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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 Superfunction ： ウィキペディア英語版 Superfunction (詳細はInfinite compositions of analytic functionsを参照) In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function ''f'' and for some variable ''x'', the superfunction could be defined by the expression :$S(z;x)\; =$ \underbrace_ . Then, ''S(z;x)'' can be interpreted as the superfunction of the function ''f(x)''. Such a definition is valid only for a positive integer index ''z''. The variable ''x'' is often omitted. Much study and many applications of superfunctions employ various ''extensions of these superfunctions to complex and continuous indices''; and the analysis of the existence, uniqueness and their evaluation. The Ackermann functions and tetration can be interpreted in terms of super-functions. ==History== Analysis of superfunctions arose from applications of the evaluation of fractional iterations of functions. Superfunctions and their inverses allow evaluation of not only the first negative power of a function (inverse function), but also of any real and even complex iterate of that function. Historically, an early function of this kind considered was $\backslash sqrt~$; the function $\backslash sqrt~$ has then been used as the logo of the Physics department of the Moscow State University. 〔Logo of the physics department of Moscow State University. (In Russian); (). V.P.Kandidov. About the time and myself. (In Russian) (). 250 anniversary of the Moscow State University. (In Russian) ПЕРВОМУ УНИВЕРСИТЕТУ СТРАНЫ - 250! ()〕 At that time, these investigators did not have computational access for the evaluation of such functions, but the function $\backslash sqrt$ was luckier than $~\backslash sqrt~~$: at the very least, the existence of the holomorphic function $\backslash varphi$ such that $\backslash varphi(\backslash varphi(u))=\backslash exp(u)$ had been demonstrated in 1950 by Hellmuth Kneser.〔 〕 Relying on the elegant functional conjugacy theory of Schröder's equation, for his proof, Kneser had constructed the "superfunction" of the exponential map through the corresponding ''Abel function'' $\backslash mathcal$, satisfying the related Abel equation : $\backslash mathcal(\backslash exp(u))=\backslash mathcal(u)+1.\backslash$ so that $\backslash mathcal(S(z;u))=\backslash mathcal(u)+z\backslash$. The inverse function Kneser found, :$S(z;u)=\backslash mathcal^(z+\backslash mathcal(u))$ is an entire super-exponential, although it is not real on the real axis; it cannot be interpreted as tetrational, because the condition $S(0;x)=x$ cannot be realized for the entire super-exponential. The real $\backslash sqrt$ can be constructed with the tetrational (which is also a superexponential); while the real $\backslash sqrt~$ can be constructed with the superfactorial. 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「Superfunction」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.