(05/03/2021, 01:58 PM)Ember Edison Wrote:(04/01/2021, 05:19 AM)JmsNxn Wrote: **Here's the arXiv link with everything**
https://arxiv.org/abs/2104.01990
I hardly understand how to read your symbols. Did you complete a brand new holomorphic Tetration in base e? Where is your superlog?
Hey, I assume the problem you are having is with the Omega-notation. The omega-notation isn't pivotal, but it saves a hell of a lot of space.
Assume we have a sequene of holomorphic functions \( \phi_j(s,z) : \mathcal{S} \times \mathcal{G} \to \mathcal{G} \) for domains \( \mathcal{S},\mathcal{G} \subseteq \mathbb{C} \). The Omega-notation has been developed repeatedly by me in about 6 papers now; and it can get a little repetitive re-introducing it in each paper, so I just ran with it in this paper.
If I write,
\(
\Omega_{j=1}^n \phi_j(s,z) \bullet z
\)
Then this is interpreted as,
\(
\phi_1(s,\phi_2(s,...\phi_n(s,z)))\\
\)
The bullet essentially binds the variable that we compose across. It's similar to how \( ds \) behaves in \( \int...ds \); it binds the operation to a specific variable. And then the \( \Omega_{j=1}^n \) just means to compose these functions across the index \( j \) from \( 1 \) to \( n \). Attached in this paper is a proof of a specific type of "Infinite composition". An infinite composition is just when we let \( n\to\infty \). These things can converge in many different ways; I only use a specific case in this paper.
\(
\Phi(s) = \lim_{n\to\infty} \Omega_{j=1}^n \phi_j(s,z)\bullet z = \lim_{n\to\infty} \phi_1(s,\phi_2(s,...\phi_n(s,z)))\\
\)
This will converge to a holomorphic function in \( s \); if the following sum converges. Let \( \mathcal{N} \subset \mathcal{S} \) be an arbitrary compact set; and let \( \mathcal{K} \subset \mathcal{G} \) be an arbitrary compact set. If there exists an \( A \in \mathcal{G} \) such for all \( \mathcal{N},\mathcal{K} \) that,
\(
\sum_{j=1}^\infty \sup_{s\in\mathcal{N},z\in\mathcal{K}} |\phi_j(s,z) - A| < \infty\\
\)
Then the sequence,
\(
\lim_{n\to\infty} \phi_1(s,\phi_2(s,...\phi_n(s,z)))
\)
Converges uniformly on \( \mathcal{N} \) and \( \mathcal{K} \); to a holomorphic function in \( s \) and a constant function in \( z \). The proof of this can be found in the appendix of this paper; but I've proven it a couple of times in other papers; specifically, it was used when I constructed the \( \phi \) method before (which only made a \( C^\infty \) tetration).
From here, I move pretty fast in the paper; again, I've done this so many times it can be exhausting to rewrite introductions in every paper.
If I define the set \( \mathbb{L} = \{(s,\lambda) \in \mathbb{C}^2\,|\, \Re \lambda > 0,\,\lambda(j-s) \neq (2k+1)\pi i,\,j,k \in \mathbb{Z},\,j\ge 1\} \); then the following sequence of functions is holomorphic on \( \mathbb{L} \times \mathbb{C} \).
\(
q_j(s,\lambda,z) = \frac{e^z}{e^{\lambda(j-s)} + 1}\\
\)
Additionally, for compact sets of \( \mathbb{L} \) and \( \mathbb{C} \); we know the sum converges,
\(
\sum_{j=1}^\infty ||q_j(s,\lambda,z)|| < \infty\\
\)
From, this we can write that,
\(
\beta_\lambda(s) = \Omega_{j=1}^\infty q_j(s,\lambda,z)\bullet z\\
\)
Is a holomorphic function for \( (s,\lambda) \in \mathbb{L} \). We can write this more explicitly as;
\(
\beta_\lambda(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{\lambda(j-s)} + 1}\bullet z\\
\)
Now, if I shift \( \beta_\lambda(s) \) forward by \( s \mapsto s+1 \) then we get a re-indexing in our infinite composition, where we start from \( j=0 \) rather than \( j=1 \).
