eta as branchpoint of tetrational

[sheldonison]

Oh I think we misunderstood each other.
If I remember you have an algorithm which computes the real-analytic (Kneser) superfunction of \( e^x \) and has the same outcome as Dimitrii's Cauchy algorithm.
Now, I guess you can generalize that algorithm to compute the superfunction of \( b^x \) for any base b, that has two non-real fixpoints.
And the question then would be if (complex) b approaches \( \sqrt{2} \) from the upper halfplane, what are the values of the superfunction (say applied to 1/2) and does the imaginary part tend to 0?

I've vaguely thought about the problem of complex bases, with two different fixed points, which aren't complex conjugates of each other, and attempting to generate a new superfunction g(z) via an f(z+theta(z)) mapping, where theta(z) decays to zero at +imag infinity, and has singularities at the integers, where the singularity gets cancelled in g(z). But thinking about it is as far as I've gotten. There are many unsolved problems....
- Sheldon