(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.