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 levy ecalle koenigs ? bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 07/24/2010, 02:59 AM (This post was last modified: 07/24/2010, 03:11 AM by bo198214.) (07/10/2010, 05:17 AM)bo198214 Wrote: Another formula (2.29 in the overview paper) that kinda combines hyperbolic and parabolic is: $\lim_{n\to\infty} \frac{f^{[n]}(v)-f^{[n]}(z)}{f^{[n+1]}(z)-f^{[n]}(z)}=w\frac{1-\lambda^{w}}{1-\lambda}$, $f^{[w]}(z)=v$, $\alpha_z(v)=w$ where $\lambda$ is the derivative at the fixed point 0, which is 1 in the parabolic case and you take the limit of lambda->1. I am in a hurry a bit. So perhaps more detailed later. (07/21/2010, 10:41 PM)tommy1729 Wrote: im waiting and hoping for those " more details " dear bo. Well, there is no more much to add, if you take the limit of the above right side: $\lim_{\lambda\to 1} w\frac{1-\lambda^{w}}{1-\lambda}$ you get $w$ if I not err, which is then the parabolic Levý formula. If you could invert $h(w)=w\frac{1-\lambda^{w}}{1-\lambda}$ for $\lambda\neq 1$ then $h^{-1}\left(\frac{f^{[n]}(v)-f^{[n]}(z)}{f^{[n+1]}(z)-f^{[n]}(z)}\right)$ would be another formula for the hyperbolic Abel function. If you can't (numerically/symbolically whatever) invert then you still have a different formula for the hyperbolic (and Lévy's formla for $\lambda\to 1$) superfunction/iteration: $f^{[w]}(z)=f^{[-n]}(h(w)*(f^{[n+1]}(z)-f^{[n]}(z)) + f^{[n]}(z))$ « Next Oldest | Next Newest »

 Messages In This Thread levy ecalle koenigs ? - by tommy1729 - 07/08/2010, 11:31 PM RE: levy ecalle koenigs ? - by bo198214 - 07/09/2010, 05:46 AM RE: levy ecalle koenigs ? - by tommy1729 - 07/09/2010, 12:27 PM RE: levy ecalle koenigs ? - by bo198214 - 07/10/2010, 05:17 AM RE: levy ecalle koenigs ? - by sheldonison - 07/09/2010, 11:48 AM RE: levy ecalle koenigs ? - by tommy1729 - 07/21/2010, 10:41 PM RE: levy ecalle koenigs ? - by bo198214 - 07/24/2010, 02:59 AM

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