Solving :f ^ [f(x)] (x) ... - sci.math.research

Message from discussion Solving :f ^ [f(x)] (x) ...

Daniel Geisler

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More options Dec 29 2004, 2:00 pm

Newsgroups: sci.math.research

From: Daniel Geisler <dan...@tetration.org>

Date: Wed, 29 Dec 2004 19:00:04 +0000 (UTC)

Local: Wed, Dec 29 2004 2:00 pm

Subject: Re: Solving :f ^ [f(x)] (x) ...

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Alain Verghote wrote:
> Dear All,

> Is this kind of equation known?
> ^[f(x)] means iteration f(x) times.
> How do we 'deal' with such a thing?

Alain, expressions like f^{f(x)}(x) have a clear meaning when
considered in the context of the Ackermann function.

Addition
f(x) = a+x
f^{n}(x) = a*n + x
f^{f(x)}(x) = a*(a + x) + x

Multiplication
f(x) = a*x
f^{n}(x) = a^n * x
f^{f(x)}(x) = a^{a*x} * x

See http://www.tetration.org/scimath/1 for an exported Mathematica
notebook. While both addition and multiplication result in solutions
for f(x), the plots of the solutions show they are not monotonic.
More generally, f^{f(x)}(x) is meaningful when using continuously
iterated functions, but then you loose the chance of arriving at a
closed form solution. I would be very surprised if anyone could find
a monotonic closed form solution.
Daniel

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