• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 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

 Possibly Related Threads... Thread Author Replies Views Last Post Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 238 09/09/2019, 10:55 PM Last Post: tommy1729 Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) Gottfried 1 402 09/09/2019, 10:50 PM Last Post: tommy1729 Half-iteration of x^(n^2) + 1 tommy1729 3 3,854 03/09/2017, 10:02 PM Last Post: Xorter Uniqueness of half-iterate of exp(x) ? tommy1729 14 14,285 01/09/2017, 02:41 AM Last Post: Gottfried [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 36,500 03/27/2015, 11:28 PM Last Post: tommy1729 [entire exp^0.5] The half logaritm. tommy1729 1 2,371 05/11/2014, 06:10 PM Last Post: tommy1729 Does the Mellin transform have a half-iterate ? tommy1729 4 3,645 05/07/2014, 11:52 PM Last Post: tommy1729 Simple method for half iterate NOT based on a fixpoint. tommy1729 2 3,233 04/30/2013, 09:33 PM Last Post: tommy1729 half-iterates of x^2-x+1 Balarka Sen 2 4,007 04/30/2013, 01:14 AM Last Post: tommy1729 Is the half iterate of 2sinh(x) analytic near R ? tommy1729 1 2,410 03/13/2013, 12:13 AM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)