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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 Solid harmonics ： ウィキペディア英語版 Solid harmonics In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates. There are two kinds: the ''regular solid harmonics'' $R^m\_\backslash ell(\backslash mathbf)$, which vanish at the origin and the ''irregular solid harmonics'' $I^m\_(\backslash mathbf)$, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately: :$$ R^m_(\mathbf) \equiv \sqrt}\; r^\ell Y^m_(\theta,\varphi) :$$ I^m_(\mathbf) \equiv \sqrt} \; \frac{r^{\ell+1}} == Derivation, relation to spherical harmonics == Introducing ''r'', θ, and φ for the spherical polar coordinates of the 3-vector r, we can write the Laplace equation in the following form :$\backslash nabla^2\backslash Phi(\backslash mathbf)\; =\; \backslash left(\backslash frac\; \backslash fracr\; -\; \backslash frac\backslash right)\backslash Phi(\backslash mathbf)\; =\; 0\; ,\; \backslash qquad\; \backslash mathbf\; \backslash ne\; \backslash mathbf,$ where ''l''2 is the square of the nondimensional angular momentum operator, :$\backslash mathbf\; =\; -i\backslash ,\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; .$ It is known that spherical harmonics Yml are eigenfunctions of ''l''2: :$$ \hat l^2 Y^m_\equiv \left(^2 +\hat l^2_y+\hat l^2_z\right )Y^m_ = \ell(\ell+1) Y^m_. Substitution of Φ(r) = ''F''(''r'') Yml into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution, :$$ \frac\fracr F(r) = \frac F(r) \Longrightarrow F(r) = A r^\ell + B r^. The particular solutions of the total Laplace equation are regular solid harmonics: :$$ R^m_(\mathbf) \equiv \sqrt}\; r^\ell Y^m_(\theta,\varphi), and irregular solid harmonics: :$$ I^m_(\mathbf) \equiv \sqrt} \; \frac{r^{\ell+1}} . 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「Solid 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.