• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Does tetration take the right half plane to itself? JmsNxn Long Time Fellow Posts: 297 Threads: 68 Joined: Dec 2010 05/10/2017, 08:22 PM (This post was last modified: 05/10/2017, 09:22 PM by JmsNxn.) Of course, as soon as I post this I've essentially proved it using the Fourier expansion. Now I'm just hoping Sheldon comes on and can enlighten me a bit more as to why this works. First and foremost $\alpha \uparrow^2 z = L - \sum_{n=1}^\infty a_n \lambda^{nz}$ where $\alpha^L = L$ and $\lambda = \log(L)$ If $h(\lambda z) = \alpha^{h(z)}$ is the inverse schroder function, normalized so that $h'(0) = 1$, then $a_n = -h^{(n)}(0) h^{-1}(1)^n$ where  $h^{-1}(1)^n(-1)^n > 0$ and, as Sheldon mentions, but I haven't seen a fully rigorous proof, $h^{(n)}(0)(-1)^{n+1} > 0$ for $n \ge 1$ therefore $a_n>0$. Now it follows that $|\alpha \uparrow^2 z - L| \le \sum_{n=1}^\infty a_n \lambda^{nx} \le \sum_{n=1}^\infty a_n = L -1$ and tetration is contained in a disk about $L$ that lies in the right half plane. NOW, all I need is $h^{(n)}(0)(-1)^{n+1} > 0$ for $n \ge 1$. Sheldon has claimed this, but in order to put it in the paper, I'll probably either need to prove it myself, or have a good reference. In a short enough, well thought out manner, that is rigorous enough to put in the finalized version of my paper (credit will be given, of course). « 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 1,247 09/09/2019, 10:55 PM Last Post: tommy1729 Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) Gottfried 1 1,855 09/09/2019, 10:50 PM Last Post: tommy1729 Half-iteration of x^(n^2) + 1 tommy1729 3 6,172 03/09/2017, 10:02 PM Last Post: Xorter Uniqueness of half-iterate of exp(x) ? tommy1729 14 22,396 01/09/2017, 02:41 AM Last Post: Gottfried [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 53,821 03/27/2015, 11:28 PM Last Post: tommy1729 [entire exp^0.5] The half logaritm. tommy1729 1 3,481 05/11/2014, 06:10 PM Last Post: tommy1729 Does the Mellin transform have a half-iterate ? tommy1729 4 5,729 05/07/2014, 11:52 PM Last Post: tommy1729 Simple method for half iterate NOT based on a fixpoint. tommy1729 2 4,789 04/30/2013, 09:33 PM Last Post: tommy1729 half-iterates of x^2-x+1 Balarka Sen 2 5,546 04/30/2013, 01:14 AM Last Post: tommy1729 Is the half iterate of 2sinh(x) analytic near R ? tommy1729 1 3,386 03/13/2013, 12:13 AM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)