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 Using a family of asymptotic tetration functions... JmsNxn Long Time Fellow Posts: 624 Threads: 102 Joined: Dec 2010 04/01/2021, 05:19 AM (This post was last modified: 04/27/2021, 06:03 AM by JmsNxn.) **Here's the arXiv link with everything** https://arxiv.org/abs/2104.01990 Hey everyone, I've spent the past nine days hammering out a construction of tetration. This is everything I wanted in $\phi$ (to such an extent that many of the proof layouts I had for $\phi$ could be transferred over). The main focus of this construction is a family of functions $\beta_\lambda(s)$ which satisfy $\log\beta_\lambda(s+1) = \beta_\lambda(s) + \mathcal{O}(e^{-\lambda s})$ as $\Re(\lambda s) \to \infty$. I worked my ass off on this for nine days, because I'm 99.99999% sure I'm correct. I'm scared there's some tiny flaw I'm not noticing. But, nonetheless, I'm happy to post this here; even more confidently than the $\phi$ method, while using much of the same principles. I've eliminated all the errors and potential errors from the $\phi$ method. I'm confident I have constructed a holomorphic tetration function $\text{tet}_\beta(s) : \mathbb{C}/(-\infty,-2] \to \mathbb{C}$ which is real-valued. I don't know anything about its behaviour at $\Re(s) = -\infty,\,\Im(s) > 0$ and no idea of its behaviour at $\Im(s) \to \infty$. I do think that as $\Re(s) \to -\infty$ is equivalent to either the julia set or fatou set of $\log$; and this is not a uniform convergence to a fixed point, like with Kneser. I'm pretty certain that as $\Im(s) \to \infty$ we'll get that $\text{tet}_\beta(s)$ should oscillate wildly and behave like orbits of $e^z$--eventually hitting infinity and back to 0. I don't know how to summarize this whole construction, but I believe I've done everything I had planned with $\phi$; I just mis-stepped in assuming I only need a single function rather than a family. And this family of functions needed to approximate the solution of tetration at infinity; and do so in a uniform manner. ... A special thanks goes to tommy for the weird infinite compositions he played with... ... And to Sheldon for running the numbers on $\phi$... ... If you want to read this paper; this requires a lot of infinite compositions, and brushes on Riemann surfaces, and a lot (I mean a lot) of complex function theory. Regards, James Here is the final update on this paper, and its final iteration.   FINAL TETRATION ASYMPTOTIC.pdf (Size: 3.66 MB / Downloads: 182) Here is the original version of the paper, which still had some errors   asymptotic_tetration__2.pdf (Size: 375.8 KB / Downloads: 202) UPDATE!! So I'm working on an update to the above paper. The things I have changed and the things I am planning to change: 1. I bettered the proof that $\beta_\lambda(s) \to \infty$ as $\Re(s) \to \infty$. I spend more time explaining this, as I realize the entire paper hinges on this. And it's exactly why $\beta_\lambda(s)$ is superior to $\phi(s)$. 2. I realized $\tau_\lambda^n(s)$ isn't an exponential series--but it looks enough like one for it not to affect the final result. I went a bit quick here in the first iteration--because I knew it didn't matter if it was either way (but I forgot to double check). The correct asymptotic is $\tau_\lambda^n(s) = - \log(1+e^{-\lambda s}) + o(e^{-\lambda s})$. The second iteration of this paper follows similarly; except when talking about convergence we have to be more careful with our compact sets. The functions $u_\lambda^n(w)$ are not holomorphic for $|w| \le \delta$, they're holomorphic for $\{0 < |w| \le \delta,\,w \neq -e^{-\lambda j}, j\ge 1\}$. But! The singularity at $0$ is removable in a specific manner--$u_\lambda^n(e^{-\lambda j}w)/e^{-\lambda j}w \to -1$ as $j\to\infty$. From which, most of the paper isn't really changed, it's just a bit more aggressive (in that there are more arguments). I've finished all this. 3. I'm trying to visualize some of the constructs. And I plan to include some graphics to help me explain some of these things--especially the domain arguments. I've already written out some functor diagrams I'm including. I'm horrible with graphics programming, though--so, it'll take me a while to develop some nice computer produced $x,y$-plane type graphs. 