eta as branchpoint of tetrational

[mike3]

I still can not completely follow your argumentation, you write:

Already we see that these are not conjugate, thus the tetrational will not be conjugate-symmetric and hence cannot be real at real heights. This contrasts with the behavior at \( b = e \). If a complex analytic function is real along some part of the real axis, but not another, there must be a branchpoint (not sure what the proof is, though.).

Ok, going with the base on the upper halfplane from e to sqrt(2).
As long as we are not landed back on the real axis,
the fixpoints are not conjugate, so the tetration value will not be real.
But this is nothing new.
Now when on the real axis: the Kneser method is not applicable to two real fixpoints.
So we must take the value there as limit from above.

Correct. I just edited the post -- I'm talking about the _extension_ of the Kneser function to the complex plane, and we take the analytic continuation to the axis for \( b < \eta \). But the idea is to imagine that in the upper half it behaves like the iteration at one fixed point, and in the lower half, like that at another.

But taking the limit, we would expect that in the upper halfplane, it should behave like the regular iteration at 2, and on the lower, like that at 4. So the resulting function will not be conjugate-symmetric, thus will not be real-valued for real heights greater than -2. On the upshot, however, we can fractionally iterate the exponential (though it'll probably be complex-valued), since \( \mathrm{tet}_b(\mathrm{slog}_b(z)) = z \) will be satisfied.

The kneser tetration \( b\mapsto b[4]p \) is real on the real axis \( b>\eta \), which implies that \( \overline{b} [4] p = \overline{b[4]p} \) (conjugation).

So approaching from above or below is just conjugate to each other.
So what one need to show imho is that the imaginary part will not tend to zero when approaching the real axis at \( b<\eta \).

_Or_ show that it is not conjugate-symmetric there.

I dont think it is 2 at \( +i\infty \) and 4 at \( -i\infty \). As mentioned before, the limits must be conjugate for the kneser tetration.
I guess there is a problem in the assumption that the Kneser tetration behaves at \( +i\infty \) like the regular iteration of the upper fixpoint in the case where the twe developing fixpoints are not conjugate.

I still agree with you that everything looks like there is a branchpoint, but I can not follow your particular arguments in the moment.

Note that in the linked-to thread I calculated a function that is apparently equivalent to the Kneser function, but with a method that could also go to a complex base. The resulting functions look like the fusion of the regular iterations at two non-conjugate fixed points. Note that since there is no proof of equivalence yet, this is not a proof, but suggestive. In any case, one could make the argument that a small deviation from the real axis at \( b > \eta \) should not radically alter the asymptotic behavior (continuity), thus it should still approach the fixed points at \( \pm i\infty \), albeit non-conjugate ones, and hence should look like a fusion of two regular iterations. Though I still don't think that's an actual proof.

It would be really great to have explicit coefficient formulas for tetration power series, etc.