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
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
