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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 Frobenius series ： ウィキペディア英語版 Frobenius method In mathematics, the Method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the form : $z^2u\text{'}\text{'}+p(z)zu\text{'}+q(z)u=0$ with : $u\text{'}\; \backslash equiv$ and $u\text{'}\text{'}\; \backslash equiv$ in the vicinity of the regular singular point $z=0$. We can divide by $z^2$ to obtain a differential equation of the form : $u\text{'}\text{'}+u\text{'}+u\; =\; 0$ which will not be solvable with regular power series methods if either ''p''(''z'')/''z'' or ''q''(''z'')/''z''2 are not analytic at ''z'' = 0. The Frobenius method enables us to create a power series solution to such a differential equation, provided that ''p''(''z'') and ''q''(''z'') are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite). == Explanation == The Method of Frobenius tells us that we can seek a power series solution of the form : $u(z)=\backslash sum\_^\backslash infty\; A\_kz^,\; \backslash qquad\; (A\_0\; \backslash neq\; 0)$ Differentiating: : $u\text{'}(z)=\backslash sum\_^\backslash infty\; (k+r)A\_kz^$ : $u\text{'}\text{'}(z)=\backslash sum\_^\backslash infty\; (k+r-1)(k+r)A\_kz^$ Substituting: :$z^2\backslash sum\_^\backslash infty\; (k+r-1)(k+r)A\_kz^\; +\; zp(z)\; \backslash sum\_^\backslash infty\; (k+r)A\_kz^\; +\; q(z)\backslash sum\_^\backslash infty\; A\_kz^$ :$=\; \backslash sum\_^\backslash infty\; (k+r-1)\; (k+r)A\_kz^\; +\; p(z)\; \backslash sum\_^\backslash infty\; (k+r)A\_kz^\; +\; q(z)\; \backslash sum\_^\backslash infty\; A\_kz^$ :$=\; \backslash sum\_^\backslash infty\; (A\_kz^\; +\; p(z)\; (k+r)\; A\_kz^\; +\; q(z)\; A\_kz^)$ :$=\; \backslash sum\_^\backslash infty\; \backslash left(+\; p(z)(k+r)\; +\; q(z)\backslash right)\; A\_kz^$ :$=\; \backslash left(r(r-1)+p(z)r+q(z)\; \backslash right)\; A\_0z^r+\backslash sum\_^\backslash infty\; \backslash left((k+r-1)(k+r)+p(z)(k+r)+q(z)\; \backslash right)\; A\_kz^$ The expression : $r\backslash left(r-1\backslash right)\; +\; p\backslash left(0\backslash right)r\; +\; q\backslash left(0\backslash right)\; =\; I(r)$ is known as the ''indicial polynomial'', which is quadratic in ''r''. The general definition of the ''indicial polynomial'' is the coefficient of the lowest power of ''z'' in the infinite series. In this case it happens to be that this is the ''r''th coefficient but, it is possible for the lowest possible exponent to be ''r'' − 2, ''r'' − 1 or, something else depending on the given differential equation. This detail is important to keep in mind because one can end up with complicated expressions in the process of synchronizing all the series of the differential equation to start at the same index value which in the above expression is ''k'' = 1. However, in solving for the indicial roots attention is focused only on the coefficient of the lowest power of ''z''. Using this, the general expression of the coefficient of ''z''''k'' + ''r'' is :$I(k+r)A\_k+\backslash sum\_^(0)\; \backslash over\; (k-j)!\}A\_j$, These coefficients must be zero, since they should be solutions of the differential equation, so :$I(k+r)A\_k+\backslash sum\_^\; (0)\; \backslash over\; (k-j)!\}\; A\_j=0$ :$\backslash sum\_^(0)\; \backslash over\; (k-j)!\}A\_j=-I(k+r)A\_k$ :$\backslash sum\_^(0)\; \backslash over\; (k-j)!\}A\_j=A\_k$ The series solution with ''A''''k'' above, :$U\_(z)=\backslash sum\_^A\_kz^$ satisfies :$z^2U\_(z)\text{'}\text{'}+p(z)zU\_(z)\text{'}+q(z)U\_(z)=I(r)z^r$ If we choose one of the roots to the indicial polynomial for ''r'' in ''U''''r''(''z''), we gain a solution to the differential equation. If the difference between the roots is not an integer, we get another, linearly independent solution in the other root. 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「Frobenius method」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.