\(
\beta_\lambda(s+1) = \Omega_{j=0}^\infty \frac{e^z}{e^{\lambda(j-s)} + 1}\bullet z\\
\)
But, this just equals,
\(
\Omega_{j=0}^\infty \frac{e^z}{e^{\lambda(j-s)} + 1}\bullet z = q_0(s,\lambda,\Omega_{j=1}^\infty \frac{e^z}{e^{\lambda(j-s)} + 1}\bullet z)\\
\)
To make a long story short; this means that,
\(
\beta_\lambda(s+1) = \frac{e^{\beta_\lambda(s)}}{e^{-\lambda s} + 1}\\
\)
I assume this is where you were having trouble; as I did just blaze through this part. The rest of the paper then focuses on solving the Abel equation at \( \Re(s) = \infty \) using this function. Where,
\(
\log \beta_\lambda (s+1) = \beta_\lambda(s) - \log(1+e^{-\lambda s})\\
\)
And we want to add in sequence of convergents \( \tau_\lambda^n \), which solve,
\(
\log (\beta_\lambda(s+1) + \tau_\lambda^n(s+1)) = \beta_\lambda(s) + \tau_\lambda^{n+1}(s)\\
\)
And we show that the limit \( \tau_\lambda = \lim_{n\to\infty} \tau_\lambda^n \) is holomorphic. This solves the Abel equation for any \( \lambda \) and large enough \( \Re(s) \); in which,
\(
F_\lambda(s) = \beta_\lambda(s) + \tau_\lambda(s)\\
\log F_\lambda(s+1) = F_\lambda(s)\\
\)
Then, the real trouble is that these tetrations are periodic and have a whole bunch of singularities; so you don't want them at all. Instead you want to take a limit \( \lim_{n\to\infty} \lim_{\lambda \to 0} \tau_\lambda^n \) where \( \lambda = \mathcal{O}(n^{-\epsilon}) \) and this will give you a real valued tetration function; which is holomorphic and whose only singularities are at the negative integers less than -1.
At this point, I'm confident this constructs holomorphic tetration for the base point \( e \); and that its real-valued. I'm working on making some Pari-GP code; but the trouble is, as I'm pulling back from infinity, we encounter overflow errors pretty quickly in the construction. But I can already achieve a precision of about 10 digits; but only when the argument is about \( \Re(s) < 3 \). I am not the best programmer; so I don't think I'll be able to improve it unless I can think of a way of discovering Taylor series. I do not know how to generalize this result to other basis though; as the dynamics are particular to \( e \); but it should work... I think, not certain.
As to the superlog; it only exists through the implicit function theorem; I have no effective way of constructing it other than just functionally inverting the tetration function.
If you have any more questions, just ask. I'm happy to answer. A lot of this is new math centering around the Omega-notation and it can be a little confusing, I understand. I have written a bunch of papers using it though; and it's just annoying to have to make a new section in each paper explaining what the notation means in depth. It's gotten to the point I assume no one really reads these things, that I just reference previous work at this point and give a quick run through.
Regards, James
PS:
If you are having trouble cognizing what \( \beta_\lambda \) looks like, you can always think of it how Tommy and Sheldon think of it.
\(
{\displaystyle\beta_\lambda(s) = \frac{e^{\displaystyle\frac{e^{\displaystyle\frac{e^{\displaystyle...}}{e^{\lambda(3-s)} + 1}}}{e^{\lambda(2-s)} + 1}}}{e^{\lambda(1-s)} + 1}}
\)
Which is just pulling the iteration all the way back to infinity; where,
\(
\beta_\lambda(-\infty) = 0\\
\)
And
\(
\beta_\lambda(s+1) = \frac{e^{\beta_\lambda(s)}}{e^{-\lambda s} + 1}\\
\)
And we are solving this in a neighborhood of negative infinity; where the value is zero; and then just iterating forward to get the whole complex plane. I don't really like this way of thinking of it; largely because if you try and prove this converges from that way, it doesn't generalize to more complicated constructions. I prefer viewing it as, if a sum converges compactly normally, then the infinite composition converges compactly normally. This generalizes well for much more exotic constructions than \( \beta_\lambda \).
