12/27/2015, 11:40 PM
(This post was last modified: 12/27/2015, 11:41 PM by sheldonison.)

(12/27/2015, 11:15 AM)andydude Wrote: Ok, so I replaced y with 1/2 and log(L) with L in the regular iteration power series to get this:Hey Andy,

as expected it's the same power series.

Thanks for your reply. Oops; I had a typo in my 2nd derivative which I fixed. I have a pari-gp program, that calculate the coefficients iteratively.

Quote:I wanted to highlight one of my findings in this paper (page 12) that is related but separate from this, which is a power series for for any analytic function with a parabolic fixed point at 0.

Substituting in we get

which I realize is a different base, but still interesting.

The parabolic case is hugely interesting. I usually work with iterating which is equivalent to iterating base . Anyway, the cool thing about the parabolic case is that the fixed point of zero for the fractional iterate is a singularity, and the formal power series is divergent at zero. References on mathoverflow: http://mathoverflow.net/questions/4347/f...ar-and-exp

For the case at hand, , my new conjecture is that the first four derivatives are continuous, but the fifth derivative at the fixed point has a singularity. And the first four derivatives would match the first four derivatives of the formal half iterate at the fixed point. I'm still not totally comfortable it yet, so I haven't posted the justification.

- Sheldon