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From: Daniel Geisler <dan...@tetration.org>
Newsgroups: sci.math.research
Subject: Re: y(f(x))=y(x)+x
Date: Mon, 10 Jan 2005 23:33:02 -0800
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References: <vz6xjmdri49p@legacy> <crqpof$oh$1@news.math.niu.edu>
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Dave Rusin wrote:
> In article <vz6xjmdri49p@legacy>, Maxim <MOsadc...@nes.ru> wrote:
>
>>Is it possible to solve equation y(f(x))=y(x)+x if f(.) is a known
>>function? (e.g., f(x)=exp(x)-1 ?)
>
>
> What do you mean by "solve"?
>
> Exactly this example is treated as a prototype for a broad family of
> similar problems in a paper by Szekeres; he argues that there is one
> particularly optimal solution y and in an accompanying paper provides
> tables of numerical values. (He uses the analysis to describe a
> one-parameter family of functions f_s with f_s o f_t = f_{s+t} and
> in particular describes a functional "square root" f_{1/2} of f = f_1.)
>
> MR0141905 (25 #5302)
> Szekeres, G.
> Fractional iteration of exponentially growing functions.
> J. Austral. Math. Soc. 2 1961/1962 301--320. MSC section 39.99
>
> dave
>
See
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A052122
for f(x) such that f(f(x)) = exp(x)-1.
Daniel