06/07/2021, 10:45 PM

ALL OF MY CODE IS DRAWN FROM MY GITHUB REPOSITORY FOR THE BETA METHOD

ALL OF SHELDON'S CODE IS DRAWN FROM fatou.gp

https://github.com/JmsNxn92/Recursive_Tetration_PARI

https://math.eretrandre.org/tetrationfor...p?tid=1017

So I thought I'd compile a list of arguments I have, that the beta method tetration is not Kneser's solution. At this point I am more than confident the beta method definitely produces a holomorphic tetration. The code is still suboptimal; I'm busy trying to make a Matrix add on to the existing code to create a grid of Taylor series of sample points which will give good enough precision for large imaginary arguments.

To begin, we initiate fatou.gp and Abel_L.gp

I have coded my superexponential at base e with \( \text{Sexp(z,n,{v=0})} \); which will work on the real line precisely; and can grab the Taylor series on the real-line precisely. It will glitch for values like \( 1 + 0.5*I \) but once the imaginary argument is greater than \( 1.1*I \) it works fine. For complex arguments near the real line; expand a Taylor series on the real-line and sum it; which is in the code dump. The variable n is the depth of iteration; for this we'll set it to 100; which is large enough for about 100 digit accuracy. The variable v, is a flag as to whether you want to pull out Taylor series or not. The explanation is in the comments of the code.

Sheldon has coded the kneser superexponential at base e with first the initialization \( \text{sexpinit(\exp(1))} \) and then \( \text{sexp(z)} \).

The first thing we'll add is a graph comparing these two solutions on the real line. This is the result of \( \text{ploth(X=-1,2,Sexp(X,100) - sexp(X))} \).

You can see immediately that Kneser's solution is going to grow a tad bit faster than the beta solution. And that they still agree on the natural numbers. This difference is less than obvious though.

Here's the beta method tetration over \( -1 \le \Re(z) \le 4 \) for \( |\Im(z)| \le 0.9 \).

And here's the Kneser tetration over \( -1 \le \Re(z) \le 4 \) for \( |\Im(z)| \le 1 \).

Both are clearly holomorphic; but they have an obvious disagreement on the real-line.

This is my first argument as to why the beta method is not Kneser's.

The second argument is topological in nature. For this we will call Kneser's solution \( \text{tet}_K(z) \) and the beta method \( \text{tet}_\beta(z) \)

Kneser's Method is similar to the standard iteration of a Schroder function about the fixed points \( L,L^* \) which are the fixed points of \( \exp(z) \) with minimal imaginary argument. Which is, we can find a pair of Schroder functions \( \Psi, \Psi^* \) and a pair of theta mappings \( \theta,\theta^* \) such that,

\(

\text{tet}_K(z) = \Psi(e^{Lz}\theta(z))\,\,\text{for}\,\,\Im(z) > 0\\

\text{tet}_K(z) = \Psi^*(e^{L^*z} \theta^*(z))\,\,\text{for}\,\,\Im(z) < 0\\

\)

Where we have the relation, \( \text{tet}_K(z^*) = \text{tet}_K(z)^* \); which allows for an analytic extension to \( z \in (-2,\infty) \) which is real-valued.

The beta method is vastly different. To begin we solve the equation \( \varphi_\lambda(z) \) for \( \Re \lambda > 0 \) such that,

\(

\varphi_\lambda(e^{-\lambda}z) = \exp \varphi_\lambda(z)\\

\varphi_\lambda(0) = \infty\\

\)

This produces a collection of tetration functions,

\(

F_\lambda(s) = \varphi_\lambda(e^{-\lambda s})\\

\)

with period \( 2 \pi i / \lambda \) with many many branchcuts/singularities. But regardless; this tetration is holomorphic on \( \mathcal{U} \subset \mathbb{C} \) with \( \overline{\mathcal{U}} = \mathbb{C} \). Which means it's holomorphic almost everywhere. Because of its periodicity we can think of \( F_\lambda \)'s domain topologically as an almost cylinder.

This means,

\(

F_\lambda(z) : \mathcal{U} \to \mathbb{C}\\

\overline{\mathcal{U}} \simeq \mathbb{C}/2\pi \mathbb{Z}\\

\)

Now we can normalize these tetrations; so that the singularities appear when \( z \) approaches the boundary of this cylinder. Therefore; we get something like this, for \( F_{1+0.1i}(s) \). Where the singularities occur on the boundary of the cylinder, but within the strip everything is normal and nice and looks like tetration.

