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 Dmitrii Kouznetsov's Tetration Extension bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 04/26/2008, 06:26 PM (This post was last modified: 04/26/2008, 06:28 PM by bo198214.) Kouznetsov Wrote:bo198214 Wrote:As for example real regular iteration/tetration is no more possible for $b>e^{1/e}$ because there is no real fixed point.I am not sure if I understand you well. At b=2 and b=e, tetration F(z) looks pretty regular (except $z\le-2$), and it is real at z>-2. Complex fixed points are easy to work with. Oh I use "regular" in the sense of "regular iteration" this is a well studied (mostly by Szekeres and Ecalle) way to compute arbitrary real or complex iterates of a function at a fixed point. There is only one solution for the iterates such that the fixed point still remains analytic or at least asymptotically analytic, this is called regular iteration. You will find the iterational formulas as well the formulas for the coefficients of the powerseries of regular iteration throughout the forum (keywords: hyperbolic and parabolic iteration). For tetration we have $b[4]t=\exp_b^{\circ t}(1)$. So if we have a fixed point of $\exp_b$ then we can just consider $\exp_b^{\circ t}$ to be the regular iteration, which gives us the regular tetration. In almost all cases the regular iteration at different fixed points give different solutions. As I now see those regular tetration (at the lower real fixed point, which is btw the only attracting fixed point of $b^x$) is cyclic along the imaginary axis: For regular iteration we have the iterational formula: $\exp_b^{\circ t}(1)=\lim_{n\to\infty} \log_b^{\circ n}(a(1-c^t) + c^t \exp_b^{\circ n}(1)),\quad c=\exp_b'(a)=\ln(a)$ where $a$ is the fixed point. We see that the regular tetration $b[4](it)=\exp_b^{\circ it}(1)$ is periodic with $2\pi/\ln(\ln(a))$ so it can not have a limit for $t->\infty$. « Next Oldest | Next Newest »

 Messages In This Thread Dmitrii Kouznetsov's Tetration Extension - by andydude - 04/16/2008, 10:16 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 04/21/2008, 10:41 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 04/22/2008, 12:59 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 04/22/2008, 08:18 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 04/24/2008, 01:02 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 04/25/2008, 04:06 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 04/26/2008, 02:12 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 04/26/2008, 06:26 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/17/2008, 03:22 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/18/2008, 05:31 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/18/2008, 05:09 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/21/2008, 12:20 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/21/2008, 06:22 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/21/2008, 11:18 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/22/2008, 07:12 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/22/2008, 10:43 PM RE: Dmitrii Kouznetsov's Tetration Extension - by andydude - 05/22/2008, 10:59 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/22/2008, 11:36 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/23/2008, 06:21 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/23/2008, 08:48 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/23/2008, 10:09 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/23/2008, 02:15 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/23/2008, 03:47 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/23/2008, 04:35 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/23/2008, 05:52 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/23/2008, 11:03 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/24/2008, 05:36 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/24/2008, 09:43 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/24/2008, 09:53 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/24/2008, 11:24 AM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/24/2008, 11:39 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/24/2008, 12:08 PM RE: Dmitrii Kouznetsov's Tetration Extension - by bo198214 - 05/26/2008, 07:01 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/26/2008, 09:03 AM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/27/2008, 03:58 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 05/28/2008, 08:58 AM compare complex plot with matrix power method - by bo198214 - 10/09/2008, 10:21 PM RE: Dmitrii Kouznetsov's Tetration Extension - by Kouznetsov - 10/10/2008, 01:17 AM [split] Taylor series of upx function - by Kouznetsov - 11/20/2008, 01:31 AM

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