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 Is this THE equation for parabolic fix ? tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 04/16/2015, 10:01 PM (03/19/2015, 09:24 PM)sheldonison Wrote: (03/19/2015, 02:31 PM)tommy1729 Wrote: What you have is not a Taylor series, so how did you find this expansion.This series uses Jean Ecalle's FPS solution; there are many other posts on mathoverflow by Will Jagy and Henryk Trapman and Gottried Helms, about Jean Ecalle's parabolic solution. (03/19/2015, 02:31 PM)tommy1729 Wrote: Also this does not answer the op. I know ways to get the Abel , but i forgot why getting to z=0 helps. Probably perturbation theory.It sounds like you're looking for the inverse of the parabolic solution for the Abel function, also developed around the fixed point, no? I haven't seen such an "inverse Abel function" formal power series; so it might be novel. The form might be similar to what was posted by Mick, on mathstack, but it would require a 1/x term to be the inverse of Ecalle's solution. Also, such an FPS would also likely have a zero radius of convergence for the same reasons that Ecalle's solution is an asymptotic series; which I briefly explained. (03/19/2015, 02:31 PM)tommy1729 Wrote: Btw for x + x^N the situation is different for Every N.True; Will Jagy explains the general case in some of his posts. I just figured I would give the FPS series for iterating the parabolic case; $f(z)=\exp(z)-1$, to help you out. Scaling this FPS solution gives the solution for iterating $\eta=\exp(1/e)\;\;\;f(z)=\eta^z\;\;\;$ which you mentioned earlier. $\alpha_\eta(z) = \alpha(\frac{z}{e}-1)\;\;\;\alpha_\eta(\eta^z)=\alpha_\eta(z)+1\;\;\;$ Abel function for $f(z)=\eta^z$ in terms of the Abel function for $f(z)=\exp(z)-1\;\;\;$ derived using $\ln(\ln(\eta^{\eta^z}))$ About that novel for the inverse Abel , A) replace in the FPS method ln with its Fps and also x^-n with its fps. Now we have a Taylor series. B) use series reversion C) you have the formal powerseries for the inverse Abel. What do you think ? This may have apps to other ideas ( continuum product etc ) Regards Tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread Is this THE equation for parabolic fix ? - by tommy1729 - 03/18/2015, 09:49 PM RE: Is this THE equation for parabolic fix ? - by sheldonison - 03/19/2015, 10:22 AM RE: Is this THE equation for parabolic fix ? - by tommy1729 - 03/19/2015, 02:31 PM RE: Is this THE equation for parabolic fix ? - by sheldonison - 03/19/2015, 09:24 PM RE: Is this THE equation for parabolic fix ? - by tommy1729 - 03/19/2015, 11:37 PM RE: Is this THE equation for parabolic fix ? - by tommy1729 - 04/16/2015, 10:01 PM

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