The manner in which we construct \( \text{tet}_\beta \) is to move these singularities to the points at \( \Im(z) = \pm \infty \). Think, we are stretching the interior of the cylinder to be the upper and lower half planes; and putting the boundaries at infinity. This is done by using an implicit function \( \lambda(s) \) which we insert into the limiting construction. I chose \( \lambda(s) = \frac{1}{\sqrt{1+s}} \) for simplicity; but any similar mapping will work.

Now, the reason this matters; is that, theoretically when we take,

\(

\lim_{\Im(z) \to \infty} \text{tet}_\beta(z)\\

\)

we are technically approaching the boundary of the cylinder. And on this boundary we have all of the singularities of \( F_\lambda \). So we should expect (I haven't been able to prove this satisfactorily yet):

\(

\lim_{\Im(z) \to \infty} \text{tet}_\beta(z) = \infty\\

\)

Or at least, that,

\(

\lim_{\Im(z) \to \infty} \text{tet}_\beta(z) \not \to L\\

\)

Whereas,

\(

\lim_{\Im(z) \to \infty} \text{tet}_K(z) \to L\\

\)

This would instantly show that the beta method produces a vastly different tetration than Kneser's method.

The third argument I'm bringing to the table is the obvious one.

Nowhere in my construction do I need to compute a theta mapping. As this is by far the most crucial ingredient to Kneser's construction (and Sheldon's fatou.gp); it would be absurd to think we can construct it without said theta mapping. It would be, surprising, to say the least. In fact, it would bypass a lot of the finesse of Kneser's construction; or even Kouznetsov's approach. Something I highly doubt is possible.

The beta method makes zero mention of the fixed points \( L,L^* \). It is constructed entirely from the exponential function's behaviour at infinity. There is no data, no Schroder function, no asymptotics, which include the values \( L,L^* \). I think it would be quite surprising and quite incredible if this were Kneser's; which is why I further doubt it is.

In conclusion; I believe the beta method produces a novel tetration. And it has divergence at \( \Im(z) = \pm \infty \); which is drastically different than Kneser. I'm still working on constructing a proof; but I am definitely getting there. I need an out of the box thought though, to complete the argument.

Regards, James

ALL OF SHELDON'S CODE IS DRAWN FROM fatou.gp

https://github.com/JmsNxn92/Recursive_Tetration_PARI

https://math.eretrandre.org/tetrationfor...p?tid=1017

So I thought I'd compile a list of arguments I have, that the beta method tetration is not Kneser's solution. At this point I am more than confident the beta method definitely produces a holomorphic tetration. The code is still suboptimal; I'm busy trying to make a Matrix add on to the existing code to create a grid of Taylor series of sample points which will give good enough precision for large imaginary arguments.

To begin, we initiate fatou.gp and Abel_L.gp

I have coded my superexponential at base e with \( \text{Sexp(z,n,{v=0})} \); which will work on the real line precisely; and can grab the Taylor series on the real-line precisely. It will glitch for values like \( 1 + 0.5*I \) but once the imaginary argument is greater than \( 1.1*I \) it works fine. For complex arguments near the real line; expand a Taylor series on the real-line and sum it; which is in the code dump. The variable n is the depth of iteration; for this we'll set it to 100; which is large enough for about 100 digit accuracy. The variable v, is a flag as to whether you want to pull out Taylor series or not. The explanation is in the comments of the code.

Sheldon has coded the kneser superexponential at base e with first the initialization \( \text{sexpinit(\exp(1))} \) and then \( \text{sexp(z)} \).

The first thing we'll add is a graph comparing these two solutions on the real line. This is the result of \( \text{ploth(X=-1,2,Sexp(X,100) - sexp(X))} \).

You can see immediately that Kneser's solution is going to grow a tad bit faster than the beta solution. And that they still agree on the natural numbers. This difference is less than obvious though.

Here's the beta method tetration over \( -1 \le \Re(z) \le 4 \) for \( |\Im(z)| \le 0.9 \).

And here's the Kneser tetration over \( -1 \le \Re(z) \le 4 \) for \( |\Im(z)| \le 1 \).

Both are clearly holomorphic; but they have an obvious disagreement on the real-line.

This is my first argument as to why the beta method is not Kneser's.

