Dmitrii Kouznetsov's Tetration Extension

[Kouznetsov]

...
I would begin with what I am best with: tetration by regular iteration.

It does not click.. May be I find it later..

"Equal to mine" is tetration that has no singularities at the right hand side of the complex halfplane.

But only if your uniqueness conjecture is correct.
Ok this demand can reduced to having no singularities in the strip \( S \) of \( 0<\Re(z)\le 1 \).

You are right, the strip is enough.

Hm, indeed I think we have only real plots available, as I am not familiar with complex function visualization.

If we work together, soon you will be.

However most tetrations considered here are given as a powerseries developed at 0.
So its not so easy to continue them to the whole strip \( S \).
However for the regular tetration I have an iterative formula so this should be possible to make a plot similar to yours.

Go ahead! I hope you can reproduce the period.

And as I already explained, it is periodic along the imaginary axis, hence has no limit at \( i\infty \). Does that mean that it is different to your's?

No, that does not. The analytic tetrations should coincide. Does your soluiton show period
\( \frac{2\pi \rm i}{ \ln\big(\ln(\sqrt{2})\Big)+\ln(2)}\approx 17.1431 \!~\rm i \)?

Which value does your \( F_{\sqrt{2}} \) assume at \( i\infty \)?
Is it the nearest complex fixed point of \( \sqrt{2}^z \)?

The previous code fails at \( b \le exp(1/\rm e) \), because it needs such a value.
I made another code for \( b=\sqrt{2} \); it does not assume the limiting value.
But it knows the asymptotic.

Well, I see, you are ready for figure 2. I am cleaning up the code. While, you can read the description I have posted. Do you already have writing access at citizendium?
For other users who watch only the last post, I repeat the url:
en.citizendium.org/wiki/Usermitrii_Kouznetsov/Analytic_Tetration
P.S. At the preview, part of url appears as "smile"; should be
U s e r : D m i t r i i _ K o u z n e t s o v without spaces