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Is this the beta method?
#1
http://apminstitute.org/recurrent-abel-f...quation-2/

I just stumbled upon it recently, looks like he describes a superfunction obtained by the beta method?
Reply
#2
(08/16/2022, 07:35 AM)bo198214 Wrote: http://apminstitute.org/recurrent-abel-f...quation-2/

I just stumbled upon it recently, looks like he describes a superfunction obtained by the beta method?



VERY VERY VERY Similar Bo!

The difference here is that they are choosing very different forms of the limit. This is very similar to how I started approaching the beta method (I wanted a Gaussian kind of representation like the \(\Gamma\)-function). But, this is a rather big but, we are adding in a convergence factor. This author has written a slightly different variation. And additionally, the author is working in the space of continuous functions.

I've found many a continuous version of this theory--I used to communicate to John Gill a fair amount; who described a very similar kind of construction. John Gill did a lot of infinite composition stuff; and is sort of the only real source for a lot of these things. And even then, it is scarce to say the least.

The difference of the beta method; and how I've built up what you see as this \(\mathcal{C}\) notation--is that it is designed for holomorphy. And holomorphy can be found quickly. And it only relies on the convergence of a sum. So the \(\beta\) method, has built into it, the following two theorems:

$$
\beta_\lambda(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{\lambda (j-s)} + 1} \bullet z
$$

Is holomorphic everywhere the composite is holomorphic, because:

$$
\sum_{j=1}^\infty |\frac{e^z}{e^{\lambda (j-s)} + 1}|\,\,\text{converges uniformly everywhere the composite is holomorphic}
$$

We can substitute my notation for the \(\mathcal{C}\) notation, yes. The main difference, and largely the difference which separates the \(\beta\) method. Is the description of it in the complex plane. And the fact we can construct these holomorphic functions.

You'll note, that they write the limit of the beta method:

$$
\lim_{N\to\infty} \log^{\circ N} \Omega_{j=1}^N\frac{e^z}{e^{\lambda (j-s-N)} + 1} \bullet z\\
$$

I write it:

$$
\lim_{N\to\infty} \log^{\circ N}\beta_\lambda(s+N)
$$

The big difference here, is that they don't acknowledge there is no holomorphy here in \(s\). It is smooth on \((x_0,\infty)\), but that's it... In the complex plane, this construction converges pointwise, but not uniformly, and is not holomorphic (it's kind of like a devil's staircase).

My form of the limit, is in my opinion superior, because we can take one first, and then the other. And we do so while retaining holomorphy.

You'll actually find a lot of random articles relating to a lot of similar things to the beta method. I am in no way the originator of infinite compositions; or the techniques used to solve these problems. But I have an edge when it comes to complex analysis; and deriving holomorphy of infinite compositions.

TL;DR This is pretty much the beta method, lol. But, as a mathematician, the devil is in the details.
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#3
(08/17/2022, 02:18 AM)JmsNxn Wrote:
(08/16/2022, 07:35 AM)bo198214 Wrote: http://apminstitute.org/recurrent-abel-f...quation-2/

I just stumbled upon it recently, looks like he describes a superfunction obtained by the beta method?



VERY VERY VERY Similar Bo!

The difference here is that they are choosing very different forms of the limit. This is very similar to how I started approaching the beta method (I wanted a Gaussian kind of representation like the \(\Gamma\)-function). But, this is a rather big but, we are adding in a convergence factor. This author has written a slightly different variation. And additionally, the author is working in the space of continuous functions.

I've found many a continuous version of this theory--I used to communicate to John Gill a fair amount; who described a very similar kind of construction. John Gill did a lot of infinite composition stuff; and is sort of the only real source for a lot of these things. And even then, it is scarce to say the least.

The difference of the beta method; and how I've built up what you see as this \(\mathcal{C}\) notation--is that it is designed for holomorphy. And holomorphy can be found quickly. And it only relies on the convergence of a sum. So the \(\beta\) method, has built into it, the following two theorems:

$$
\beta_\lambda(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{\lambda (j-s)} + 1} \bullet z
$$

Is holomorphic everywhere the composite is holomorphic, because:

$$
\sum_{j=1}^\infty |\frac{e^z}{e^{\lambda (j-s)} + 1}|\,\,\text{converges uniformly everywhere the composite is holomorphic}
$$

We can substitute my notation for the \(\mathcal{C}\) notation, yes. The main difference, and largely the difference which separates the \(\beta\) method. Is the description of it in the complex plane. And the fact we can construct these holomorphic functions.

You'll note, that they write the limit of the beta method:

$$
\lim_{N\to\infty} \log^{\circ N} \Omega_{j=1}^N\frac{e^z}{e^{\lambda (j-s-N)} + 1} \bullet z\\
$$

I write it:

$$
\lim_{N\to\infty} \log^{\circ N}\beta_\lambda(s+N)
$$

The big difference here, is that they don't acknowledge there is no holomorphy here in \(s\). It is smooth on \((x_0,\infty)\), but that's it... In the complex plane, this construction converges pointwise, but not uniformly, and is not holomorphic (it's kind of like a devil's staircase).

