Bo, again. I think you are misinterpreting me.
Let's say that \(z_0 = \inf_{z \in \mathbb{C}} \{|\Im(z_0)|\,|\, b^{z_0} = z_0\}\). Then within the shell thron region for \(b > e^{-e}\) the infimum is \(z_0\) and it is real valued. When you let \(b < e^{-e}\), the value \(z_0\) is split into fixed point pairs \(z_0^{\pm}\) where \(|\Im(z_0)|> 0\).
This is what I believed Leo was referring to, when I made my typo; that we no longer can take the P-iteration--or the Schroder iteration with \(\cos(\pi x)\) (as you did), because we have to map between two fixed points; as opposed to the single real valued fixed point when \(z_0 \in \mathbb{R}^+\).
I mean, I'm not sure what you mean by that question. Primary, was as I defined it in the post; which is the infimum I just drew out. I guess I thought that was apparent. I really am just saying, that it's going to be a bit more involved outside of the Shell thron region, and I agree with everything you've said and within the thread. I apologize if I've said something off.
Regards.
Let's say that \(z_0 = \inf_{z \in \mathbb{C}} \{|\Im(z_0)|\,|\, b^{z_0} = z_0\}\). Then within the shell thron region for \(b > e^{-e}\) the infimum is \(z_0\) and it is real valued. When you let \(b < e^{-e}\), the value \(z_0\) is split into fixed point pairs \(z_0^{\pm}\) where \(|\Im(z_0)|> 0\).
This is what I believed Leo was referring to, when I made my typo; that we no longer can take the P-iteration--or the Schroder iteration with \(\cos(\pi x)\) (as you did), because we have to map between two fixed points; as opposed to the single real valued fixed point when \(z_0 \in \mathbb{R}^+\).
I mean, I'm not sure what you mean by that question. Primary, was as I defined it in the post; which is the infimum I just drew out. I guess I thought that was apparent. I really am just saying, that it's going to be a bit more involved outside of the Shell thron region, and I agree with everything you've said and within the thread. I apologize if I've said something off.
Regards.