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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 Elliptic rational functions ： ウィキペディア英語版 Elliptic rational functions In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name). Rational elliptic functions are identified by a positive integer order ''n'' and include a parameter ξ ≥ 1 called the selectivity factor. A rational elliptic function of degree ''n'' in ''x'' with selectivity factor ξ is generally defined as: :$R\_n(\backslash xi,x)\backslash equiv\; \backslash mathrm\backslash left(n\backslash frac\backslash ,\backslash mathrm^(x,1/\backslash xi),1/L\_n\backslash right)$ * cd() is the Jacobi elliptic cosine function. * K() is a complete elliptic integral of the first kind. * $L\_n(\backslash xi)=R\_n(\backslash xi,\backslash xi)$ is the discrimination factor, equal to the minimum value of the magnitude of $R\_n(\backslash xi,x)$ for $|x|\backslash ge\backslash xi$. For many cases, in particular for orders of the form ''n'' = 2''a''3''b'' where ''a'' and ''b'' are integers, the elliptic rational functions can be expressed using algebraic functions alone. Elliptic rational functions are closely related to the Chebyshev polynomials: Just as the circular trigonometric functions are special cases of the Jacobi elliptic functions, so the Chebyshev polynomials are special cases of the elliptic rational functions. == Expression as a ratio of polynomials == For even orders, the elliptic rational functions may be expressed as a ratio of two polynomials, both of order ''n''. :$R\_n(\backslash xi,x)=r\_0\backslash ,\backslash frac)\}$      (for n even) where $x\_i$ are the zeroes and $x\_$ are the poles, and $r\_0$ is a normalizing constant chosen such that $R\_n(\backslash xi,1)=1$. The above form would be true for even orders as well except that for odd orders, there will be a pole at x=∞ and a zero at x=0 so that the above form must be modified to read: :$R\_n(\backslash xi,x)=r\_0\backslash ,x\backslash ,\backslash frac\; (x-x\_i)\}\; (x-x\_)\}$      (for n odd) 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「Elliptic rational functions」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.