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Pictures of the Chi-Star
#11
James,

I had to edit my posts to remove any references to Kneser's function.  I can get as far as Kneser's RiemannMapping region, which exactly matches Jay's post.  And, if you reread my edited posts,  I showed that you can get from the RiemannMapping region as follows:

 
Then you can use the complex valued inverse Abel function to get Tetration as follows:


But I don't understand Kneser's function which does not seem to be  .   Kneser is using the RiemannMapping  result in a different way than I am.  Also, Kneser finishes by constructing the real valued slog.... This thread is still good and the pictures are really cool, but I am discouraged that after all these years I still don't understand Kneser as much as I would like.  I'm sure that in time, I will understand more, or perhaps someone can step in and further enlighten me.

I think maybe I got it, but I will need to reread Henryk's post a few more times.  The only thing I can figure, that makes any sense at all is:
Kneser's equation for the inverse of Tetration in terms of tau
This shows the inverse of tau in terms of my z+theta(z)
But then is the end result of the inverse of the RiemannMapping, which totally I don't get from Henryk's post, and it still confuses me....

can also be expressed as a different 1-cyclic mapping I'm not sure this matter much though
- Sheldon
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#12
The periodic manner makes a bit more sense, but I'm confused how we get since the original super function never equals zero. Is this just playing tricks with an essential singularity making a zero pop out? But then, how is it holomorphic in a neighborhood of zero? There must be some trick I'm missing.

I guess what I don't like about Kneser's solution is that there is no way to apply the same techniques to get from tetration to pentation. We inherently use that is entire, so the same idea won't work for since it isn't entire. For example, it won't have an entire inverse Schroder function.

I think I'm more prone to Kouznetsov's method, since it appears we can get to pentation from it, and so on and so forth. Sadly the fact Kouznetsov wasn't able to prove the representation actually converges is a real downer.

I think an important facet of solving for tetration is that the same technique can be used to solve for pentation, then hexation, so on and so forth. That's probably why I just love the bounded case so much! To get from tetration to pentation we literally just do what we did to get from exponentiation to tetration.
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#13
(06/07/2017, 08:18 PM)JmsNxn Wrote: The periodic manner makes a bit more sense, but I'm confused how we get since the original super function never equals zero. Is this just playing tricks with an essential singularity making a zero pop out? But then, how is it holomorphic in a neighborhood of zero? There must be some trick I'm missing.

I think the Schwarz reflection (since the Tetration is real valued at the real axis) guarantees there are no singularities.  We started by mapping the entire real axis with the and then the functions to generate the Riemann mapping region.  As far as the limit of the singularity, the composition of the complex valued superfunction  with the result of the Riemann mapping approaches arbitrarily close to zero.  I'm not sure what details of the singularity matter to Kneser's proof of the construction, but the Riemann mapping region itself takes an extraordinarily complicated path.

Quote:I guess what I don't like about Kneser's solution is that there is no way to apply the same techniques to get from tetration to pentation

Tetration has real valued fixed points, so you don't need another Kneser Mapping for pentation, but such pentation functions don't seem all that interesting.  Personally, Kneser's Tetration holds a special place, and seems much more fun than the bounded Tetration solutions for bases<=eta, and also more interesting then pentation.
- Sheldon
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#14
(06/08/2017, 01:19 PM)sheldonison Wrote: Tetration has real valued fixed points, so you don't need another Kneser Mapping for pentation, but such pentation functions don't seem all that interesting.  Personally, Kneser's Tetration holds a special place, and seems much more fun than the bounded Tetration solutions for bases<=eta, and also more interesting then pentation.

Really? Does the same happen for pentation? Does it have a real fixed point? Has anybody bothered to go around and actually construct using Kneser's tetration as a base? I mean, if tetration has a repelling real fixed point then it's easy to get to pentation. If pentation has the same thing, it's easy to get hexation. So on and so forth. Though of course, computationally it'd probably be exhausting and impractical. I mean, it would probably be the most taxing thing on a computer known to man to compute something like . But if they are well behaved a nice proof by induction may work. That's all you need to get . That to me is the holy grail of mathematics, constructing .