The second argument is topological in nature. For this we will call Kneser's solution \( \text{tet}_K(z) \) and the beta method \( \text{tet}_\beta(z) \)

Kneser's Method is similar to the standard iteration of a Schroder function about the fixed points \( L,L^* \) which are the fixed points of \( \exp(z) \) with minimal imaginary argument. Which is, we can find a pair of Schroder functions \( \Psi, \Psi^* \) and a pair of theta mappings \( \theta,\theta^* \) such that,

\(

\text{tet}_K(z) = \Psi(e^{Lz}\theta(z))\,\,\text{for}\,\,\Im(z) > 0\\

\text{tet}_K(z) = \Psi^*(e^{L^*z} \theta^*(z))\,\,\text{for}\,\,\Im(z) < 0\\

\)

Where we have the relation, \( \text{tet}_K(z^*) = \text{tet}_K(z)^* \); which allows for an analytic extension to \( z \in (-2,\infty) \) which is real-valued.

The beta method is vastly different. To begin we solve the equation \( \varphi_\lambda(z) \) for \( \Re \lambda > 0 \) such that,

\(

\varphi_\lambda(e^{-\lambda}z) = \exp \varphi_\lambda(z)\\

\varphi_\lambda(0) = \infty\\

\)

This produces a collection of tetration functions,

\(

F_\lambda(s) = \varphi_\lambda(e^{-\lambda s})\\

\)

with period \( 2 \pi i / \lambda \) with many many branchcuts/singularities. But regardless; this tetration is holomorphic on \( \mathcal{U} \subset \mathbb{C} \) with \( \overline{\mathcal{U}} = \mathbb{C} \). Which means it's holomorphic almost everywhere. Because of its periodicity we can think of \( F_\lambda \)'s domain topologically as an almost cylinder.

This means,

\(

F_\lambda(z) : \mathcal{U} \to \mathbb{C}\\

\overline{\mathcal{U}} \simeq \mathbb{C}/2\pi \mathbb{Z}\\

\)

Now we can normalize these tetrations; so that the singularities appear when \( z \) approaches the boundary of this cylinder. Therefore; we get something like this, for \( F_{1+0.1i}(s) \). Where the singularities occur on the boundary of the cylinder, but within the strip everything is normal and nice and looks like tetration.

The manner in which we construct \( \text{tet}_\beta \) is to move these singularities to the points at \( \Im(z) = \pm \infty \). Think, we are stretching the interior of the cylinder to be the upper and lower half planes; and putting the boundaries at infinity. This is done by using an implicit function \( \lambda(s) \) which we insert into the limiting construction. I chose \( \lambda(s) = \frac{1}{\sqrt{1+s}} \) for simplicity; but any similar mapping will work.

Now, the reason this matters; is that, theoretically when we take,

\(

\lim_{\Im(z) \to \infty} \text{tet}_\beta(z)\\

\)

we are technically approaching the boundary of the cylinder. And on this boundary we have all of the singularities of \( F_\lambda \). So we should expect (I haven't been able to prove this satisfactorily yet):

\(

\lim_{\Im(z) \to \infty} \text{tet}_\beta(z) = \infty\\

\)

Or at least, that,

\(

\lim_{\Im(z) \to \infty} \text{tet}_\beta(z) \not \to L\\

\)

Whereas,

\(

\lim_{\Im(z) \to \infty} \text{tet}_K(z) \to L\\

\)

This would instantly show that the beta method produces a vastly different tetration than Kneser's method.

The third argument I'm bringing to the table is the obvious one.

Nowhere in my construction do I need to compute a theta mapping. As this is by far the most crucial ingredient to Kneser's construction (and Sheldon's fatou.gp); it would be absurd to think we can construct it without said theta mapping. It would be, surprising, to say the least. In fact, it would bypass a lot of the finesse of Kneser's construction; or even Kouznetsov's approach. Something I highly doubt is possible.

The beta method makes zero mention of the fixed points \( L,L^* \). It is constructed entirely from the exponential function's behaviour at infinity. There is no data, no Schroder function, no asymptotics, which include the values \( L,L^* \). I think it would be quite surprising and quite incredible if this were Kneser's; which is why I further doubt it is.

In conclusion; I believe the beta method produces a novel tetration. And it has divergence at \( \Im(z) = \pm \infty \); which is drastically different than Kneser. I'm still working on constructing a proof; but I am definitely getting there. I need an out of the box thought though, to complete the argument.

Regards, James