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+--- Thread: A conjectured uniqueness criteria for analytic tetration (/showthread.php?tid=1102)
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RE: A conjectured uniqueness criteria for analytic tetration - bo198214 - 01/25/2017
It seems like Szekeres considered this problem already but didnt come to some explicit conlusion.
Look at the paper in the attachment (hopefully not getting some copyright lawsuit ...)
RE: A conjectured uniqueness criteria for analytic tetration - JmsNxn - 01/25/2017
It looks like we are in good company investigating this property as a uniqueness criterion!
RE: A conjectured uniqueness criteria for analytic tetration - tommy1729 - 02/03/2017
(01/25/2017, 06:26 PM)bo198214 Wrote: It seems like Szekeres considered this problem already but didnt come to some explicit conlusion.
Look at the paper in the attachment (hopefully not getting some copyright lawsuit ...)
Quote from paper :
[img=595x842]file:///page4image256[/img]
As for the main subject of this thread that is ok.
But i want to point out a few things.
1) the majority of the ideas have actually been discussed here before.
2) in particular the idea of lim a(x) / b(x) where a , b are the derivatives of the 2 abel functions.
3) the derivative of the Abel equation is called the Julia equation and its solution Julia function.
When that solution has a limit towards x -> oo we call it a smooth Julia function.
4) the Smooth Julia function condition was considered b4 as no zero's for its second derivative ( Tommy 2 sinh and elsewhere both by me and others ).
5) the smooth Julia is NOT a uniqueness criterion that makes exp(x) - 1 the best method.
In fact i think 2sinh also has a smooth Julia , and lim a(x)/b(x) =1 also.
6) not sure if it has been shown that koenigs Function ( for hyperb fix ) always produces a smooth Julia function. That might be intresting.
7) again Julia stuff : the Julia set of exp - 1 , exp , 2sinh has not yet been compared !!
Not Trying to be hostile.
Regards
Tommy1729
RE: A conjectured uniqueness criteria for analytic tetration - JmsNxn - 02/17/2017
So I posted here the most solvable version of your question I could think of on MO http://mathoverflow.net/questions/262444/interesting-stipulation-about-completely-monotone-functions
If this statement turns out to be true then it unequivocably proves that not only is the schroder iteration method the only completely monotone solution to tetration; but also for pentation, sexation, and so on. I'm confident this statement follows too. This would give a really good real valued criterion for the uniqueness of \( \alpha \uparrow^n x \) for \( \alpha \in (1,\eta) \) and \( x \in \mathbb{R}^+ \). I hope someone in the more general mathematics community might be able to help us.