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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 Associated Legendre polynomial ： ウィキペディア英語版 Associated Legendre polynomials In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation :$(1\; -\; x^2)\; \backslash frac\; P\_\backslash ell^m(x)\; -\; 2\; x\; \backslash frac\; P\_\backslash ell^m(x)\; +\; \backslash left(\backslash ell\; (\backslash ell\; +\; 1)\; -\; \backslash frac\; \backslash right)\; P\_\backslash ell^m(x)\; =\; 0$, or equivalently :$\backslash frac\; \backslash left((1\; -\; x^2)\; \backslash frac\; P\_\backslash ell^m(x)\; \backslash right)\; +\; \backslash left(\backslash ell\; (\backslash ell\; +\; 1)\; -\; \backslash frac\; \backslash right)\; P\_\backslash ell^m(x)\; =\; 0$, where the indices ℓ and ''m'' (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on () only if ℓ and ''m'' are integers with 0 ≤ ''m'' ≤ ℓ, or with trivially equivalent negative values. When in addition ''m'' is even, the function is a polynomial. When ''m'' is zero and ℓ integer, these functions are identical to the Legendre polynomials. In general, when ℓ and ''m'' are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when ''m'' is odd. The fully general class of functions with arbitrary real or complex values of ℓ and ''m'' are Legendre functions. In that case the parameters are usually labelled with Greek letters. The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics. ==Definition for non-negative integer parameters ℓ and ''m''== These functions are denoted $P\_\backslash ell^(x)$, where the superscript indicates the order, and not a power of ''P''. Their most straightforward definition is in terms of derivatives of ordinary Legendre polynomials (''m'' ≥ 0) :$P\_\backslash ell^(x)\; =\; (-1)^m\backslash \; (1-x^2)^\backslash \; \backslash frac\backslash left(P\_\backslash ell(x)\backslash right)\backslash ,$ The (−1)''m'' factor in this formula is known as the Condon–Shortley phase. Some authors omit it. The functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ℓ and ''m'' follows by differentiating ''m'' times the Legendre equation for ''P''ℓ:〔.〕 :$(1-x^2)\; \backslash fracP\_\backslash ell(x)\; -2x\backslash fracP\_\backslash ell(x)+\; \backslash ell(\backslash ell+1)P\_\backslash ell(x)\; =\; 0.$ Moreover, since by Rodrigues' formula, :$P\_\backslash ell(x)\; =\; \backslash frac\; \backslash \; \backslash frac\backslash left(),$ the ''P'' can be expressed in the form :$P\_\backslash ell^(x)\; =\; \backslash frac\; (1-x^2)^\backslash \; \backslash frac\}(x^2-1)^\backslash ell.$ This equation allows extension of the range of ''m'' to: −ℓ ≤ ''m'' ≤ ℓ. The definitions of ''P''ℓ±''m'', resulting from this expression by substitution of ±''m'', are proportional. Indeed, equate the coefficients of equal powers on the left and right hand side of :$$ \frac} (x^2-1)^ = c_ (1-x^2)^m \frac}(x^2-1)^, then it follows that the proportionality constant is :$$ c_ = (-1)^m \frac , so that :$$ P^_\ell(x) = (-1)^m \frac P^_\ell(x). 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「Associated Legendre 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.