Re: Real Tetration Solution

> Hi Andrew,
>
> The good news is that such a solution is very
> interesting if indeed you've
> got it. Much more interesting that a simple
> continuous solution. The bad
> news is that an infinitely differentiable solution
> has already been
> constructed and what's worse, in that same paper the
> author proves that
> there are infinitely many infinitely differentiable
> solutions (as well as
> infinitely many simply continuous solutions), in a
> paper which has already
> been submitted for publication:
>
> http://users.forthnet.gr/ath/jgal/math/ExtensionsPaper
> .html
>
> Consequently, you still cannot talk about "the" value
> of, for example,
> e^^pi, or things similar.
> The really hard part, is finding a _real analytic_
> solution, which, as far
> as I know, has not been done yet analytically, but
> can certainly be done
> numerically, using series approximations.
>
> For some good attempts on the later, you can check
> out David Rusin's pages
> [ref 29] on the above paper.
>
> Cheerio,
> --
> I. N. Galidakis
> http://users.forthnet.gr/ath/jgal/
> Eventually, _everything_ is understandable
>

You are right in saying there are an infinite number of extensions to tetration which are infinitely differentiable, but I think there is only one that is not oscillating. Or in other words, all derivatives are positive where the order is positive. Or in other words all derivatives are monotonic increasing, where the order is positive. I also believe that the inversion of the tetralog function I found makes an infinitely differentiable AND non-oscillating extension of tetration. So I believe I have found that one non-oscillating infinitely differentiable extension.

Andrew Robbins

PS. I've read your paper with your two extensions of tetration, and neither are non-oscillating.
.

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