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Derivative of exp^[1/2] at the fixed point?
#5
(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:

as expected it's the same power series.
Hey Andy,

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
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RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/27/2015, 11:40 PM

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