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 I thought I'd compile some of the things I know... JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 01/26/2021, 04:39 AM (01/25/2021, 11:54 PM)sheldonison Wrote: I look forward to seeing your updated paper. Thanks, Sheldon. I'm working on making it as air-tight as I possibly can. And on top of that, being as explanatory as possible. I also have been working on better explaining why $\phi$ is holomorphic by using similar analogies I've used in other papers. Where I didn't bother to initially, because I'd already done it a bunch of times in separate papers. It can be a drag to re-explain everything from previous papers in the next paper... I just worked with an $\epsilon/\delta$ construction of $\phi$ and didn't bother with the years of intuition which led me to construct this. I'm glad you're at least convinced $\phi$ is holomorphic... Hope to zoom call with you again soon. Give me a couple more days to hammer out this paper a bit more and hopefully get closer to convincing you of holomorphy. Regards, James sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 01/29/2021, 02:58 PM (This post was last modified: 01/29/2021, 02:58 PM by sheldonison.) (01/26/2021, 04:39 AM)JmsNxn Wrote: (01/25/2021, 11:54 PM)sheldonison Wrote: I look forward to seeing your updated paper. Thanks, Sheldon. I'm working on making it as air-tight as I possibly can. And on top of that, being as explanatory as possible. I also have been working on better explaining why $\phi$ is holomorphic by using similar analogies I've used in other papers. Where I didn't bother to initially, because I'd already done it a bunch of times in separate papers. It can be a drag to re-explain everything from previous papers in the next paper... I just worked with an $\epsilon/\delta$ construction of $\phi$ and didn't bother with the years of intuition which led me to construct this. I'm glad you're at least convinced $\phi$ is holomorphic... Hope to zoom call with you again soon. Give me a couple more days to hammer out this paper a bit more and hopefully get closer to convincing you of holomorphy. Regards, JamesHi James, If anyone else wants to join the zoom call, send me an email at shel@sheltx.com with Tetration zoom call in the header; math.eretandre handle, and time zone. I'll set it up once James is ready, during a weekend. - Sheldon JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 01/30/2021, 12:46 AM (This post was last modified: 01/30/2021, 07:09 AM by JmsNxn.) So, I think I have it complete. I've added about 15 pages to this paper trying to make it air-tight, and drawing out every string. The proofs have increased in length, and all the things I took for granted have been better explained. I'm going to sit on it for maybe 5 more days, and go into proof-reading mode; making sure I haven't made any glaring mistakes. But I thought I'd share the shift in my thinking which allows for a more elegant approach. Now, as we've done it, we've written our solution: $ \lim_{n\to\infty}\log \log \cdots(n\,\text{times})\cdots \log \phi(s+n+\omega) = e \uparrow \uparrow s = \phi(s+\omega) + \tau(s + \omega)\\$ Now, as $\Re(s) \to -\infty$ we can write $\phi(s) \to 0$ and $\tau(s) \to L$--both geometrically--and $e^L = L$. This can be done, given that $\tau(s)$ is holomorphic on $0 < \Im(s) < 2\pi$ upto a nowhere dense set--which is the real challenge of the paper. Then, we can write, $ e \uparrow \uparrow s = \lim_{n\to\infty}\exp \exp \cdots(n\,\text{times})\cdots \exp \phi(s+\omega-n) + \tau(s+\omega-n)\\$ Again, almost everywhere. But this converges geometrically to, $ \lim_{n\to\infty}\exp \exp \cdots(n\,\text{times})\cdots \exp \phi(s+\omega-n) + L\\$ And it's fairly routine to show that this function converges uniformly on compact sets for $\Re(s) < -T$ and $0 < \Im(s) < 2\pi$ as $n\to\infty$. So in an essence, to get this construction to work, we want to look at the recursion from the opposite end to fill in the blanks that are missing. Now this form is definitely not ideal for the real line, but it works well on strips way off in the left hand plane. It also clearly shows why our construction CANNOT be holomorphic in a strip larger than $2\pi$--it would be periodic if it were and tetration can't be periodic. So branch cuts arise and cluster in the iteration at the lines $\mathbb{R} + 2\pi i k$ because otherwise we'd get a periodic tetration (which is absolute nonsense). And with it different $L$ appear, as otherwise we'd achieve periodicity. Some quick notes on $L$; not too sure which $L$ we converge to, but each strip $2 \pi k < \Im(s) < 2\pi(k+1)$ should garner a different $L$. The key fact being, there is some $L$ and $\overline{L}$ in which this iteration will be real valued on the real line; which would correspond to the strips $-2\pi < \Im(s) < 0$ and $0 < \Im(s) < 2\pi$. I'll set a deadline for myself on when I'll post this updated version. I'll release it on wednesday whether I feel it's perfect or not. If I keep hacking at this sooner or later it'll end up being a 50 page paper saying every possible thing I can say; including alternative representations... that may be a bit much for a quick update... EDIT: I think a zoom call for next saturday will be best. I'll send the request later--would 1pm be okay? Not to do it too late, but our 9am zoom call isn't optimal. Even though I live in toronto, I am on australia time, Sheldon. Lol. sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 01/31/2021, 12:10 AM (01/30/2021, 12:46 AM)JmsNxn Wrote: ... I think a zoom call for next saturday will be best. I'll send the request later--would 1pm be okay? Not to do it too late, but our 9am zoom call isn't optimal. Even though I live in toronto, I am on australia time, Sheldon. Lol. James, Next Saturday Feb 6th, 1pm eastern, 1800 GMT works great for me!  I'll also post the zoom link separately to invite all Tetration forum members to join!  My plan it to make this a monthly meeting with your paper being the first meeting!   https://us02web.zoom.us/j/89038247428?pwd=UFo1dmVGT21YTHpSbTNqUjMyazUzQT09 Meeting ID: 890 3824 7428 Passcode: 322183 I have many ideas for subsequent meetings; for example I'd like to cover Peter Walker's 1991 paper with the $C\infty$ proof, as well as the conjecture that this is a nowhere analytic function. Discuss Ecalle's asymptotic series for the Abel function for $z \mapsto \exp(z)-1$.  Peter Walker's paper also includes the simultaneous equation solution, which was later reinvented as Andrew's slog.  I think this could be a lot of fun! - Sheldon JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 02/01/2021, 09:31 PM I'd love to talk about those ideas! I always liked Andy's slog, but I'm horrible with matrices so it never made much sense (other than the broad motions). Has it ever actually been shown to be analytic? I really enjoyed Peter Walker's paper too. tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 02/01/2021, 11:20 PM (02/01/2021, 09:31 PM)JmsNxn Wrote: I'd love to talk about those ideas! I always liked Andy's slog, but I'm horrible with matrices so it never made much sense (other than the broad motions). Has it ever actually been shown to be analytic? I really enjoyed Peter Walker's paper too. As far as we know (here) it has not been shown to be analytic. On the other hand if it converges to a C^oo function then it is analytic because it is a taylor series. I mentioned at least 1 method that forces to solve infinite systems of linear equations however it has not been proven to give a nonzero-radius hence not proven c^oo or even convergeant let alone analytic. I even discovered a set of equations that turned out to be truncated equations equivalent to Andy/Peter , hence a rediscovery in some way. Unfortunately my other viewpoint was not helpful. Analytic tetration is a difficult subject ! Btw I posted a correction to the NBLR method ( the recent closed form based on phi and lambertW). You should probably want to see it. regards tommy1729 « Next Oldest | Next Newest »

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