\backslash scriptstyle\backslash cos^(\backslash theta)\backslash right)\; d\; x\}$ giving :$I\; =\; \backslash int\; \_0^\}\backslash frac\; +\; a\; b\}\}$ gives the desired result (in the algebraic form) : & = \int _0^\infty\frac\right)^2\right) \left(t^2 + \left(\sqrt\right)^2\right)}} \, dt \end This latter step is facilitated by writing the radical as :$\backslash sqrt\; =\; 2x\; \backslash sqrt\backslash right)^2\}$ and the infinitesimal as :$dx\; =\; \backslash frac$ is easily recognized and cancelled between the two factors. ==Arithmetic-geometric mean and Legendre's first integral== If the transformation is iterated a number of times, then the parameters $\backslash scriptstyle$ and $\backslash scriptstyle$ converge very rapidly to a common value, even if they are initially of different orders of magnitude. The limiting value is called the arithmetic-geometric mean of $\backslash scriptstyle$ and $\backslash scriptstyle$, $\backslash scriptstyle$. In the limit, the integrand becomes a constant, so that integration is trivial :$I\; =\; \backslash int\; \_0^\}\; \backslash frac\}\backslash frac\; \backslash ,\; d\backslash theta\; =\; \backslash frac$ The integral may also be recognized as a multiple of Legendre's complete elliptic integral of the first kind. Putting $\backslash scriptstyle$ :$I\; =\; \backslash frac\; \backslash int\; \_0^\}\; \backslash frac\; F\backslash left(\; \backslash frac,k\backslash right)\; =\; \backslash frac\; K(k)$ Hence, for any $\backslash scriptstyle$, the arithmetic-geometric mean and the complete elliptic integral of the first kind are related by :$K(k)\; =\; \backslash frac)\}$ By performing an inverse transformation (reverse arithmetic-geometric mean iteration), that is :$a\_\; =\; a\; +\; \backslash sqrt\; \backslash ,$ :$b\_\; =\; a\; -\; \backslash sqrt\; \backslash ,$ :$\backslash operatorname(a,b)\; =\; \backslash operatorname(a\; +\; \backslash sqrt,a\; -\; \backslash sqrt)\; \backslash ,$ the relationship may be written as :$K(k)\; =\; \backslash frac\; \backslash ,$ which may be solved for the AGM of a pair of arbitrary arguments; :$\backslash operatorname(u,v)\; =\; \backslash frac\backslash right)\}.$ :''The definition adopted here for ''$\backslash scriptstyle$'', differs from that used in the arithmetic-geometric mean article, such that ''$\backslash scriptstyle$'' here is ''$\backslash scriptstyle$'' in that article.'' 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「Landen's transformation」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.