I'd also like to add that the bounded case has some pretty insane properties. I think it's special in a different sense compared to our usual tetration. It's just as interesting. It is bounded. It is exponentially decaying. It is periodic. It has a Fourier series expansion.



for some Dirichlet series . Which is pretty cool. They're also Newton summable. They're totally monotone (as we've so recently uncovered for tetration, and I've recently come to believe for any arbitrary bounded analytic hyper operator).

All in all they do some pretty crazy stuff.
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#15
(06/08/2017, 09:20 PM)JmsNxn Wrote: Really? Does the same happen for pentation? Does it have a real fixed point? Has anybody bothered to go around and actually construct using Kneser's tetration as a base? I mean, if tetration has a repelling real fixed point then it's easy to get to pentation. If pentation has the same thing, it's easy to get hexation...
Just a quick overview on pentation for bases>eta.  I have not spent any time thinking about hexation for these bases.   How far have you gotten in thinking about pentation and hexation for bases<eta?

This is a graph of Tet( 1.63532449671528 )  Tetration bases greater than eta, and less than this base have three  real valued fixed points.  Tetration bases greater than this base have one real fixed point.   The most straightforward way to generate pentation is from the lower fixed point, somewhere between -2 and -1, using that real valued repelling fixed point.  Nuinho first pointed out this Tetration base, with a parabolic upper fixed point.

   

And here is a graph of pentation base(2), from Tet_2's fixed point of -1.743909176132.  This pentation base2 graph does not have any real valued fixed points.  There are singularities in Pent(z+1) wherever Pent(z) is a negative integer<=-2; this should happen in the complex plane arbitrarily close to the real axis, as z gets bigger.  

   
- Sheldon
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#16
(06/09/2017, 01:11 PM)sheldonison Wrote:
(06/08/2017, 09:20 PM)JmsNxn Wrote: Really? Does the same happen for pentation? Does it have a real fixed point? Has anybody bothered to go around and actually construct using Kneser's tetration as a base? I mean, if tetration has a repelling real fixed point then it's easy to get to pentation. If pentation has the same thing, it's easy to get hexation...
Just a quick overview on pentation for bases>eta.  I have not spent any time thinking about hexation for these bases.   How far have you gotten in thinking about pentation and hexation for bases<eta?

This is a graph of Tet( 1.63532449671528 )  Tetration bases greater than eta, and less than this base have three  real valued fixed points.  Tetration bases greater than this base have one real fixed point.   The most straightforward way to generate pentation is from the lower fixed point, somewhere between -2 and -1, using that real valued repelling fixed point.  Nuinho first pointed out this Tetration base, with a parabolic upper fixed point.



And here is a graph of pentation base(2), from Tet_2's fixed point of -1.743909176132.  This pentation base2 graph does not have any real valued fixed points.  There are singularities in Pent(z+1) wherever Pent(z) is a negative integer<=-2; this should happen in the complex plane arbitrarily close to the real axis, as z gets bigger.  
It's fairly straight forward to go from tetration to pentation in the bounded case. And quite literally, we can do the same thing to get to hexation from pentation. I have a proof by induction that constructs the bounded hyper-operators, it's a little bulky, but it's rather straight forward. I have a nice expression for them, but computing them is rather difficult. I only ever graphed tetration, seeing as graphing pentation would require a higher accuracy of tetration then I was willing to do. And graphing hexation would require a super high accuracy of both pentation and hexation. The formula is recursive and slowly converging. I'm finalizing the paper, just triple checking and making sure all the proofs don't have holes, and it's clear and concise.

That's disappointing that pentation has singularities arbitrarily close to the real line. But I've long since had a hunch that getting for  would be next to impossible, and one would have to settle with for . Pentation also looks a little wonky, it's got a weird curvature to it. It looks like there could be fixed points, so maybe hexation is possible.

Sadly, how would one show these fixed points are repelling? I have no idea. How would one actually show they exist, I also have no idea. I doubt they exist for all bases though, I wonder if was large enough then pentation doesn't have a fixed point.
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