4. I clarified a lot of the language--at least part way. I haven't fixed everything yet. 5. I'm working on trying to develop a Taylor series at zero for computational purposes, but I lack far too much computational knowledge (especially because pari-gp is too unfamiliar to me, and the only language I ever really worked with was C and python). Everything I do just hits overflow before I'm even at $\beta_\lambda$ for large arguments. Which says more about my coding than my method... Again, the main result hasn't changed at all. Just how we get there is a tad more difficult; but I'm explaining it much better. If you felt confused from the first iteration, I apologize. I'll clear everything up. I got ahead of myself--again, I wrote that in 9 days in a light-bulb moment. The proof schema is still the exact same; just some of the brickwork needs to be patched, is all. Regards, James. UPDATE! #2 Please see post #3 for all the information I changed in this update. I, about, doubled the length of this paper. But it's much more solid now.   asymptotic_tetration__4.pdf (Size: 3.07 MB / Downloads: 179) UPDATE #3 Please see post #4 for all the information I changed in this update. This is the final update, and as far as I see, constructs a holomorphic tetration. I added about 19 graphs, and I think that's the most I can do. I think I got out everything that needed to be shown. It's still a little rough around the edges, but I believe this constructs tetration. And further, constructs it pretty well.   FINAL TETRATION ASYMPTOTIC.pdf (Size: 3.66 MB / Downloads: 182) Attached Files   asymptotic_tetration__4(1).pdf (Size: 3.59 MB / Downloads: 153) JmsNxn Long Time Fellow Posts: 624 Threads: 102 Joined: Dec 2010 04/05/2021, 08:43 AM Hey guys, I've written some supplemental literature to the original paper. This short analysis is meant to explain the general case better. It's fairly short; and only serves to elaborate on everything above. What I'm attaching here; is nothing more than an explanation of variable changes in an infinite composition. And the manner one can understand this around $f(z) = e^z$. This brief notice is about solving Schroder equations at infinity; and in a well enough manner. Which explains a lot of this paper; but done in a more blank pattern. Regards, James   Mock_Schroder.pdf (Size: 272.65 KB / Downloads: 186) JmsNxn Long Time Fellow Posts: 624 Threads: 102 Joined: Dec 2010 04/11/2021, 01:01 AM Hey, guys! So I've fixed the error in the first iteration, which was pretty silly of me actually. I had written that the process, $ \tau_\lambda^{n+1}(s) = \log(\beta_\lambda(s+1) + \tau_\lambda^n(s+1)) - \beta_\lambda(s)\\$ produces an exponential series. This is not true at all. It just looks like an exponential series (enough for the main theorem to be unaffected); but there are a bunch of singularities. It was really silly of me. The correct statement is that, $ \tau_\lambda^n(s) = -\log(1+ e^{-\lambda s}) + o(e^{-\lambda s})\,\,\text{as}\,\,\Re(s)\to \infty\\$ This doesn't affect the main theorem at all though. I just have to be more careful when applying Banach's Fixed Point theorem. For the most part, it's much of the same proof. I've added a total of 12 figures; which took a lot out of me. I am horrible at working out graphs. And since this construction is to do with super-exponential behaviour at infinity; I don't know how to code around the overflow errors. I'm still working on making graphs of the actual tetration (which looks like I might be able to do soonish). I just have to put on my coding hat and find an efficient manner at computing this. I've also added a bunch of commutative diagrams, in an attempt to better explain some of the morphisms I use. I've expanded a bunch of the arguments and clarified as much of the language as I think I can. Particularly, I made sure the proof that, $ \beta_\lambda(s) \to \infty\,\,\text{as}\,\,\Re(s) \to \infty\\$ Was as solid as I could make it. I had to reference three people to make this argument--all from Milnor's book. This theorem is the crux of the method though. Where, the number one reason the construction with $\phi$ failed was because it oscillated between $0$ and $\infty$ very rapidly. I must say, it's very satisfying to see the graphs concur with the divergence of $\beta_\lambda$ on paper. I tried to explain the variable changes more clearly in this iteration. I did it a tad off-hand initially. But, I imagine much