eta as branchpoint of tetrational

[sheldonison]

But what if we start not with the Kneser tetration, but with the regular tetration?

We want to continue regular iteration with respect to its base along a path in the upper (resp. lower) half plane. For example along the path \( b=\gamma(t) \), \( t\in (0,1) \) in my previous post.
.... So at the arriving fixpoint regular tetration is not real anymore, and hence different from the Kneser tetration.
......I think Sheldon mentioned somewhere something where \( \theta \) vanishes towards imaginary infinity, which could give a new tetration.

I can't comment intelligently about continuing from a real base>eta to b=sqrt(2), using a path with complex bases, or directly, through the Shell-Thron boundary. Its something I'll need to spend some time on. Perhaps continuing along that path arrives at the alternative different superfunction, that I calculated. As Henryk said, for base=sqrt(2), I used a "Kneser" mapping from the upper entire function to calculate a \( \text{NewSuperFunction}_{\sqrt{2}}(z)=\text{usexp}_{\sqrt{2}}(z+\theta(z)) \). See this post, where theta(z) exponentially decays as imag(z) grows larger. And this would be a different superfunction than either of the two superfunctions previously developed from the upper and lower fixed points.

fyi, I also succeeded in doing another different modified Kneser mapping to accurately calculate \( \theta(z) \) for \( \text{sexp}_{\sqrt{2}}(z)=\text{usexp}_{\sqrt{2}}(z+\theta(z)) \), where in this case \( \theta(z) \) is a real valued analytic function which doesn't decay as imag(z) grows, and instead has unit singularities as imag(z) grows larger positive and negative. The results are in another post in the same thread.
- Sheldon