(09/16/2021, 07:23 PM)sheldonison Wrote: Your construction sounds very interesting, but I don't understand it. Actually any construction that yields an analytic Tetaratiin for bases greater than exp(1/e) sounds interesting! Can varying lambda to avoid singularities also yield different analytic solutions?
Also, your earlier iterated function method lead to a conjectured nowhere analytic function, and this seems somewhat similar, but I don't remember the details. Perhaps the two methods are related?
Hey, Sheldon. Not a problem if you're confused. It's fairly difficult. I'll answer your questions as I best can.
1.) Can varying lambda to avoid singularities also yield different analytic solutions?
The way I defined my solution was--which will all be equivalent (up to a normalization constant),
\(
\text{tet}_\beta(z -x_\epsilon) = \lim_{n\to\infty} \lim_{\lambda\to 0^+} \log^{\circ n} \beta_\lambda(z+n)\,\,\text{while}\,\,\lambda = \mathcal{O}(n^{-\epsilon})\,\,\text{for}\,\,0 < \epsilon < 1\\
\)
I just use \( \lambda = 1/\sqrt{1+z} \) for simplicity; but any such limit will result in the same function, due to my use of Banach's fixed point theorem. This is a very subtle argument though. But no matter how I code these solutions, they all look the same. So varying epsilon graphically has no effect (sometimes barely even changing the normalization constant).
2. Perhaps the two methods are related?
They are absolutely related. Let me refresh your memory. Originally I had considered,
\(
\phi(s) = \Omega_{j=1}^\infty e^{s-j+z}\,\bullet z\\
\phi(s) = e^{\displaystyle s-1+e^{\displaystyle s-2+e^{s-3+...}}}\\
\)
which satisfies the equation:
\( \phi(s+1) = e^{s+\phi(s)} \)
I constructed this using "infinite compositions", but the manner you did it is equally as valid. In fact, the manner you did it is a tad more natural, and definitely helped me in constructing the beta method.
As per your derivation--we have a function,
\(
h(0) = 0\\
h(e\cdot w) = we^{h(w)}\\
\)
and we can iteratively define the taylor series at zero; then the phi function is just,
\(
\phi(s) = h(e^s)\\
\)
Then we took \( \lim_{n\to\infty} \log^{\circ n} \phi(s+n) = \text{tet}_\phi(s-x_0) \) for a normalization constant \( x_0 \). And I had erroneously thought this would be holomorphic; it is not, it's only \( \mathcal{C}^\infty \) on \( \mathbb{R}^+ \)--(at least as conjectured by you, which is 1000% confirmed by numerical calculations).
This sort of put a wrench in the gears, and had me a little downtrodden. Then tommy kept on experimenting with more functions and more infinite compositions; and a lightbulb went off. The phi method fails because,
\(
h(\infty) = \infty \cdot e^{h(\infty)}\\
\)
We have a dangling infinity which over shoots past tetration. So what I did to remedy this was use a function \( g \) such that,
\(
g_\lambda(0) = 0\\
g_\lambda(e^\lambda \cdot w) = \frac{w}{w+1}e^{g_\lambda(w)}\\
\)
This produces an analytic function for \( |w| < e^\lambda \) and when extended to the complex plane it has singularities at \( w = - e^{\lambda j} \) for \( j \ge 1 \). Now why this function is better, is because it approximates tetration for large arguments much better. Then, at infinity,
\(
g_\lambda(\infty) = 1 \cdot e^{g_\lambda(\infty)}\\
\)
so it acts as a "quasi fixed point" at infinity. Then our beta function is just,
\(
\beta_\lambda(s) = g_\lambda(e^{\lambda s})\\
\)
Which satisfies,
\(
\beta_\lambda(s+1) = e^{\beta_\lambda(s)}/(1+e^{-\lambda s})\\
\)
And note that this gets closer and closer to tetration's functional equation as we increase the real argument! These are what I call "asymptotic solutions to the tetration equation"; phi is not an asymptotic solution--tommy's function and beta are both asymptotic solutions. And from this we can think of \( \Re(s) = \infty \) as a (kind of) fixed point on the Riemann sphere. And this allows us to make Schroder functions about infinity, in which we can discover
\(
\varphi_\lambda(e^{\lambda}w) = \exp (\varphi_\lambda(w))\\
\lim_{w\to\infty*} \varphi_\lambda(w) = \infty\\
\)
Where this limit must be taken in a specific manner avoiding the poles and the such; but is otherwise correct to say \( \varphi_\lambda(\infty) = \infty \).
This gives us a family of Schroder functions about infinity. The trouble is, these solutions are dependent on \( \lambda \) and the singularities that vary for each \( \lambda \). So we want to "dodge the singularities." In my (remember its a preprint) paper I used Banach's fixed point theorem; in which any function \( \lambda : \mathbb{R}^+ \to \mathbb{R}^+ \) and \( \beta_{\lambda(s)}(s) \) is holomorphic on a sector including \( \mathbb{R}^+ \); and satisfies \( \lambda(s) = \mathcal{O}(s^{-\epsilon}) \) for \( 0 < \epsilon < 1 \) (this is the most crucial fact for my proof); any lambda like this will produce the same function.
I just chose, again, \( \lambda(s) = 1/\sqrt{1+s} \) for simplicity.
So to answer your question, this is very much the phi method--which you dubbed \( \text{tet}_\phi \); but a much more refined version. And remember that \( \beta_\lambda \) actually looks like tetration for large arguments. The phi function DOES NOT. It overshot tetration. This neatly approaches tetration. And all we're doing is calculating a small error between it and tetration. We're actually trying to find something small.
In which its easy to find a function,
\(
\tau_\lambda(s) : S \to \mathbb{C}\\
\)
in which \( \lim_{\Re s\to\infty} \tau_\lambda(s) \to 0 \) and,
\(
F_\lambda(s) = \beta_\lambda(s) + \tau_\lambda(s)\\
\exp F_\lambda(s) = F_\lambda(s+1)\\
\)
If you have any more questions please feel free to ask! I have a lot of notes and literature on how this works; and the best way to sus it out is for people to ask me questions. I can forget what is obvious and what is not. Glad you're back, sheldon.
Sincere regards, James.