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
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