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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 Laguerre polynomials ： ウィキペディア英語版 Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are solutions of Laguerre's equation: : $xy\text{'}\text{'}\; +\; (1\; -\; x)y\text{'}\; +\; ny\; =\; 0$ which is a second-order linear differential equation. This equation has nonsingular solutions only if ''n'' is a non-negative integer. More generally, the name Laguerre polynomials is used for solutions of : $xy\text{'}\text{'}\; +\; (\backslash alpha+1\; -\; x)y\text{'}\; +\; ny\; =\; 0~.$ Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonin polynomials, after their inventor Nikolay Yakovlevich Sonin). The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form : $\backslash int\_0^\backslash infty\; f(x)\; e^\; \backslash ,\; dx.$ These polynomials, usually denoted ''L''0, ''L''1, ..., are a polynomial sequence which may be defined by the Rodrigues formula, :$L\_n(x)=\backslash frac\backslash frac\backslash left(e^\; x^n\backslash right)\; =\backslash frac\; \backslash left(\; \backslash frac\; -1\; \backslash right)\; ^n\; x^n,$ reducing to the closed form of a following section. They are orthogonal polynomials with respect to an inner product : $\backslash langle\; f,g\; \backslash rangle\; =\; \backslash int\_0^\backslash infty\; f(x)\; g(x)\; e^\backslash ,dx.$ The sequence of Laguerre polynomials is a Sheffer sequence, :$\backslash frac\; L\_n\; =\; \backslash left\; (\; \backslash frac\; -\; 1\; \backslash right\; )\; L\_.$ The Rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi–Carlitz polynomials. The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator. Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of ''n''! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.) == The first few polynomials == These are the first few Laguerre polynomials: } (x^2-4x+2) \, |- | align="center" | 3 | $\}\; (-x^3+9x^2-18x+6)\; \backslash ,$ |- | align="center" | 4 | $\}\; (x^4-16x^3+72x^2-96x+24)\; \backslash ,$ |- | align="center" | 5 | $\}\; (-x^5+25x^4-200x^3+600x^2-600x+120)\; \backslash ,$ |- | align="center" | 6 | $\}\; (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720)\; \backslash ,$ |} 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「Laguerre polynomials」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.