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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 Chebyshev equation ： ウィキペディア英語版 Chebyshev equation Chebyshev's equation is the second order linear differential equation :$$ (1-x^2) - x + p^2 y = 0 where p is a real constant. The equation is named after Russian mathematician Pafnuty Chebyshev. The solutions are obtained by power series: :$y\; =\; \backslash sum\_^\backslash infty\; a\_nx^n$ where the coefficients obey the recurrence relation :$a\_\; =\; a\_n.$ These series converge for x in $(1)$, as may be seen by applying the ratio test to the recurrence. The recurrence may be started with arbitrary values of a0 and a1, leading to the two-dimensional space of solutions that arises from second order differential equations. The standard choices are: :a0 = 1 ; a1 = 0, leading to the solution :$F(x)\; =\; 1\; -\; \backslash fracx^2\; +\; \backslash fracx^4\; -\; \backslash fracx^6\; +\; \backslash cdots$ and :a0 = 0 ; a1 = 1, leading to the solution :$G(x)\; =\; x\; -\; \backslash fracx^3\; +\; \backslash fracx^5\; -\; \backslash cdots.$ The general solution is any linear combination of these two. When p is an integer, one or the other of the two functions has its series terminate after a finite number of terms: F terminates if p is even, and G terminates if p is odd. In this case, that function is a pth degree polynomial (converging everywhere, of course), and that polynomial is proportional to the pth Chebyshev polynomial. :$T\_p(x)\; =\; (-1)^\backslash \; F(x)\backslash ,$ if p is even :$T\_p(x)\; =\; (-1)^\backslash \; p\backslash \; G(x)\backslash ,$ if p is odd 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「Chebyshev equation」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.