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Nixon-Banach-Lambert-Raes tetration is analytic , simple and “ closed form “ !!
#11
(02/02/2021, 04:40 AM)JmsNxn Wrote: Hey, Tommy!

So I had noticed that you implicitly assumed in your original analysis. I didn't notice it right away, but I noticed it afterwards when I saw it would imply 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 for . This is where our function will recycle, and a cluster of singularities will force non-analycity of

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 would imply non-analytic tetration right ?
But you have 2 papers claiming analyticity and intend to explain it further.

Regards

tommy1729
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#12
(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 in your original analysis. I didn't notice it right away, but I noticed it afterwards when I saw it would imply 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 for . This is where our function will recycle, and a cluster of singularities will force non-analycity of

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 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,



Is a holomorphic function upto a nowhere dense set . Now this, I believe is technically correct, but I had implicitly assumed that it is analytic on . Sheldon, thoroughly convinced me that this probably doesn't happen. What I believe now, which is essentially the above statement, except,



Which explicitly states where it is holomorphic. This is to say, it is still holomorphic upto a nowhere dense set; but seems to be in this set. The mistake I made was pretty foolhardy,

I had assumed that,



so that 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 for large enough --and from here the paper continues as it did before with the construction of .  The problem being,



Has solutions which cluster towards as . This causes our function to dip towards small values, causing to hit a singularity. Essentially we hit a wall of singularities at the real line. But in the strip we have no such problem because grows and acts like a minimum in the strip ; forcing our construction of to converge.

All in all; I was incorrect to think I showed analycity on --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 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.
Reply


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