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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 Zonal spherical harmonics ： ウィキペディア英語版 Zonal spherical harmonics In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group. On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented in spherical coordinates by :$Z^(\backslash theta,\backslash phi)\; =\; P\_\backslash ell(\backslash cos\backslash theta)$ where ''P''ℓ is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by $Z^\_)$, where x is a point on the sphere representing the fixed axis, and y is the variable of the function. This can be obtained by rotation of the basic zonal harmonic $Z^(\backslash theta,\backslash phi).$ In ''n''-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (''n''−1)-sphere. Define $Z^\_)$ in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ. In other words, the following reproducing property holds: :$Y(\backslash mathbf)\; =\; \backslash int\_\_)Y(\backslash mathbf)\backslash ,d\backslash Omega(y)$ for all ''Y'' ∈ Hℓ. The integral is taken with respect to the invariant probability measure. ==Relationship with harmonic potentials== The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in R''n'': for x and y unit vectors, :$\backslash frac|^n\}\; =\; \backslash sum\_^\backslash infty\; r^k\; Z^\_),$ where $\backslash omega\_$ is the surface area of the (n-1)-dimensional sphere. They are also related to the Newton kernel via :$\backslash frac|^\}\; =\; \backslash sum\_^\backslash infty\; c\_\; \backslash frac\}Z\_|\}^(\backslash mathbf/|\backslash mathbf|)$ where x,y ∈ R''n'' and the constants ''c''''n'',''k'' are given by :$c\_\; =\; \backslash frac.$ The coefficients of the Taylor series of the Newton kernel (with suitable normalization) are precisely the ultraspherical polynomials. Thus, the zonal spherical harmonics can be expressed as follows. If α = (''n''−2)/2, then :$Z^\_)\; =\; \backslash frac(\backslash mathbf\backslash cdot\backslash mathbf)$ where ''c''''n'',ℓ are the constants above and $C\_\backslash ell^$ is the ultraspherical polynomial of degree ℓ. 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「Zonal spherical harmonics」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.