06/10/2011, 09:36 PM
(06/10/2011, 01:54 PM)sheldonison Wrote: Looks great! Thanks for making the graph. I'm not sure I understand all of your comments though... regular tetration at a base>eta would be a complex superfunction, with no real valued real axis, right?
Regular iteration is complex for complex bases, but I'm talking about the bipolar/perturbed-Fatou/Kneser iteration. If this "merge" tetrational can be considered as the analytic continuation of the Kneser tetrational halfway around the branch point at \( b = \eta \) (the two functions graphed, then, are, respectively, the continuation around the branch point through the lower and upper half-plane to the real axis where \( b < \eta \), specifically \( b = \sqrt{2} \)), then since it is sooooooo close to being real-valued for real heights, the branch point at \( b = \eta \) must be extremely slight, at least for real heights greater than -2 and when the cut is placed at the real axis for \( b < \eta \). It is like how there is a "mild" singularity/branch point in \( \exp^{1/2}(z) \) at the conjugate fixed point pair, but perhaps even "milder".