Dmitrii Kouznetsov's Tetration Extension

[bo198214]

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

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

Regular iteration is definitely something you have to learn about.
The usual starting article for doing so is
G. Szekeres, Regular iterations of real and complex functions. Acta Math. 100 (195, 203-258

However I think it is written somewhat cumbersome.
If you speak French I advice you to read
Jean Écalle, Théorie des invariants holomorphes, Publ. Math. Orsay No. 67–7409

Lars Kinderman has a very good collection of literature for the topic.

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
\( \frac{2\pi \rm i}{ \ln\big(\ln(\sqrt{2})\Big)+\ln(2)}\approx 17.1431 \!~\rm i \)?

Hm, not exactly, what I derived some posts earlier would give merely \( \frac{2\pi i}{\ln(\ln(\sqrt{2}))} \), yours is \( \frac{2\pi i}{\ln(\ln(2))} \).

Btw. there are infinitely many analytic functions that have no singularity on right halfplane: If \( F \) has no singularity there, also \( F(x+c\sin(2\pi x)) \), \( 0<c<\frac{1}{2\pi} \), is a superexponential with no singularities on the right halfplane and any 1-periodic function \( p \) with \( p(x)>-x \) in \( 0<x\le 1 \) would do.

More interesting seems here the boundedness on the strip \( 0<\Re(z)\le 1 \), which seems to be an universal uniqueness criterion.

I made another code for \( b=\sqrt{2} \); it does not assume the limiting value.
But it knows the asymptotic.

ah, ok.

Do you already have writing access at citizendium?

Hm, I signed up but did not get any response yet ...

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

You can include clickable urls into your post with the construction:

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[url=http://....]how the link appears[/url]