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 Fundamental Principles of Tetration sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 03/11/2016, 08:50 AM (This post was last modified: 03/11/2016, 08:50 AM by sheldonison.) (03/10/2016, 04:55 AM)Daniel Wrote: (03/10/2016, 03:06 AM)sheldonison Wrote: You are familiar with the Schroeder equation, lets call the Schroeder solution $S(z)$. Call the Abel equation $\alpha(z)$, and its inverse, $\alpha^{-z}(z)$ $\alpha^{-1}(z) = S^{-1}(L^z)\;\;\;$ This is the complex valued superfunction, which you are familiar withUsing the Classification of Fixed Points, there are several types of tetration. Hyperbolic tetration given by Schroeder's equation which accounts for almost all values. Parabolic tetration given by Abel's equation is only for $a=e^{1/e}$, rationally neutral tetration $a=e^{-e}$, super attracting tetration $a=1$. While I have looked for a simple way to move between Abel's and Schroeder's equations, they are not topologically conjugate. Kneser's would take the Schroeder function of the real valued tetration function, to get what he called the Chi-Star function; but I was actually showing my construction, and its equivalence to Kneser's; see my last post showing the Riemann mapping to a circle. As far as the topologically conjugate issue, yes the $\alpha^{-1}(z)=S^{-1}(L^z)\;\;$ complex valued superfunction is Periodic in the complex plane with period~= 4.447 + 1.058i, so it is not one to one. You have to be consistent with your branch to get the real valued sexp(z) via the theta(z) function. Knesser's Riemann mapping for that matter, must also be consistent. But the periodicity of the complex valued superfunction is not an issue. And in fact, Kneser's tetration solution is psuedo periodic with a pseudo period approaching arbitirarily closely to the period of the complex valued superfunction as imag(z) increases. - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Fundamental Principles of Tetration - by Daniel - 03/08/2016, 03:58 AM RE: Fundamental Principles of Tetration - by sheldonison - 03/08/2016, 10:18 AM RE: Fundamental Principles of Tetration - by Daniel - 03/10/2016, 02:14 AM RE: Fundamental Principles of Tetration - by sheldonison - 03/10/2016, 03:06 AM RE: Fundamental Principles of Tetration - by sheldonison - 03/10/2016, 04:16 AM RE: Fundamental Principles of Tetration - by Daniel - 03/10/2016, 04:55 AM RE: Fundamental Principles of Tetration - by sheldonison - 03/11/2016, 08:50 AM RE: Fundamental Principles of Tetration - by marraco - 03/08/2016, 06:58 PM RE: Fundamental Principles of Tetration - by Daniel - 03/09/2016, 10:33 AM RE: Fundamental Principles of Tetration - by Gottfried - 03/11/2016, 08:52 AM RE: Fundamental Principles of Tetration - by tommy1729 - 03/12/2016, 01:27 PM RE: Fundamental Principles of Tetration - by tommy1729 - 03/15/2016, 12:11 AM

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