Alternate solution of tetration for "convergent" bases discovered

[sheldonison]

Hi.

I think I may have discovered an alternate solution, not the regular iteration(!)

When you compare to the regular iteration solution, are you developing the regular solution from a fixed point at -real infinity?
\( b = 2.33 + 1.28i \)
Generating the "regular" solution, I get fixed point=
0.45507 + 0.88875i, which matches one of your two fixed points! The period=3.98+0.23*I. But wouldn't the solution generated from the fixed point (if repelling) be an entire solution? Here is a parametric plot, from z=-100 to z=0, at the real axis. The function is nicely developing from the fixed point.

   

Here, the graph continues from z=0 to z=8.4, with the beginning of superexponential growth seen. This doesn't really look like any of your solutions..... I also tried your second base=1.33+1.28i. and it also seems to work nicely via regular tetration, with a steeper "slope". By the way, the color graphics in your post are really nice! Is there a way to generate the "png" color graphics file from pari-gp?

   

....
Consider the complex base \( b = 2.33 + 1.28i \), of which such a solution was constructed via the continuum sum method with a Fourier approximation (have had no luck using the Cauchy integral at complex bases).....

The parametric graph at the complex plane from z==100 to z=0 looks like:

So, this solution is no longer periodic in the complex plane. I wonder if you're developing something analogous to using the other fixed attracting point (assuming the other fixed point is attracting, and there is an attracting fixed point). Or else its somehow equivalent to a Schwarz reflection, with some sort of 1-cyclic transformation of the regular sexp solution, that creates the second fixed point, along with singularities for f(z)=log(0), log(log(0)), log(log(log(0))) ....
- Sheldon