tetration from alternative fixed point

[sheldonison]

....To generate these contours lines in the SuperFunction, I used the inverse super function with img(z)=-3pi*i contour with real(z) varying from \( +/-\infty \)...

.... With the fix, the alternative contours no longer fits snugly... I'm still interested in seeing if there's any way to Riemann map the contours back to a well defined real axis, but I still assume that it is not possible.
- Sheldon

I'm pretty sure it is possible to generate an analytic sexp(z) function from the alternative fixed point after all! Looking at my older posts, I was also very very very close to seeing the solution 15 months ago. The problem is that there is more than one way to unwrap the inverse Schroder function into the complex plane, to generate the complex superfunction. From the best graph I previously posted, the correction for how to to unwrap the inverse Schroder function is to rotate the graph, and shrink it, so that superf(z+1)=exp(super(z)). But it would be better to just start over! Anyway, I have much prettier pictures this time, because I'm using Mike/Andy's complex graph coloring scheme.

Let's start with the complex superfunction from the secondary fixed point, L=2.0623 + 7.5886i. Here is a color plot of the complex superfunction. The period of the complex superfunction from the secondary fixed point is approximately 1.3769+2.17514i.
   
If you notice the light grey contour, you're looking at where the complex superfunction traces out the real number line from roughly -infinity to 4,000,000, or roughly from sexp(-2) to sexp(3), or five periods of z+theta(z). The negative real numbers are graphed in cyan. One more unit to the left of the cyan/grey contour, would be the 3pi i imaginary contour. Notice, that its \( 3\pi i \) instead of \( \pi i \), because for this alternative solution, \( \Im(\text{sexp}(-3..-2))=3\pi i \). The grey contour needs to get z+theta(z) mapped, so that this grey line becomes the real axis of the alternative sexp(z). I don't know how to calculate the theta(z) mapping, or the mathematically equivalent Riemann mapping, because this alternative sexp(z) function is not nearly as well behaved as the sexp(z) from the primary fixed point. Notice how quickly the function starts misbehaving as real(z) increases and imag(z) increases, above the grey contour. But in theory, it should be possible to calculate a theta/Riemann mapping, which would generate a 1 to 1 bijection between the superfunction from the secondary fixed point, and the upper half of the complex plane. I have some ideas for how to calculate it, although the existing Kneser.gp algorithm will not converge.

For comparison, here is the sexp(z) from the primary fixed point. Notice how nicely it is behaved, especially as imag(z) increases away from the real axis, and the function quickly converges to the primary fixed point! I also included the equivalent grey contour, for the real number line from roughly -infinity to 4,000,000, or sexp(-2) to sexp(3).
   

Here is the path from log(0.5) to the secondary fixed point. I didn't even try to include all of the path from 0.5 vertical to the fixed point. The path is even more chaotic than in my earlier post, with the rotated graph superfunction graph.
   

Finally, here is what the alternative sexp(z) graph would probably look like. As I said, I haven't calculated it yet, but this would have the requisite sexp(-3)..sexp(-2)=3pi i contour, where sexp(z) at integer values of z has the "z" term coefficient equal to 0, and the z^2 term coefficient also equal to zero.
- Sheldon