My form of the limit, is in my opinion superior, because we can take one first, and then the other. And we do so while retaining holomorphy.

You'll actually find a lot of random articles relating to a lot of similar things to the beta method. I am in no way the originator of infinite compositions; or the techniques used to solve these problems. But I have an edge when it comes to complex analysis; and deriving holomorphy of infinite compositions.

TL;DR This is pretty much the beta method, lol. But, as a mathematician, the devil is in the details.

Im still puzzled by the beta method and this post of yours captures all the puzzles ( i think i get all the rest )

First , I think both limit definitions give the same result. Although maybe at different speeds.

Not sure if you accept that or not.

Secondly and more important ;

Are you saying the beta method is not analytic ? near the real line ? or nowhere ?

And proven ? or just conjectured ?

And why ?

sure it has poles.

And one could even argue that those poles can go to infinity if one allows and accepts decreasing bases.

and one has many logs.  but also many branches to pick.

So im still puzzled.

Saying it converges pointwise on the complex plane is even more puzzling to me !


I though a sequence of (bounded) univalent ( locally biholomorphic ) functions that converges , is also analytic in the limit ?

( I write (bounded) because that is somewhat tautologie by the property of convergent )

Isnt that a theorem ??



And how could it converge at the points that were originally poles ??

ln ln ln pole does not converge pointwise ???!!!???



How does one prove the function f(z) is not analytic in lets say

z in  [a + b i , a - b i] 

where a > 1 , b < 1/a ?

***

Forgive me for a somewhat unrelated question that might be better asked elsewhere but 

you talked about infinite compositions as solutions to difference equations.
in particular related to parabolic fixpoints and iterations of z + ...

But i was thinking about a popular equation here 

the binary partition function 

f(n+1) = f(n) + f(n/2)
or
f(n+1) = f(n) + f((n-1)/2)

and similar ones

( such as f ' (x) = f(x/2) , f(x+1) = f(x) + f(x/2) )

so 

what does infinite compositions and " compositional calculus " have to say about it ??

f(s+1) = f(s) + f(s/2)

f(s+1) = id(f(s)) + Q(f(s))

with Q( f(s) ) = f(s/2)

what looks like a superfunction type equation already.

So we have infinite compositions involving

B(z) = z + Q(z)

or similar we investigate the dynamics of iterations of B(z).

We already know alot about the binary partition function but i wonder what this approach can teach us.

see : https://math.eretrandre.org/tetrationfor...hp?tid=911


regards

tommy1729
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#4
Hey, tommy, so essentially I glossed over the details there on "converges pointwise." It's very hard to describe.

First of all the beta function itself is a holomorphic function. And the iteration \(\log^{\circ n} \beta(s+n)\) does converge almost everywhere in \(\mathbb{C}\). But the taylor data does not converge.

It's pretty hard to explain how this happens, and the best I can do is point you to my arxiv paper describing it. Sad 

But essentially:

$$
\lim_{n\to\infty} \log^{\circ n} \beta(s+n) = \sum_{n=0}^\infty a_n (s-s_0)^n\\
$$

Is divergent, and at best creates an asymptotic series. Which satisfies the functional equation.

You are correct about your statements of bounded continuously differentiable (locally biholomorphic); but this object doesn't converge to a bounded continuously differentiable function. (We get something like a devil's staircase, but in the complex plane).

That out of the way, the beta method is holomorphic when the base \(b = e^{\mu}\) is within Shell thron. And it is also holomorphic for some \(b = e^{\mu}\) outside of shell thron. But when we use the beta method for \(b > \eta\), it's nowhere holomorphic. This is largely because you can show that:

$$
\log^{\circ n} \beta(s+n)\\
$$

Converges. BUT! for any compact set \(N\) about a point \(s_0\):

$$
\log^{\circ n} \beta(s+n)\,\,\text{does not converge uniformly on}\,N\\
$$

And this forces us to not have holomorphy. Despite getting a pointwise value (which doesn't happen all the time, but happens almost everywhere in \(\mathbb{C}\)). So it really is quite the anomaly when \(b > \eta\).


Also note, this doesn't apply to the gaussian method you propose. I'm pretty sure this can be massaged to converge to Kneser. So it's not that this is never holomorphic. Just when we use the logistic function, it isn't. So if we try to make a \(2 \pi i / \lambda\) periodic tetration base \(b=e\)--the best we can do is make asymptotic solutions. Which means that \(\log^{\circ n} \beta(s+n)\) is littered with branch cuts for fixed \(n\), and they become more dense as we increase \(n\). But point wise, it still converges to a value. And we can uncover an asymptotic series. It's pretty hard to explain. I spent about 90 pages trying to explain how and when we derive holomorphy for the beta method. So I can't really cover it here.

A lot of the work was done by Sheldon and myself, and there are still threads on here where sheldon argues pretty strongly that \(2 \pi i\) periodic \(b = e\) tetration is only smooth on \(\mathbb{R}\) and the taylor data doesn't converge. So what you get are asymptotic expansions at everypoint. We get something similar in the complex plane, asymptotic expansions; The taylor series at a point diverges.
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