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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 Jacobi–Anger expansion ： ウィキペディア英語版 Jacobi–Anger expansion In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and cylindrical waves), and in signal processing (to describe FM signals). This identity is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger. The most general identity is given by:〔Colton & Kress (1998) p. 32.〕〔Cuyt ''et al.'' (2008) p. 344.〕 :$$ e^ = \sum_^ i^n\, J_n(z)\, e^, where $J\_n(z)$ is the $n$-th Bessel function of the first kind and $i$ is the imaginary unit, $i^2=-1.$ Consequently: :$$ e^ = \sum_^ J_n(z)\, e^. Using the relation $J\_(z)\; =\; (-1)^n\backslash ,\; J\_(z),$ valid for integer $n$, the expansion becomes:〔〔 :$e^=J\_0(z)\backslash ,\; +\backslash ,\; 2\backslash ,\; \backslash sum\_^\backslash ,\; i^n\backslash ,\; J\_n(z)\backslash ,\; \backslash cos\backslash ,\; (n\; \backslash theta).$ ==Real-valued expressions== The following real-valued variations are often useful as well:〔Abramowitz & Stegun (1965) (p. 361, 9.1.42–45 )〕 :$$ \begin \cos(z \cos \theta) &= J_0(z)+2 \sum_^(-1)^n J_(z) \cos(2n \theta), \\ \sin(z \cos \theta) &= -2 \sum_^(-1)^n J_(z) \cos\left(\theta\right ), \\ \cos(z \sin \theta) &= J_0(z)+2 \sum_^ J_(z) \cos(2n \theta), \\ \sin(z \sin \theta) &= 2 \sum_^ J_(z) \sin\left(\theta\right ). \end 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「Jacobi–Anger expansion」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.