(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))
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  = 2\pi k)
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,
 \to \mathbb{C}\\<br />
)
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,
| = 0\,\,\text{iff}\,\,y = k\pi\\<br />
)
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,
| = 0\\<br />
)
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
 < 2\pi)
we have no such problem because
grows and acts like a minimum in the strip
 < 2\pi - \delta)
; 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.
