(01/15/2021, 02:25 AM)tommy1729 Wrote: Good news everyone.
I just logged in and haven’t read the paper yet but from the comments, I can confirm this tetration is ( or can be made ) analytic and in fact has more or less a closed form !!!
Im sleepy now but will explain more tomorrow.
I immediately realized this function phi(s) was one that satisfied properties that I was Looking for years, hence my quick analysis.
Knowledge of perturbation theory and complex analysis helped too.
This paper by James Nixon also convinced me my sinh method can Probably also be made analytic.
The “ probably “ comes from the idea that I’m not sure if fixed point theorems like that of banach can be used and the higher complexity of the superfunction of 2sinh.
Apart from those 2 single obstacles I see no reason why analog arguments would fail.
Hey, thanks Tom. I know your sentiment, of just how precisely \( \phi \) allows for the limiting method you used, to work. It started a lot with your Tommy Sexp (at least the inception of the idea). I was always frustrated that \( 2\sinh \) didn't just work--or at least as far as proving it goes. Needless to say it took a very long time to find \( \phi \) and a very long time to condense its construction in a bite sized paper. I've been working a lot on difference equations in the complex plane, and this sort of just popped out.
I'm excited to see if you can make some of the techniques from this paper work for \( 2\sinh(s) = h(s) \)
And if you wanna add a Banach Theorem; I think it prolly looks like this,
\(
\log \log \cdots(n\,\text{times})\cdots \log H(s+n)\\
\)
Where \( H \) looks like this,
\(
H(s) = \lim_{n\to\infty}e^{s-1}h(e^{s-2}h(e^{s-3}h(...e^{s-n})))\\
\)
This, I imagine is a whole 'nother uniqueness problem.