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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 Denjoy–Carleman–Ahlfors theorem ： ウィキペディア英語版 Denjoy–Carleman–Ahlfors theorem The ''Denjoy–Carleman–Ahlfors theorem'' states that the number of asymptotic values attained by a non-constant entire function of order ρ on curves going outwards toward infinite absolute value is less than or equal to 2ρ. It was first conjectured by Arnaud Denjoy in 1907.〔 〕 Torsten Carleman showed that the number of asymptotic values was less than or equal to (5/2)ρ in 1921.〔 〕 In 1929 Lars Ahlfors confirmed Denjoy's conjecture of 2ρ.〔 〕 Finally, in 1933, Carleman published a very short proof.〔 〕 The use of the term "asymptotic value" does not mean that the ratio of that value to the value of the function approaches 1 (as in asymptotic analysis) as one moves along a certain curve, but rather that the function value approaches the asymptotic value along the curve. For example, as one moves along the real axis toward negative infinity, the function $\backslash exp(z)$ approaches zero, but the quotient $0/\backslash exp(z)$ does not go to 1. ==Examples== The function $\backslash exp(z)$ is of order 1 and has only one asymptotic value, namely 0. The same is true of the function $\backslash sin(z)/z,$ but the asymptote is attained in two opposite directions. A case where the number of asymptotic values is equal to 2ρ is the sine integral $\backslash text(z)=\backslash int\_0^z\backslash frac\backslash ,d\backslash zeta$, a function of order 1 which goes to −π/2 along the real axis going toward negative infinity, and to +π/2 in the opposite direction. The integral of the function $a\backslash sin(z^2)/z+b\backslash sin(z^2)/z^2$ is an example of a function of order 2 with four asymptotic values (if ''b'' is not zero), approached as one goes outward from zero along the real and imaginary axes. More generally, $f(z)=\backslash int\_0^z\backslash fracd\backslash zeta,$ with ρ any positive integer, is of order ρ and has 2ρ asymptotic values. It is clear that the theorem applies to polynomials only if they are not constant. A constant polynomial has 1 asymptotic value, but is of order 0. 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「Denjoy–Carleman–Ahlfors theorem」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.