Pictures of the Chi-Star

[sheldonison]

The periodic manner makes a bit more sense, but I'm confused how we get \( ^{-1} e = 0 \) since the original super function never equals zero. Is this just playing tricks with an essential singularity making a zero pop out? But then, how is it holomorphic in a neighborhood of zero? There must be some trick I'm missing.

I think the Schwarz reflection (since the Tetration is real valued at the real axis) guarantees there are no singularities.  We started by mapping the entire real axis with the \( \Psi(z) \) and then the \( \alpha(z) \) functions to generate the Riemann mapping region.  As far as the limit of the singularity, the composition of the complex valued superfunction \( \alpha^{-1}(z) \) with the result of the Riemann mapping approaches arbitrarily close to zero.  I'm not sure what details of the singularity matter to Kneser's proof of the construction, but the Riemann mapping region itself takes an extraordinarily complicated path.

I guess what I don't like about Kneser's solution is that there is no way to apply the same techniques to get from tetration to pentation

Tetration has real valued fixed points, so you don't need another Kneser Mapping for pentation, but such pentation functions don't seem all that interesting.  Personally, Kneser's Tetration holds a special place, and seems much more fun than the bounded Tetration solutions for bases<=eta, and also more interesting then pentation.