of you are new to this infinite composition stuff; so the idea of changing variables in an infinite composition may seem odd. I tried to make it simpler to understand by contrasting it with commutative diagrams. I don't think the paper is quite done yet. I am looking to revamp it one more time once I somehow manage to get some workable code to evaluate these tetrations (of which we want to limit to find the right tetration). At least, as long as I can evaluate it in a rudimentary way. The more I work with this though, the more I'm convinced this actually constructs tetration. Plus the graphs are behaving exactly as I expected them too (even if they're not the tetration graphs yet). Regards, James.   asymptotic_tetration__4.pdf (Size: 3.07 MB / Downloads: 164) JmsNxn Long Time Fellow Posts: 624 Threads: 102 Joined: Dec 2010 04/23/2021, 04:26 AM (This post was last modified: 04/23/2021, 04:42 AM by JmsNxn.) HEY EVERYONE! I've got some pictures to share! I've further written my paper (it's still not perfect yet, the next time I share will be the last time); but in the process I ran more numbers. I *ahem* acquired matlab; and familiarized myself with the code. I had some help from a fellow on Stack overflow; where I wasn't calling my recursion properly. And I got a couple of tetration graphs to display. If we call, $ F_\lambda(s) = \lim_{n\to\infty}\log \beta_\lambda(s+n)\\$ Which, $ F_\lambda(s+1) = e^{F_\lambda(s)}\\$ Then, when $\lambda =\log(2)$ and $-1\le\Re(s)\le 1,\,-1\le\Im(s)\le1$; the function $|F_\lambda(s)|$ looks like, after about 7 iterations,     Where these quick spikes and jumps move further and further together as you do more iterations. And the center of these spikes is Tetration at $-2$. We still haven't shifted our argument yet, so that $F_\lambda(s+x_0)$ is tetration. The sharp drops to infinity is the clustering of singularities about $\text{tet}(-2)$ in the iteration. But, this graph looks more and more level, as you iterate further and further, excepting where the singularities are. Here is a second graph when $\lambda = 1/2 + i$, so it models $|F_\lambda(s)|$ again, but with a complex multiplier (about the logarithmic singularity at $-2$).     Now, the errors occur in one half plane and not the other. This can be related (naively) to $\beta_\lambda(s) \to \infty$ when $e^{-\lambda s} \to 0$. And we can rotate the plane as we move $\lambda$. Using a different modeling technique, we get convergence in a different way,     This converges to the same function eventually; but is more satisfactory for local values about small numbers. We just take the limit slightly different in this case. I'm in the process of finalizing this paper; but these graphs confirm, in my mind, that these tetrations are analytic. And one can paste them together. I'll post this paper in maybe a week or so. I'm trying to make sure every nail is hammered at this point. I think this really is analytic tetration! JmsNxn Long Time Fellow Posts: 624 Threads: 102 Joined: Dec 2010 04/24/2021, 08:26 AM Here's a better picture of what this tetration looks like; where we've dodged all the short circuits in the code:     JmsNxn Long Time Fellow Posts: 624 Threads: 102 Joined: Dec 2010 04/24/2021, 08:54 PM Hey, everyone. So I finished this paper. I added more details on the error term, and tightened the bounds I used. I tried to explain every little detail, and I tried to dot all my i's and cross all my t's. At this point, there's not much else I can do with this paper. It's as good as I can possibly make it at this point. I wouldn't be able to explain it any better if I tried. I'm a little exhausted, so I won't be touching this paper anymore; unless it's for an actual publisher. I feel I managed to explain everything that I could. Though the formatting of the paper is a little bleh (I'm not the best at making a well dressed paper). Anyway, here's the final iteration of this paper. And I'm certain this constructs tetration; but I may have lapsed a bit in some of my arguments; but then, it's the best I could do.   