tetration base conversion, and sexp/slog limit equations

[mike3]

The calculation was done using the Pari/GP package,
....
And that's interesting about the ringing, it confirms the lack of a viable asymptotic here even without looking at the complex graph.

So it would seem that the "cheta" method is not a good method for extending tetration.

I'll have to try out that Pari/GP package. I agree, the "cheta" method is not a good method for extending tetration, and I also agree that \( \theta(x) \) is not analytic. But it tells us something about tetration, that fact that tetration for two different bases can be made to ring. There is an approximate sexp base conversion constant for large numbers, but the ringing means it is only approximate. The ringing for base e, and base eta, it is 0.08% from min to max. For cheta, and e, (or any other two sexp bases greater than \( \eta \)) there is a 1-cyclic conversion factor, such that as integer "n" increases,
\( \text{sexp}_e(x+n+\theta(x)) = \text{cheta}(x+n) \)
for sexp_e, cheta \( \theta(x) \) varies in the range -0.58432+/-0.0004

And there's still a couple of more interesting things I'm trying to figure out. But actually, the only bases for which I know how to generate reasonably accurate slog/sexp are base eta, and base e. Also, I can generate pretty good approximations for sexp for bases between eta and maybe 1.6 or so. It would help a lot if I had a Taylor series, for example for base 2, to verify some of the patterns I'm seeing. The phase and amplitude patterns hold for bases between eta, and 1.6, and also for base e. I'm sure someone must have a link to Andy's slog solution Taylor series results...
- Sheldon

The graph I gave for \( \mathrm{tet}(z) \) was done via the Cauchy integral. It should be possible also to use the Cauchy integral at other bases greater than \( \eta \). I'll see if I could try one for \( \mathrm{tet}_2(z) \) to get a graph and Taylor series approximation.

I do wonder though, even if \( \check{\eta}(z) \) cannot be used to approximate \( \mathrm{tet}(z) \), whether it is still possible that maybe \( \mathrm{tet}_{b_1}(z) \) and \( \mathrm{tet}_{b_2}(z) \) can be used to approximate each other, for real \( b_1 \) and \( b_2 \) greater than \( \eta \). However, the fine detail in that fractal thingy in the graph makes it seem questionable.