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 Complex Tetration, to base exp(1/e) Ember Edison Fellow Posts: 67 Threads: 7 Joined: May 2019 05/08/2019, 12:25 PM (05/07/2019, 04:17 PM)sheldonison Wrote: (05/05/2019, 11:38 PM)Ember Edison Wrote: Hi, I was reading the article[1] and i can't reproduce it in mathematica. I need some help, and very much need some code. Edison [1]https://arxiv.org/abs/1105.4735 Equation 18 is the key, which is the asymptotic ecalle formal power series Abel function for iterating $z\mapsto\exp(z)-1$ which is congruent to iterating $\eta=\exp(1/e);\;\;\;y\mapsto\eta^y;\;\;\;z=\frac{y}{e}-1;$ The asymptotic series for the Abel equation for iterating z is given by equation 18.  I have used this equation to also get the value of Tetration or superfunction for base $\eta=\exp(1/e)$, by using a good initial estimate, and then Newton's method.  If you use pari-gp or are interested in downloading pari-gp, I can post the pari-gp code here, including the logic to generate the formal asymptotic series equation for equation 18, and the logic to generate the two superfunctions and their inverses.   $\alpha(z)=-\frac{2}{z}+\frac{1}{3}\log(\pm z)-\frac{1}{36}z+\frac{1}{540}z^2+\frac{1}{7776}z^3-\frac{71}{435456}z^4+...$ If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.   $\alpha(z)\approx\alpha(\exp(z)-1)+1$ To get arbitrarily accurate results, we iterate $z\mapsto\exp(z)-1$ enough times or for the repellilng flower, we can iterate $z\mapsto\log(z+1)$ enough times so that z is small and the asymptotic series works well. Yes, I need it! I think just has something wrong when i am definiting function. Source code will be helpful. « Next Oldest | Next Newest »

 Messages In This Thread Complex Tetration, to base exp(1/e) - by Ember Edison - 05/05/2019, 11:38 PM RE: Complex Tetration, to base exp(1/e) - by sheldonison - 05/07/2019, 04:17 PM RE: Complex Tetration, to base exp(1/e) - by Ember Edison - 05/08/2019, 12:25 PM RE: Complex Tetration, to base exp(1/e) - by sheldonison - 05/08/2019, 04:50 PM RE: Complex Tetration, to base exp(1/e) - by Ember Edison - 05/08/2019, 06:20 PM RE: Complex Tetration, to base exp(1/e) - by Ember Edison - 08/06/2019, 05:22 PM RE: Complex Tetration, to base exp(1/e) - by bo198214 - 08/13/2019, 08:27 PM RE: Complex Tetration, to base exp(1/e) - by sheldonison - 08/14/2019, 09:15 AM

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