Nixon-Banach-Lambert-Raes tetration is analytic , simple and “ closed form “ !!

[tommy1729]

Hey, Tommy!

Wow! Never would've guessed the correction term for \( \phi \) to make it into tetration (the function V) would be expressible using Lambert! That definitely makes talking about branch cuts much easier. Gives us something concrete to fiddle with!

My uniqueness condition (at least the one I like) is the exponential decay uniqueness.

If \( f \) is continuous on \( \mathbb{R} \) and

\(
f(x+1) = e^{x+f(x)}\\
\lim_{x\to-\infty} f(x) = 0\\
\)



Then, \( f(x) = \phi(x+q) \). Also, we can note instantly, if it's asymptotic to \( e^{x-1} \), it means that \( q = 0 \). This really isn't too hard to prove.

I haven't tried doing anything with \( \phi^{-1} \) but I have been fiddling with the slogarithm. The conjecture I was thinking is that, \( \text{slog} \) is holomorphic on \( \mathbb{C} \) minus a nowhere dense set (i.e: a whole bunch of branch cuts) and these branch cuts are located about all the fixed points of \( e^L = L \). This is reasoned because I have a rough argument that \( e \uparrow \uparrow s \neq L \) for all these fixed points, but as \( \lim_{|s|\to\infty} e \uparrow \uparrow s = L \) so long as \( \pi \ge \arg(s)\ge \pi/2 \). I.e: that if we limit to infinity in the left half plane we approach a fixed point of exp. I haven't gone back to trying to figure this out lately, as I inevitably bump into the question--where the hell are the branch cuts of \( e\uparrow\uparrow s \)? Which amounts to, where the hell are the zeroes of \( e\uparrow\uparrow s \)? In my wildest dreams there is only one zero at \( -1 \), but I can't prove that .

You seem to also have stumbled upon the thesis of my work lately.

If,

\(
\sum_{j=0}^\infty ||h_j(s,z) - z|| < \infty
\)

Then,

\(
\lim_{n\to\infty} h_0(h_1(...h_n(s,z))) = H
\)

Is holomorphic in both variables.

For the function \( \phi \) it's a special case but the hearty theorem is, if:

\(
\sum_{j=0}^\infty ||h_j(s,z) - A|| < \infty
\)

For a constant A, then,


\(
\lim_{n\to\infty} h_0(h_1(...h_n(s,z))) = H
\)

is holomorphic in \( s \) but constant in \( z \).

These \( ||...|| \) are all supremum norms of compact subsets of wherever the hell these things are holomorphic.

I've been detailing a lot of what I like to call compositional analysis. No more sums and products; everything is composition. I did a whole bunch on the integral too and switching that up. Again, I only really came to this tetration in passing; it kind of just popped out after doing all this stuff. You might find my first paper on the subject interesting, I solve the equation \( y(s+1) - y(s) = e^{sy(s)} \) in the complex plane. It's a lot harder to construct than \( \phi \) though...

Thanks for contributing Tommy! I'm excited to see what you uncover!

Regards, James

Thank you for your kind and interesting reply James.

I hope that means you agree with the name Nixon-Banach-Lambert-Raes tetration (NBLR) then ? 

I considered the questions you posted too but did not have time to solve them yet.
They are interesting though.

I have not read any of your papers completely yet. Not even the one I discussed here. 
I knew it worked from the comments given in the other thread. I read about 2 pages to get about the same notation for clarity.

Ofcourse reading your papers is on my to do list !!

I could be wrong but does your uniqueness condition not fail ? 
I mean when I plug in the 1 periodic function , the limit is still zero I think ...
Maybe I missed a detail.

I also wonder if any other solution exists with all derivatives positive ( for 0 < s ) ... periodic functions tend to plug in some negatives.

I have not stumbled on your thesis.

Nor have I read the paper about  \( y(s+1) - y(s) = e^{sy(s)} \) .

I do not even know where to find them ? Tell me

It seems your skills have improved dramatically.


You seem to also have stumbled upon the thesis of my work lately. 

If,

\(
\sum_{j=0}^\infty ||h_j(s,z) - z|| < \infty
\)

Then,

\(
\lim_{n\to\infty} h_0(h_1(...h_n(s,z))) = H
\)

Is holomorphic in both variables. 

For the function \( \phi \) it's a special case but the hearty theorem is, if:

\(
\sum_{j=0}^\infty ||h_j(s,z) - A|| < \infty
\)

For a constant A, then,


\(
\lim_{n\to\infty} h_0(h_1(...h_n(s,z))) = H
\)

is holomorphic in \( s \) but constant in \( z \). 

This feels very intuitive to me !

Are you the first to discover this and prove it formally ? Is that your thesis ?

It looks familiar especially in 1 variable (instead of the 2 you use).

If that is completely new , my congratulations.

I conjecture that :
 the plots made by sheldon by comparing the fake semi exponential and the same function by using kneser method will look the same as

comparing the fake semi expontential and the same function based on NBLR tetration.

Although I must add that the fake is also computed based on kneser. But I think the fake based on kneser will be similar to that of NBLR.



Regards

tommy1729

Tom Marcel Raes