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 Nixon-Banach-Lambert-Raes tetration is analytic , simple and “ closed form “ !! tommy1729 Ultimate Fellow Posts: 1,700 Threads: 374 Joined: Feb 2009 02/03/2021, 11:44 PM (02/02/2021, 04:40 AM)JmsNxn Wrote: Hey, Tommy! So I had noticed that you implicitly assumed $V(s+1) = V(s)$ in your original analysis. I didn't notice it right away, but I noticed it afterwards when I saw it would imply $2 \pi i$ periodicity. I looked at it some more, and was pretty sure you were on to something--but never been too good with the Lambert function. This makes much much more sense quite frankly. Especially if we think of the branch cuts appearing at $\Im(s) = 2\pi k$ for $k \in \mathbb{Z}$. This is where our $\phi$ function will recycle, and a cluster of singularities will force non-analycity of $\tau$.  Now I am confused. You say non-analytic here. And you also wrote 2 papers claiming analytic ? Im aware of Sheldon's arguments and the complexity of tetration. But the point is I am confused about your viewpoint. I mean non-analycity of $\tau$ would imply non-analytic tetration right ? But you have 2 papers claiming analyticity and intend to explain it further. Regards tommy1729 JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 02/04/2021, 03:47 AM (This post was last modified: 02/04/2021, 03:50 AM by JmsNxn.) (02/03/2021, 11:44 PM)tommy1729 Wrote: (02/02/2021, 04:40 AM)JmsNxn Wrote: Hey, Tommy! So I had noticed that you implicitly assumed $V(s+1) = V(s)$ in your original analysis. I didn't notice it right away, but I noticed it afterwards when I saw it would imply $2 \pi i$ periodicity. I looked at it some more, and was pretty sure you were on to something--but never been too good with the Lambert function. This makes much much more sense quite frankly. Especially if we think of the branch cuts appearing at $\Im(s) = 2\pi k$ for $k \in \mathbb{Z}$. This is where our $\phi$ function will recycle, and a cluster of singularities will force non-analycity of $\tau$.  Now I am confused. You say non-analytic here. And you also wrote 2 papers claiming analytic ? Im aware of Sheldon's arguments and the complexity of tetration. But the point is I am confused about your viewpoint. I mean non-analycity of $\tau$ would imply non-analytic tetration right ? But you have 2 papers claiming analyticity and intend to explain it further. Regards tommy1729 Hey, Tommy. I'll clarify my stance. I initially thought I had showed that, $ e \uparrow \uparrow s : \mathbb{C} / \mathcal{L} \to \mathbb{C}\\$ Is a holomorphic function upto a nowhere dense set $\mathcal{L}$. Now this, I believe is technically correct, but I had implicitly assumed that it is analytic on $\mathbb{R}^+$. Sheldon, thoroughly convinced me that this probably doesn't happen. What I believe now, which is essentially the above statement, except, $ e \uparrow \uparrow s: \mathbb{C} / (\mathbb{R}+2\pi i k) \to \mathbb{C}\\$ Which explicitly states where it is holomorphic. This is to say, it is still holomorphic upto a nowhere dense set; but $\mathbb{R}^+$ seems to be in this set. The mistake I made was pretty foolhardy, I had assumed that, $ \frac{d}{dy}|\phi(t+iy)| = 0\,\,\text{iff}\,\,y = k\pi\\$ so that $|\phi(t+\pi i)|$ is a global minimum. But this isn't so. What I believe I've shown now, is that it is only a local minimum, but the domain in which it is a minimum eventually grows to $\delta < y < 2\pi - \delta$ for large enough $t\ge T$--and from here the paper continues as it did before with the construction of $\tau$.  The problem being, $ \frac{d}{dy}|_{y=y_k} |\phi(t+iy)| = 0\\$ Has solutions $y_k$ which cluster towards $\mathbb{R}^+$ as $t\to \infty$. This causes our function $|\phi(s)|$ to dip towards small values, causing $\log \log ... \log \phi(s+n)$ to hit a singularity. Essentially we hit a wall of singularities at the real line. But in the strip $0 < \Im(s) < 2\pi$ we have no such problem because $|\phi(t+\pi i)|$ grows and acts like a minimum in the strip $\delta < \Im(s) < 2\pi - \delta$; forcing our construction of $\tau$ to converge. All in all; I was incorrect to think I showed analycity on $\mathbb{R}^+$--but I do believe this is still holomorphic; just unfortunately not for real values. At best I can show is continuously differentiable, but I don't think a $C^{\infty}$ proof is that out of reach. All in all I was half-right at best. Also, the second paper, is just the same paper accounting for this foolhardy mistake--and trying to correct it--and state a stronger version of what I had originally stated. « Next Oldest | Next Newest »

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