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 Pictures of the Chi-Star sheldonison Long Time Fellow Posts: 640 Threads: 22 Joined: Oct 2008 06/04/2017, 02:07 PM (This post was last modified: 06/07/2017, 08:46 PM by sheldonison.) James, I had to edit my posts to remove any references to Kneser's $\tau$ 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 $z+\theta(z)$ from the RiemannMapping $U(z)$ region as follows: $z+\theta(z)=\frac{\ln(U(e^{2\pi i z}))}{2\pi i}$   Then you can use the complex valued inverse Abel function to get Tetration as follows: $\text{Tet}(z)=\alpha^{-1}(z+\theta(z))$ But I don't understand Kneser's $\tau(z)$ function which does not seem to be  $z+\theta(z)$.   Kneser is using the RiemannMapping $U(z)$ 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: $\text{Tet}^{-1}(z)=\text{slog}(z)=\tau(\alpha(z))=\tau\Big(\frac{\ln(\Psi(z))}{\lambda}\Big)\;\;\;$Kneser's equation for the inverse of Tetration in terms of tau $\tau^{-1}(z)=\frac{\ln(U(e^{2\pi i z}))}{2\pi i}=z+\theta(z)\;\;\;$ This shows the inverse of tau in terms of my z+theta(z) But then $\tau$ 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.... $\tau$ can also be expressed as a different 1-cyclic mapping $\tau=z+\theta_\alpha(z)\;\;\;$ I'm not sure this matter much though - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Pictures of the Chi-Star - by sheldonison - 05/28/2017, 01:13 AM RE: Pictures of the Chi-Star - by JmsNxn - 05/28/2017, 08:46 PM RE: Pictures of the Chi-Star - by sheldonison - 05/29/2017, 09:14 AM RE: Pictures of the Chi-Star - by sheldonison - 05/31/2017, 02:40 AM RE: Pictures of the Chi-Star - by JmsNxn - 05/29/2017, 05:07 PM RE: Pictures of the Chi-Star - by JmsNxn - 05/31/2017, 05:38 PM RE: Pictures of the Chi-Star - by sheldonison - 06/01/2017, 12:49 AM RE: Pictures of the Chi-Star - by JmsNxn - 06/01/2017, 07:29 AM RE: Pictures of the Chi-Star - by sheldonison - 06/01/2017, 01:41 PM RE: Pictures of the Chi-Star - by JmsNxn - 06/07/2017, 08:18 PM RE: Pictures of the Chi-Star - by sheldonison - 06/08/2017, 01:19 PM RE: Pictures of the Chi-Star - by JmsNxn - 06/08/2017, 09:20 PM RE: Pictures of the Chi-Star - by sheldonison - 06/09/2017, 01:11 PM RE: Pictures of the Chi-Star - by JmsNxn - 06/09/2017, 05:25 PM

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