• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Fundamental Principles of Tetration sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 03/10/2016, 04:16 AM (This post was last modified: 03/10/2016, 04:35 AM by sheldonison.) (03/10/2016, 03:06 AM)sheldonison Wrote: (03/10/2016, 02:14 AM)Daniel Wrote: .... In what manner is Kneser's solution preferred? Please explain the mathematics behind it or provide links. My understanding is that use of Schroeder's equation ... Now, it turns out one can write an equation for $\theta(z)$ exactly in terms of Kneser's Riemann mapping. Actually, I find it easiest to write an equation for Kneser's Riemann mapping in terms of my $\theta(z)$ mapping; Here are the relevant equations: If you start with the real valued solution, sexp(z), then you can generate Kneser's Riemann mapping as follows in terms of the complex valued Abel function, $\alpha(z)$ $f(z) = \alpha(\text{sexp}(z)) = z +\theta(z)\;\;$ here, $\theta(z)$ is my theta(z) mapping with trivial algebra since $\text{sexp}(z) = \alpha^{-1}(z+\theta(z)) \;\;\;$ this is my theta mapping for sexp(z) Then Kneser's Riemann mapping results in $\exp(2\pi i \cdot f(z))$ where z has been wrapped around a unit circle by using the substitution $z = \frac{\log(y)}{2\pi i}$. Then Kneser's Riemann mapping results in this unit circle function in terms of $f(z) = z +\theta(z)$ $\exp \left[ 2\pi i \cdot f\left(\frac{\log(y)}{2\pi i} \right) \right] \;\;\;$ This is Kneser's Riemann mapping on a unit circle in terms of y and $f(z) = z +\theta(z)$ From there, Kneser works backwards to $f(z) = z+\theta(z)\;\;\;\text{sexp}(z)=\alpha^{-1}(f(z))\;\;\;$ Kneser's real valued sexp(z) function using my notation - 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

 Possibly Related Threads... Thread Author Replies Views Last Post A fundamental flaw of an operator who's super operator is addition JmsNxn 4 7,496 06/23/2019, 08:19 PM Last Post: Chenjesu

Users browsing this thread: 1 Guest(s)