Arguments for the beta method not being Kneser's method

[JmsNxn]

Kneser's tetration was conjectured to have no singularities in the upper half plane ( for Im(z) > 0 ).
Or was that Kouznetsov ? If I recall correctly ; both ... implying they are identical and unique.

I could be wrong though.

After all these years, I see little public info about singularities. ( apart from trivial log singularities at expected places )

Not that it is easy though. For a taylor series we try to prove the radius of convergeance for a function expansion somewhere... usually by special properties or by patterns and asymptotics in the n th derivatives.

But usually we do not get a nice taylor series with proven trends.

We get something for which no efficient radius of convergeance method is known or is efficient.
Converting to a taylor series often does not help immediately and feels like a restatement of the problem or a nonconstructive circular logic.

Apart from uniqueness by singularities there is the idea of uniqueness by bounds.
( sometimes proven, sometimes conjectures ... im not even sure without thinking about it first )

Those are just my impressions though and I could be wrong.

Also I just scratched the surface of the large amount of things that could have been said about it.


regards

tommy1729

What I was pointing out is that Samuel Cogwill and William Paulsen proved a uniqueness condition.

\(
F(z)\,\,\text{is holomorphic for}\,\,\Im(z) >0\\
F(0) = 1\\
F(z^*) = F(z)^*\\
F(z+1) = \exp(F(z))\\
\lim_{\Im(z) \to \infty} F(z) = L\\
\exp(L) = L\,\,\text{and it has minimal imaginary argument of all of exp's fixed points}\\
F : \mathbb{R}^+ \to \mathbb{R}^+\\
\Rightarrow\,\,F\,\,\text{is Kneser's Tetration}\\
\)

This implies that Kouznetsov is Kneser's solution; unless it has singularities in the upper half plane. The \( \beta \)-method changes only one thing, that \( \lim_{\Im(s) \to \infty} F(s) = \infty \). Kouznetsov has just found a different construction to Kneser. But it is Kneser's. At least, according to Paulsen and Cogwill. And their paper is peer reviewed and absolutely phenomenal. They really bring it down to earth. I suggest reading it.

The main difference in my method is divergence at imaginary infinity. If it doesn't diverge it's just Kneser.

As to Taylor Series; it isn't quite backwards as you may be thinking. I'm mostly referring to it as a programming strategy in pari-gp; not actually calculating the Taylor series analytically. It makes the programming somewhat more accurate and avoids the nasty hairs I keep seeing everywhere.

Regards.