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 Does tetration take the right half plane to itself? JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 05/16/2017, 08:46 PM (This post was last modified: 05/16/2017, 08:48 PM by JmsNxn.) (05/16/2017, 03:34 PM)sheldonison Wrote: (05/16/2017, 04:09 AM)JmsNxn Wrote: ...  It easily follows from this that $\log^{\circ n}(\lambda^n z + L)$ is fully monotone, it uniformly converges to the inverse Schroder function, and therefore the inverse Schroder function is a fully monotone function...Good luck with your paper.  We need more rigorous iterated exponentiation papers. I just wondered what your approach to show the sequence "uniformly converges".  Is there an easy theorem, or did you want to go with a more complicated approach something like my lemma5?  Just curious.  Also, you probably meant  $\log_b^{\circ n}(\lambda^n z + L)$ where b is the tetration base. Yeah, I meantt $\log_b$. This is actually a well known result on how to represent the inverse Schroder function. Bo used it before, and that's how I first learnt about it.  Essentially if $f(z_0) = z_0$ and $0 < |f'(z_0)| < 1$, then not only does $\lim_{n\to\infty} \frac{f^{\circ n}(z) - z_0}{\lambda^n} \to \Psi(z)$ for $|z- z_0|<\delta$, where $\Psi$ is the Schroder function.  We also get that, if $g = f^{-1}$ $\lim_{n\to\infty} g^{\circ n}(\lambda^n z + z_0) \to \Psi^{-1}(z)$ for $|z-z_0| < \delta$, for $\delta$ sufficiently small. This is all you really need. « Next Oldest | Next Newest »

 Messages In This Thread Does tetration take the right half plane to itself? - by JmsNxn - 05/10/2017, 07:46 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/10/2017, 08:22 PM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/15/2017, 08:16 PM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/15/2017, 09:00 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 04:09 AM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/16/2017, 03:34 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 08:46 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 04:46 AM

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