# Tetration Forum

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I came across an article here https://math.eretrandre.org/tetrationfor...p?aid=1187 which has some information about tetration, but it doesn't appear to be published through an official journal or arXiv. There's this result in the MSC database, but there is no document accessible anywhere, so it can't be verified. https://mathscinet.ams.org/msc/msc.html?...earch&ls=s

If you look around on page 15, it almost looks like it is providing an analytic continuation of tetration for any height of tetration (although for very limited bases), even complex numbers or non-integer numbers. Is that actually correct? Do we finally have a way to use tetration for any height?

Even if it is correct though, if it can't be attributed with a reputable source (which neither wikipedia nor a random forum on the internet is), then it can't be cited for anything, so hopefully there's some way to find it through some official outlet.
Wow, the home base for tetration has no input, that's very surprising.
(08/09/2018, 07:44 PM)Chenjesu Wrote: [ -> ]I came across an article here https://math.eretrandre.org/tetrationfor...p?aid=1187 which has some information about tetration, but it doesn't appear to be published through an official journal or arXiv. There's this result in the MSC database, but there is no document accessible anywhere, so it can't be verified. https://mathscinet.ams.org/msc/msc.html?...earch&ls=s

If you look around on page 15, it almost looks like it is providing an analytic continuation of tetration for any height of tetration (although for very limited bases), even complex numbers or non-integer numbers. Is that actually correct? Do we finally have a way to use tetration for any height?

Even if it is correct though, if it can't be attributed with a reputable source (which neither wikipedia nor a random forum on the internet is), then it can't be cited for anything, so hopefully there's some way to find it through some official outlet.

Dr. William Paulson has published a couple of recent paper's on Kneser's analytic tetration solution for bases>exp(1/e).
http://myweb.astate.edu/wpaulsen/tetration.html

James Nison's approach only applies to real bases less than exp(1/e) where Tetration is bounded.  For real valued bases<exp(1/e), you can use the real fixed point and use Schröder's equation to define the behavior of the iterated function in the neighborhood of the fixed points.  This is a much simpler problem then Kneser's solution.
It doesn't quite look like it's arguing for an interpolation of any fractional heights, it looks simply like it's toying around with complex *bases*. I see an interesting looking limit within a paper on the site for what appears to be an inverse relationship for the recursive relation for tetration of a given base, but I don't see a solution in the form of ^{n}b. So, what exactly does that paper accomplish? I also am not concearned with constant basis, I want to know what works for a variable base, like ^{a}x = x^(x^(x^(...))) "a" times.
(08/14/2018, 05:51 PM)Chenjesu Wrote: [ -> ]It doesn't quite look like it's arguing for an interpolation of any fractional heights, it looks simply like it's toying around with complex *bases*. I see an interesting looking limit within a paper on the site for what appears to be an inverse relationship for the recursive relation for tetration of a given base, but I don't see a solution in the form of ^{n}b. So, what exactly does that paper accomplish? I also am not concearned with constant basis, I want to know what works for a variable base, like ^{a}x = x^(x^(x^(...))) "a" times.

You might also want to see Henryk Trapmann's paper, Uniqueness of Holomorphic Abel functions at a Complex Fixed Point Pair

At first,  I was going to limit my discussion to published references given the Ops initial statement, "if it can't be attributed with a reputable source (which neither wikipedia nor a random forum on the internet is), then it can't be cited for anything".

But, yes, there is an accepted unique extension of the function $\exp_b^{[\circ n]}(1)$ to arbitrary real and complex valued heights.  $\text{tet}_b(z)=\exp_b^{[\circ z]}(1)$.  This is for real bases>exp(1/e), and even for complex bases.  I have written programs like fatou.gp which is available on this website that will calculate it.

Hopefully, the Op will catch up on the mathematical background, perhaps here on this website, or using some of the published links.  The mathematics to understand Kneser's analytic tetration solution is pretty high level.  You need a comfort level with both Complex Dynamics and Complex Analysis.

Edit: One more published link. I would use the graduate level book by Lennart Carleson and Tehodore Gamelin, "Complex Dynamics", instead of James Nixon's paper since it contains the generic solution for iterating any real valued function with a real valued fixed point and a positive multiplier at the fixed point.  Nixon's paper applies only to bounded tetration with real valued bases, 1<b<exp(1/e).  I would use Kneser's solution for bases b>exp(1/e) which is a much more difficult problem.
But you're not addressing the issue that the reason to cite it would be the premise that it interpolates tetration for fractional heights.

The claim in the paper J.N. paper appears to be that any complex height z can be represented with a standard analytic series and integral, there's nothing particularly tricky about that beyond calculus. The question is whether or not it can actually be interpreted as defining tetration for any complex fractional height. This is supposed to be THE home site for tetration, and so far it's been extraordinarily disappointing.

The recent paper states in (15) a definition of fractional iterates of the exponential function. Okay, great, I'd love if there was such a thing. The problem is it defines them in terms of newly defined functions ksexp and kslog which isn't even conventional notation to begin with.

Then, when I look at how the paper defines ksexp and kslog, it refuses to. What's even worse is I see in the paper " Unfortunately it lacks a proof of convergence ". And then even in your post you make up yet another notation "tet_b."

Hopefully, you will catch up and understand that it is the author's burden to effectively communicate their ideas. Even if someone comes up with a proof for the Riemann hypothesis, it's completely useless if no one can interpret it.

