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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 Sommerfeld identity ： ウィキペディア英語版 Sommerfeld identity The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves, :$$ \frac = \int\limits_0^\infty I_0(\lambda r) e^ \frac where :$$ \mu = \sqrt is to be taken with positive real part, to ensure the convergence of the integral and its vanishing in the limit $z\; \backslash rightarrow\; \backslash pm\; \backslash infty$ and :$$ R^2=r^2+z^2 . Here, $R$ is the distance from the origin while $r$ is the distance from the central axis of a cylinder as in the $(r,\backslash phi,z)$ cylindrical coordinate system. The function $I\_0$ is a Bessel function. Here the notation for Bessel functions follows the German convention, to be consistent with the original notation used by Sommerfeld. In English literature it is more common to use :$I\_n(\backslash rho)=J\_n(i\; \backslash rho)$. This identity is known as the Sommerfeld Identity (). An alternative form is :$$ \frac = i\int\limits_0^\infty Where :$$ k_z=(k_0^2-k_\rho^2)^ (). The notation used here is different form that above: $r$ is now the distance from the origin and $\backslash rho$ is the axial distance in a cylindrical system defined as $(\backslash rho,\backslash phi,z)$. The physical interpretation is that a spherical wave can be expanded into a summation of cylindrical waves in $\backslash rho$ direction, multiplied by a plane wave in the $z$ direction; see the Jacobi-Anger expansion. The summation has to be taken over all the wavenumbers $k\_\backslash rho$. == References == # Sommerfeld, A.,''Partial Differential Equations in Physics'',Academic Press,New York,1964 # Chew, W.C.,''Waves and Fields in Inhomogeneous Media'',Van Nostrand Reinhold,New York,1990 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「Sommerfeld identity」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.