base holomorphic tetration

[mike3]

that would be awesome. And this would mean that there is no singularity at \( b=e^{1/e} \)?

Yes, that would be correct.

with "use this" you mean the powerseries development you just derived at \( z_0 \)? But how do you know that it converges and that there is no branchpoint at \( z=e^{1/e} \)?

That's correct. Though as I've mentioned, there is as of yet no rigorous proof that the series converges even though the first 25 terms given do, hence why I mentioned it suggests, not proves, the hypothesis.

That it takes on a real value here suggests this continuation cannot be interpreted as regular iteration at these bases, which would take on complex values.

*nods* but at least it is already known that the regular iteration \( f(z)=\exp_{e^{1/e}}^{\circ t}(z) \) is not analytic at \( z=e^{1/e} \). However it is currently not clear to me what this states about the regularity of \( f(z)=\exp_z^{\circ t}(1) \) at \( z=e^{1/e} \).

Not analytic at \( e^{1/e} \)? That's strange. Not analytic at \( e \) would make more sense. If you look at a graph of \( ^{x} \left(e^{1/e}\right) \) for \( x > -2 \), it is a smooth, monotone curve that has a horizontal asymptote at \( e \). If we mirror it to get the shape of the inverse function \( \mathrm{slog}_{e^{1/e}}(x) \), we see the singularity is at \( e \) and \( e^{1/e} \) is a smooth point. Then we have \( \exp_{e^{1/e}}^t(z)\ =\ {}^{\mathrm{slog}_{e^{1/e}}(z) + t} \left(e^{1/e}\right) \), and as the slog is smooth at \( e^{1/e} \), and the tet is smooth at 1, it should be smooth there, no?

Ya I will try it with the series formula (or perhaps a mixture with limit formulas).
Actually it seems that no-one posted pictures of tetra-powers yet!!!
(yes Bat I mean tetra-powers.)
So it will be time that we have some pictures at least, as the theoretic consideration seems utmost complicated to me.

Can you tell me what this series formula is, by the way? I am only familiar with the limit formulas. Series seems easier to analyze theoretically than a limit with the whole "log log log log log log log ..." thing, and maybe also easier to analyze numerically, in that it may not have the increasing numerical precision requirement.