The problem has nothing to do with level, it's the problem of consistency and clarity as the fundamental reason that tetration is not widely used. Who do you think would end up using such a formula? Physicists, computer scientists, astronomers, chemists, etc, they don't specialize in pure math because they're busy doing research on physical phenomena and it's the job of mathematicians to explain their tools for them.
(08/14/2018, 08:34 PM)Chenjesu Wrote: [ -> ]The claim in the paper J.N. paper appears to be that any complex height z can be represented with a standard analytic series and integral...
Why are you so focused on nit-picking one unpublished paper?  I have listed peer reviewed published papers and books, applicable to solutions of $\exp_b^{[\circ z]}(1)$.  I'm not saying James Nixon's paper is incorrect, I'm just saying it isn't relevant given that you are interested in peer reviewed rigorous work.
• Dr. William Paulson's two recent papers
• Henryk Trapmann's Uniqueness paper
• Lennart Carleson's book "Complex Dynamics"
• By indirect extension Kneser's 1950 paper
(08/14/2018, 10:25 PM)sheldonison Wrote: [ -> ]
(08/14/2018, 08:34 PM)Chenjesu Wrote: [ -> ]The claim in the paper J.N. paper appears to be that any complex height z can be represented with a standard analytic series and integral...
Why are you so focused on nit-picking one unpublished paper?  I have listed peer reviewed published papers and books, applicable to solutions of $\exp_b^{[\circ z]}(z)$.  I'm not saying James Nixon's paper is incorrect, I'm just saying it isn't relevant given that you are interested in peer reviewed rigorous work.
• Dr. William Paulson's two recent papers
• Henryk Trapmann's Uniqueness paper
• Lennart Carleson's book "Complex Dynamics"
• By indirect extension Kneser's 1950 paper

Yet again, you intentionally dodged the discussion. I went on to discuss Trappmann's iterated exponential formula beside the fact that you're the one who suggested a comparison between the two papers.

Let's look at the Trappmann claim for the simple case of the half-exponential function.

$f(z)= \exp_b^{z}(1)=s \exp_{b}(z)= \underbrace{b^{b^{b^{...}}}}_{\text{z times}}$, this looks interesting, now let us extend it to non-integers:

$\exp_b^{c}(z)= s \exp_{b}(c+s \log_{b}(z))$

We want to know what raising something to the "1/2" height means, at least according to this conjecture, and z in this case is the height of tetration:

$\exp_b^{ \frac{1}{2}}(1)=s \exp_{b}(\frac{1}{2})= \underbrace{b^{b^{b^{...}}}}_{\text{\frac{1}{2} times}}$

How should this result be interpreted to yield a final answer in terms of standard mathematical functions for b=e?
(08/14/2018, 11:09 PM)Chenjesu Wrote: [ -> ]... We want to know what raising something to the "1/2" height means, at least according to this conjecture, and z in this case is the height of tetration:

$\exp_b^{ \frac{1}{2}}(1)=s \exp_{b}(\frac{1}{2})= \underbrace{b^{b^{b^{...}}}}_{\text{\frac{1}{2} times}}$

How should this result be interpreted to yield a final answer in terms of standard mathematical functions for b=e?

This is a very good question.  William Paulsen's paper from earlier this year, "figure 1" would be a good starting place;

You might also want to view my post showing pictures of the Riemann mapping: https://math.eretrandre.org/tetrationfor...p?tid=1172 But this figure is hard to explain; the short explanation of this figure is as follows:
• Start with the Schroeder function for the complex fixed point L~=0.318131505 + 1.33723570I; then e^L=L
• Turn the Schroder function into an Abel function, Abel(exp(z))=Abel(z)+1.
• But this is a complex valued Abel function for the real number line!
• It has singularities at z=0,1,e,e^e....
• Take the Abel function of the real axis ... then you get the repeating function above in the darkened line, along with the requisite singularities, starting with -infinity
• Now wrap the repeating 1-cyclic pattern around the real axis via the mapping $y \mapsto \exp(2\pi\i z)$
• Take the Rieman mapping of that circle function so that it maps to the unit circle.
• Put the singularity at z=1.  Now, z=-1 corresponds to 0.5, exp(0.5), exp(exp(0.5)) etc.  Unwrap it onto the original drawing, and magically you have a real valued tetration of the real axis.
Unfortunately, the Riemann mapping is hard to compute, so realistically, all that we have is the proof of the existence of a nicely behaved solution.  In general, Riemann mappings cannot be expressed in terms of elementary functions or series and therefore Kneser's solution probably isn't expressable in terms of elementary functions.

As far as I know, nobody has a rigorously proven computation technique published; Paulsen's paper gives numerical results just like Trapmann's paper, along with heuristic arguments for why it should equal Kneser's solution.  There are a few really good programs that can easily calculate sexp(-0.5) for any base, like my pari-gp program, fatou.gp   In the future, I expect someone will show how to make one of these computation schemes mathematically rigorous ...
I don't see how those results directly translate into the previously mentioned functions, it seems Trappmann already defined and graphed half-iterates of the exponential function, there should then be a way to plot half-iterates of heights of tetration. I suppose we should start with the half-exponential itself.
This value L, uncoincidentally is a reflection of the first fixed point of e^x at lambertw(-1) or otherwise log(ssrt(e^(-1))), http://www.wolframalpha.com/input/?i=lambertw(-1),  L~=0.318131505 + 1.33723570I  It also looks awfully close to the (maximum curvature) vertex of the e^x function itself. The star curves you presented also look very similar to the -1 branch of the real W function, though that is possibly a coincidence.

But more importantly, what does the half-exponential function actually look like in terms of functions? Trappman suggests something that looks like a basic composite of logs and exponentials, like you just exponentiate an iterated logarithm summed with a constant, yet I have not found a simplification that proves f(x) for f(f(x))=e^x.
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