Dmitrii Kouznetsov's Tetration Extension

[bo198214]

Could you register as citizendium? It seems to be the only wiki that allows original researches.

Hm, yes I will do. However I am not convinced about the concept of Citizendium. I mean if you work at an expert level probably publishing in peer reviewed journals is a more desirable and established way to ensure quality. And if you work on an amateur level then you can put it into Wikipedia.
If I had to design and start a project like Citizendium then I just make a wikipedia with the additional feature of digitally signing articles and members. So if someone puts his signature below (a version) of an article then he states by best knowledge that the contents is thorough. And if he puts his signature below a person then he states by best knowing that the person is reliable. Based on whom signs whom and what you can compute a ranking of trustworthiness for articles and persons.
Ok, but if someone wants to continue this discussion then please start a new thread in "General discussions and questions".

2. There are two regular superexponentials at base \( b \) such that \( 1<b<\exp(1/{\rm e}) \).

What do you mean by regular? I would deprecate the term regular superexponential except in the sense of "the superexponential retrieved by regular iteration": \( \text{sexp}_b(x)=\exp_b^{\circ x}(1) \).
Otherwise we impose major confusion on this forum.
I think your use of "regular" can be replaced by "analytic", or something similar?

I have plotted the only one, \( F \) such that \( F(0)=1 \).
At \( b=\sqrt{2} \), for example,
\( \lim_{x \rightarrow \infty} F_{\sqrt{2}}(x+{\rm i}y)=2 \);
\( \lim_{x \rightarrow -\infty} F_{\sqrt{2}}(x+{\rm i}y)=4 \).

This contradicts \( F(z^\ast)=F(z)^\ast \)? How does it come anyway that you now can compute tetration also for bases \( <e^{1/e} \)?

There is another one, \( G \) that grows up along the real axis faster than any exponential and aproaches its limiting values in the opposite direction.

But if we use base \( \sqrt{2} \) it can not grow to infinity, it has to be limited by \( 2 \) on the real axis. Or what exactly do you mean?

I am writing source for its evaluation.

Did you read my previous post? I asked you to send me some code of your computations.

3. Then we have covered the ranges \( 1<b<\exp(1/\rm e) \) and \( b>\exp(1/\rm e) \); and I think about cases \( b=\exp(1/\rm e}) \) and \( b<1 \).

Yes, we would have covered them in one way. However if you read through this forum we have established several other (at least 3) ways to compute analytic tetration. And we even dont know yet which are equal or which are equal to yours.

I suggest that you use the same idea: first, find the asymptotics and periodicity (if any); then recover the analytic function with these properties. Could you calculate some pictures (similar to those I have posted) for these cases?

As I said: for the case of tetration by regular iteration, which is applicable in the range \( e^{-e}<b<e^{1/e} \), I can (and did) compute pictures.
Apropos pictures: I think you have to explain something more about your second posted picture.