rumor result

[sheldonison]

hi all

i dont have time to explain or post alot.

but i managed to get some partial results about tetration i believe.

i seem to have gotten a similar equation as andrew's slog ( the infinite matrix ) for my tommysexp(tommyslog(x)+k) in base e.

also this seems to relate and possibly answer many uniqueness and existance questions that seemed somewhat unrelated and unattackable before.

im still working on it but it seems promising and i might even solve some of those hard matrix questions.

in a sense i also seem to have found a trick to attach a fixpoint to an infinite matrix equation. ( yes this is vague and informal , i dont have time to explain fully )

regards

tommy1729

minor edit
I spent some time analyzing the derivatives of the basechange function, and the work can also be applied to tommysexp. I didn't post the results because they're complicated, and I didn't think they were good enough to provide a proof of nowhere analytic, but the results were very interesting. Here's a summary. The derivatives of the basechange/tommysexp functions act somewhat like a Russian stacking doll. For tommysexp, centered at x=0, with tommysexp(0)=1, the first 5 million or so derivatives are consistent with a radius of convergence of ~0.46, and appear to be well behaved. But then somewhere between the 5 millionth and the 6 millionth derivative, the radius of convergence abruptly changes to ~0.035. Then, again abruptly around the sexp(4)th derivative, the radius of convergence changes again, to approximately 0.0000000058. Near the sexp(5)th derivative, the radius of convergence abruptly changes to approximately 1/sexp(4). This goes on forever, with the radius of convergence getting arbitrarily small, but only for superexponentially huge derivatives.
- Shel