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.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.

I know ways to get the Abel , but i forgot why getting to z=0 helps.

Probably perturbation theory.

(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; , to help you out. Scaling this FPS solution gives the solution for iterating which you mentioned earlier.

Abel function for in terms of the Abel function for derived using

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