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 Using a family of asymptotic tetration functions... JmsNxn Long Time Fellow Posts: 379 Threads: 78 Joined: Dec 2010 04/01/2021, 05:19 AM (This post was last modified: 04/11/2021, 06:24 AM by JmsNxn.) 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   asymptotic_tetration__2.pdf (Size: 375.8 KB / Downloads: 33) 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: 11) JmsNxn Long Time Fellow Posts: 379 Threads: 78 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: 23) JmsNxn Long Time Fellow Posts: 379 Threads: 78 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: 12) « Next Oldest | Next Newest »

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