(01/13/2021, 04:19 PM)sheldonison Wrote: James,Hey Sheldon! You seem to be piecing together the whole argument!
Looks like I have some catching up to do. I haven't gone back to your paper yet, because I wanted to think about \( \phi(s) \) on my own more first. I wanted to generate a formal power series for f which leads tor \phi. I started to generate the formal series for f, so I am convinced enough that there is a unique definition for \( \phi \) that is indeed well defined and entire and is \( 2\pi i \) periodic.
Back to real work, and then back to your paper when I am able. Again, thanks for posting James, it is a delight to see some well thought out results from your work.
\( f=x+\sum_{n=2}^{\infty}a_n\cdot\.x^n\;\;\;\phi(s)=f(e^s)\;\;\;\phi(s+1) = e^{s+\phi(s)} \)
So for Re(z) negative enough, psi(z) looks more or less like exp(z), and as z at the real axis increases, psi(z) looks like a super exponential. Psi is 2 pi i periodic.
So how does psi(z) behave at Im(z)=pi*i as the real(z) increases? When Im(z)=Pi*i, I think it is always 0>psi(z)>abs(exp(z), but psi will eventually oscillate between negative values approaching the magnitude of exp(z), and nearly zero values ...
Also, I'm curious to is if phi might lead to an analytic solution for tet_e(x), instead of one only defined at the real axis, which is typically what we see for Tommy's iterated logarithm approach.
Yes the worst \( \phi(s) \) behaves is when \( \Im(s) = (2k+1)\pi \) for integer \( k \). It starts to get all loopy here. The hardest part of this paper is arguing that, eventually, \( |\phi(t+\pi i)| \ge 1+\epsilon \) for \( t > T \) very large, and some \( \epsilon > 0 \). So, we say that "eventually" it grows past this point, but it bounces around and about and as you've guessed shrinks and grows. But luckily, this is sufficient for us to employ the Banach Fixed Point Theorem.
As a little demonstration for you, call \( \psi(t) = - \phi(t+\pi i) \); then \( \psi(t+1) = e^{t-\psi(t)} \)
If \( \psi(t) > t \) for \( t>T \) then \( \psi(t+1) <1 \)--contradiction. So if \( \psi \) grows, expect it to grow \( o(t) \). If it grows, it grows slower than \( t \). And if it grows too fast, it gets arbitrarily close to zero; which will kaput the entire construction. The argument to prove that it "grows" and eventually stays past \( 1+\epsilon \) is indeed the most complicated part of the paper. Mostly because it uses infinite compositions; and it's something I'm very familiar with; but there's probably only a handful of people who know what an "infinite composition" is. I've been spending the past 2 years creating criterion for things like,
\( \lim_{n\to\infty} h_0(s,h_1(s,h_2(s,...h_n(s,z)))) \)
To converge to holomorphic functions (and I've developed a good feel for handling asymptotics of these things as well). In particular, if we look at,
\(
\psi_m(t,z) = e^{\displaystyle t-1-e^{\displaystyle t-2-e^{\displaystyle ...e^{t-2m-1-z}}}\\
\)
Where there are an odd number of exponentials; you can note that each of these grow exponentially. Now \( \lim_{m\to\infty}\psi_m \to \psi \); so we can show that for very large m and very large T, that this is greater than 1; and then we can show as we increase m it can't shrink below 1. It's a little tricky, but doable. Essentially we group the functions in pairs. Because \( a(t) = e^{t-k-e^{t-k-1-z}} \) has decay to zero as \( k\to\infty \) and \( t\to\infty \), we know that \( e^{t-a(t)} \) still has exponential growth; similarly does \( e^{t-e^{\displaystyle t-1-e^{t-2-a(t)}}} \) and so on and so forth. An odd number of exponentials composed with \( a(t) \) essentially teeter out more manageably--and exhibit growth... But the whole thing still approaches \( \psi \)
Now this doesn't let us prove \( \psi \) has exponential growth (and it can't by nature), but it can show us that it stays above \( 1+\epsilon \).... eventually.
As to your last point; this paper actually focuses on proving that \( F \) (our tetration function) is holomorphic on \( \mathbb{C} \) (not just the real line), excluding a nowhere dense set in \( \mathbb{C} \) (I.e: where-ever the branch cuts pop-up). Which is to say there are most definitely branch cuts (at least the one at \( (-\infty,-2] \) and perhaps elsewhere in the complex plane), but I couldn't derive where, just that it only accounts for a nowhere dense portion of \( \mathbb{C} \). Proving it's analytic on \( (-2,\infty) \) is actually fairly elementary thanks to the super-exponential nature of \( \phi \) on a neighborhood of \( \mathbb{R} \). Getting it holomorphic on \( \mathbb{C} \) is the real hard part. Mostly because of \( \phi \)'s pesky behaviour when \( \Im(s) = \pi \).
No rush, on reading it; I might do a bit more editing to it (to clarify some of the arguments, clean it up a bit). When you get to it, you get to it.
Regards, James