Exact and Unique solution for base e^(1/e)

[sheldonison]

Well, I've been off with projects (experimenting with random number generators, learning lots about primes, 2-adic numbers, galois fields, spectral tests, lattice reduction, lattice covering and packing, and the like).

But I've decided to turn my interests back to tetration and related subjects.

One particular subject that I stopped investigating was my "cheta" function, which is essentially a scaled version of the continuous iteration of \( \exp(x)-1 \). (I stopped work on it mainly because I was using it to find a general solution to tetration using a change of base formula, and I unfortunately found that the results don't match those of the "natural" solution.)

So I've started tinkering with it again, slowly remembering old insights and forming new ones. Before I invest too much time, I was wondering how much has been discussed on this subject in my 1-2 year absence. Can someone point me to anything new that I should read up on before I get too far along? Relevant subjects include not only base \( e^{1/e} \), a.k.a. eta, but also the problem of non-convergence for non-integer iterates of functions, etc.

Jay,

I become interested in tetration last fall, long after you left this forum, but followed many of the same lines of reasoning that you did, often without realizing it. I eventually went through every one of your posts, and thoroughly enjoyed reading them. I realized you had already explored almost every idea I had, and many more. My math education is somewhat lacking (25 years ago I got an BS in computer engineering), but it has been fun to read about the progress made in tetration.

Dimitrii Kouznetsov has some graphs published for \( \text{sexp}_\eta \), at http://en.citizendium.org/wiki/Tetration . There was also a lot of interesting posts about the upper super exponential, which I found very interesting, http://math.eretrandre.org/tetrationforu...260&page=2 Henryk and Dimitrii coauthored a paper on that. And there's my own thread, http://math.eretrandre.org/tetrationforu...hp?tid=236, where I explored some of the same base conversion ideas you had earlier explored. I stopped, after realizing that the base conversion definition of the super exponential is probably not analytic, although I don't have the math background to give it a rigorous treatment.

I haven't actually seen a published graph of the upper super exponential for base e^(1/e). I personally think its a very interesting base to work with. I don't have the sophisticated math software to make the nice complex contour graphs.
- Sheldon Levenstein