FINAL TETRATION ASYMPTOTIC.pdf (Size: 3.66 MB / Downloads: 192) MphLee Fellow Posts: 219 Threads: 22 Joined: May 2013 04/24/2021, 11:09 PM Well done, it seems a long, hard but satisfying journey. I can't offer you technical comments yet. I don't think I'll be qualified to help reviewing the technical arguments because I'm very weak in analysis. What I can say is that I enjoy the pictures a lot and I can clearly see the effort you put into this and that I'm glad that some of my drawings did inspire you. If my comment seems poor, I'm sorry: here I can add an extra bonus comment on the commutative diagrams. I don't want to be THAT annoying guy but.... Those are not exactly the kind of diagrams that the "algebraic guy" would expect or is familiar with. But as a good side, I already saw them in Robbin's FAQ looong ago and they look interesting. Oh btw. Before learning how to properly use diagrams, it took me 5 years, I used diagrams exactly in that way. MathStackExchange account:MphLee Fundamental Law $(\sigma+1)0=\sigma (\sigma+1)$ JmsNxn Long Time Fellow Posts: 624 Threads: 102 Joined: Dec 2010 04/26/2021, 01:28 AM (This post was last modified: 04/26/2021, 01:55 AM by JmsNxn.) Hey, MphLEE! Yes, these diagrams would be very basic to a trained mathematician who specializes in algebra, lol. I was inspired by your diagrams... to use diagrams. Not to actually use complex graphs; like the stuff you do. Not even to broach the commutative diagram nethers. I'm not a diagrammatical person (by nature, in my mathematics, so it'd probably be worse if I did). But In John Milnor's book Dynamics In One Complex Variable; he uses graphs of about the same complexity as mine (though I will say, mine were a little weird.) The entire purpose of the graphs is to view the morphism from a domain in $\mathbb{L}$ to $\mathbb{D}^\times$; while respecting the functional equation. I was  thinking of my diagrams as morphisms between $\mathbb{L} \to \mathbb{D}^\times$ while respecting $s\mapsto s+1$ gets mapped to $w\mapsto e^{-\lambda}w$. And then considering the $\log$ map; as on either/or space. Where we have a nice bounded argument on $\tau_\lambda$ in $\mathbb{L}$; and we can now visualize this bounded argument on $u_\lambda$ in $\mathbb{D}^\times$ with a different functional equation. I may not have expressed it perfectly. I only wrote the diagrams as a visual aid, not as a method of proof It was entirely supplemental. I apologize if they're not to your standard though, lol. As far as I used them, was as a visual aid. Which, was essentially how Milnor was to me. Thanks though, for giving the paper enough of a time of day to reply to. I hope the paper seems, at least intuitive, if not rigorous to you. By which, if you can't corroborate the technical aspects; at least, I hope the motions make sense, lol. Thanks again, James sheldonison Long Time Fellow Posts: 684 Threads: 24 Joined: Oct 2008 04/26/2021, 02:11 AM (This post was last modified: 04/26/2021, 02:16 AM by sheldonison.) Thanks James, Your latest paper looks really impressive; I will comment more later; I've been too busy to spend any time on tetration over the last month and a half. - Sheldon - Sheldon MphLee Fellow Posts: 219 Threads: 22 Joined: May 2013 04/26/2021, 10:24 AM (04/26/2021, 01:28 AM)JmsNxn Wrote: Hey, MphLEE! ... not as a method of proof It was entirely supplemental. I apologize if they're not to your standard though, lol. As far as I used them, was as a visual aid. Which, was essentially how Milnor was to me. Thanks though, for giving the paper enough of a time of day to reply to. I hope the paper seems, at least intuitive, if not rigorous to you. By which, if you can't corroborate the technical aspects; at least, I hope the motions make sense, lol. Thanks again, James Don't worry, I got the goals of your diagrams and I wasn't asking for standards to be respected. When I'll have time to go thru it, I'll automatically convert to category theory everything that can be "diagrammed" anyways hahaha (that's how my mind works). Also I got that atm this is as far into diagram as you can go, just like the the formal definition of limit is as far as I can go in epsilon-delta proofs xD. The paper is not totally intuitive to me only because I need more exercise in the parts about convergence, but its my fault of course. I still have to find some time to put into it in order to get used to how the pieces fit together. Overall, the motion of the argument still makes sense. ps: lately got really slow on all of this since the more I worked in polishing the Sup.Func.Sp.ces' paper and the more I fell into rabbit holes. 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