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