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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 Laguerre–Pólya class ： ウィキペディア英語版 Laguerre–Pólya class The Laguerre–Pólya class is the class of entire functions consisting of those functions which are locally the limit of a series of polynomials whose roots are all real. 〔("Approximation by entire functions belonging to the Laguerre–Pólya class" ) by D. Dryanov and Q. I. Rahman, ''Methods and Applications of Analysis" 6 (1) 1999, pp. 21–38.〕 Any function of Laguerre–Pólya class is also of Pólya class. The product of two functions in the class is also in the class, so the class constitutes a monoid under the operation of function multiplication. Some properties of a function $E(z)$ in the Laguerre–Pólya class are: *All roots are real. *$|E(x+iy)|=|E(x-iy)|$ for ''x'' and ''y'' real. *$|E(x+iy)|$ is a non-decreasing function of ''y'' for positive ''y''. A function is of Laguerre–Pólya class if and only if three conditions are met: *The roots are all real. *The nonzero zeros ''zn'' satisfy :$\backslash sum\_n\backslash frac$ converges, with zeros counted according to their multiplicity) * The function can be expressed in the form of a Hadamard product :$z^m\; e^\backslash prod\_n\; \backslash left(1-z/z\_n\backslash right)\backslash exp(z/z\_n)$ with ''b'' and ''c'' real and ''c'' non-positive. (The non-negative integer ''m'' will be positive if ''E''(0)=0. Note that if the number of zeros is infinite one may have to define how to take the infinite product.) ==Examples== Some examples are $\backslash sin(z),\; \backslash cos(z),\; \backslash exp(z),\; \backslash exp(-z),\; \backslash text\backslash exp(-z^2).$ On the other hand, $\backslash sinh(z),\; \backslash cosh(z),\; \backslash text\; \backslash exp(z^2)$ are ''not'' in the Laguerre–Pólya class. For example, :$\backslash exp(-z^2)=\backslash lim\_(1-z^2/n)^n.$ Cosine can be done in more than one way. Here is one series of polynomials having all real roots: :$\backslash cos\; z=\backslash lim\_((1+iz/n)^n+(1-iz/n)^n)/2$ And here is another: :$\backslash cos\; z=\backslash lim\_\backslash prod\_^n\; \backslash left(1-\backslash frac)\backslash pi)^2\}\backslash right)$ This shows the buildup of the Hadamard product for cosine. If we replace ''z''2 with ''z'', we have another function in the class: :$\backslash cos\; \backslash sqrt\; z=\backslash lim\_\backslash prod\_^n\; \backslash left(1-\backslash frac\; z)\backslash pi)^2\}\backslash right)$ Another example is the reciprocal gamma function 1/Γ(z). It is the limit of polynomials as follows: :$1/\backslash Gamma(z)=\backslash lim\_\backslash frac\; 1(1-(\backslash ln\; n)z/n)^n\backslash prod\_^n(z+m).$ 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「Laguerre–Pólya class」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.