• 1 Vote(s) - 5 Average
• 1
• 2
• 3
• 4
• 5
 Inverse super-composition sheldonison Long Time Fellow Posts: 631 Threads: 22 Joined: Oct 2008 12/25/2016, 04:16 AM (This post was last modified: 12/25/2016, 11:17 AM by sheldonison.) (12/24/2016, 09:53 PM)Xorter Wrote: I am interested in all the working methods, it can be easy way or not. Please, if you do not mind, it would be really helpful for me and for the community if you tried to find it. Thank you very much. Your question is too general, since you don't identify what f(x) you are interested in. In general, the type of solution depends on the behavior at the fixed point. I assume you are interested in real valued functions. Some iterated functions have an attracting point. Then we look at the slope at the fixed point. If the slope at the fixed point is equal to 1, then we have the parbolic case, which is where the method of Ecalle works. This is the method that James was refering too. Ecalle's method can be used to find the solution for the slog inverse of the iterated function for $f(z) \mapsto \eta^z$ where $\eta=\exp(1/e)$, and the fixed point is "e". Then the method of Ecalle generates the Abel function at the fixed point. See http://mathoverflow.net/questions/45608/...x-converge and look for the $\alpha(z)$ formal power series definition where Will Jagy writes, "Now, given a specific x....it is a result of Jean Ecalle at Orsay that we may take". The algebra for the method of Ecalle is easiest if the fixed point is moved to zero, by solving the Abel function for the equivalent probem, $f(y) \mapsto \exp(y)-1$ instead of $f(z) \mapsto \eta^z$ where $y=\frac{z}{e}-1$ If the slope is less than 1, then we can use Koenig's Schröder's function solution; see https://en.m.wikipedia.org/wiki/Schröder's_equation If there are no real valued fixed points, then we have Kneser's solution for tetration. There are various numerical solutions for Kneser's slog such as mine: http://math.eretrandre.org/tetrationforu...p?tid=1017 - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Inverse super-composition - by Xorter - 11/24/2016, 12:53 PM RE: Inverse super-composition - by JmsNxn - 11/25/2016, 08:55 PM RE: Inverse super-composition - by Xorter - 12/23/2016, 01:33 PM RE: Inverse super-composition - by JmsNxn - 12/23/2016, 08:12 PM RE: Inverse super-composition - by Xorter - 12/24/2016, 09:53 PM RE: Inverse super-composition - by sheldonison - 12/25/2016, 04:16 AM RE: Inverse super-composition - by Xorter - 12/25/2016, 04:38 PM RE: Inverse super-composition - by sheldonison - 12/25/2016, 08:35 PM RE: Inverse super-composition - by Xorter - 12/25/2016, 10:23 PM RE: Inverse super-composition - by sheldonison - 12/26/2016, 07:10 AM RE: Inverse super-composition - by Xorter - 01/12/2017, 04:19 PM RE: Inverse super-composition - by Xorter - 05/26/2018, 12:00 AM

 Possibly Related Threads... Thread Author Replies Views Last Post Is bugs or features for fatou.gp super-logarithm? Ember Edison 10 1,941 08/07/2019, 02:44 AM Last Post: Ember Edison A fundamental flaw of an operator who's super operator is addition JmsNxn 4 6,605 06/23/2019, 08:19 PM Last Post: Chenjesu Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 2,480 06/10/2019, 04:29 AM Last Post: Ember Edison Inverse Iteration Xorter 3 2,200 02/05/2019, 09:58 AM Last Post: MrFrety The super 0th root and a new rule of tetration? Xorter 4 3,621 11/29/2017, 11:53 AM Last Post: Xorter the inverse ackerman functions JmsNxn 3 5,693 09/18/2016, 11:02 AM Last Post: Xorter Uniterated composition Xorter 2 3,016 09/15/2016, 05:17 PM Last Post: MphLee Solving tetration using differintegrals and super-roots JmsNxn 0 1,813 08/22/2016, 10:07 PM Last Post: JmsNxn The super of exp(z)(z^2 + 1) + z. tommy1729 1 2,471 03/15/2016, 01:02 PM Last Post: tommy1729 Super-root 3 andydude 10 10,255 01/19/2016, 03:14 AM Last Post: andydude

Users browsing this thread: 1 Guest(s)