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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 Hermite polynomial ： ウィキペディア英語版 Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * probability, such as the Edgeworth series; * in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * in numerical analysis as Gaussian quadrature; * in finite element methods as shape functions for beams; * in physics, where they give rise to the eigenstates of the quantum harmonic oscillator; * in systems theory in connection with nonlinear operations on Gaussian noise. Hermite polynomials were defined by 〔P.S. Laplace: ''Théorie analytique des probabilitte és'' 1812 livre 2, 321-323; Oeuvres VII〕 though in scarcely recognizable form, and studied in detail by Chebyshev (1859).〔P.L.Chebyshev: ''Sur le développement des fonctions à une seule variable'' Bull. Acad. Sci. St. Petersb. I 1859 193-200; Oeuvres I 501-508.〕 Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new.〔C. Hermite: ''Sur un nouveau développement en série de fonctions'' C. R Acad. Sci. Paris 58 1864 93-100; Oeuvres II 293-303〕 They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials. ==Definition== There are two different ways of standardizing the Hermite polynomials: * The "probabilists' Hermite polynomials" are given by ::$\backslash mathit\_n(x)=(-1)^n\; e^\}\backslash frace^\}=\backslash left\; (x-\backslash frac\; \backslash right\; )^n\; \backslash cdot\; 1$, * while the "physicists' Hermite polynomials" are given by ::$H\_n(x)=(-1)^n\; e^\backslash frace^=\backslash left\; (2x-\backslash frac\; \backslash right\; )^n\; \backslash cdot\; 1$. These two definitions are not exactly identical; each one is a rescaling of the other, :$H\_n(x)=2^\}\; \backslash ,x),\; \backslash qquad\; \backslash mathit\_n(x)=2^\}H\_n\backslash left(\backslash frac\; x\}$ is the probability density function for the normal distribution with expected value 0 and standard deviation 1. The first eleven probabilists' Hermite polynomials are: :$\}\_1(x)=x\backslash ,$ :$\}\_3(x)=x^3-3x\backslash ,$ :$\}\_5(x)=x^5-10x^3+15x\backslash ,$ :$\}\_7(x)=x^7-21x^5+105x^3-105x\backslash ,$ :$\}\_9(x)=x^9-36x^7+378x^5-1260x^3+945x\backslash ,$ :$(x)=x^-45x^8+630x^6-3150x^4+4725x^2-945\backslash ,$ and the first eleven physicists' Hermite polynomials are: :$H\_0(x)=1\backslash ,$ :$H\_1(x)=2x\backslash ,$ :$H\_2(x)=4x^2-2\backslash ,$ :$H\_3(x)=8x^3-12x\backslash ,$ :$H\_4(x)=16x^4-48x^2+12\backslash ,$ :$H\_5(x)=32x^5-160x^3+120x\backslash ,$ :$H\_6(x)=64x^6-480x^4+720x^2-120\backslash ,$ :$H\_7(x)=128x^7-1344x^5+3360x^3-1680x\backslash ,$ :$H\_8(x)=256x^8-3584x^6+13440x^4-13440x^2+1680\backslash ,$ :$H\_9(x)=512x^9-9216x^7+48384x^5-80640x^3+30240x\backslash ,$ :$H\_(x)=1024x^-23040x^8+161280x^6-403200x^4+302400x^2-30240\backslash ,$ 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「Hermite 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.