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 Derivative of exp^[1/2] at the fixed point? sheldonison Long Time Fellow Posts: 633 Threads: 22 Joined: Oct 2008 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: $\exp^{1/2}(z + L) = L + \sqrt{L} z + \frac{\sqrt{L}}{2(1 + \sqrt{L})} z^2 + \frac{\sqrt{L} - 3L + 4L^{3/2} - 3L^2 + L^{5/2}}{6(1 + L)(1 - L)^2} z^3 + \cdots$ 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 $f^{1/x}(x)$ for any analytic function $f$ with a parabolic fixed point at 0. $f^{1/x}(x) = \frac{x}{1 - f_2} + \left(f_2 - \frac{f_3}{f_2}\right)\frac{\log(1 - f_2)}{(1 - f_2)^2} x^2 + \cdots$ Substituting in $f(z) = \exp_{\eta}(z) = \exp(z/e)$ we get $\exp_{\eta}^{1/z}(z) = e + 2(z-e) - \frac{2 \log(2)}{3e} (z - e)^2 + \frac{(1 + \log(4))^2}{18e^2} (z - e)^3 + \cdots$ which I realize is a different base, but still interesting. The parabolic case is hugely interesting. I usually work with iterating $f(x)=\exp(x)-1$ which is equivalent to iterating base $\eta$. 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, $\exp(L+x)$, 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 « Next Oldest | Next Newest »

 Messages In This Thread Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/23/2015, 04:39 PM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/24/2015, 03:25 AM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/25/2015, 04:05 PM RE: Derivative of exp^[1/2] at the fixed point? - by andydude - 12/27/2015, 11:15 AM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/27/2015, 11:40 PM RE: Derivative of exp^[1/2] at the fixed point? - by andydude - 12/29/2015, 10:51 PM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/29/2015, 10:25 PM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 12/31/2015, 11:02 AM RE: Derivative of exp^[1/2] at the fixed point? - by tommy1729 - 12/31/2015, 01:25 PM RE: Derivative of exp^[1/2] at the fixed point? - by sheldonison - 01/01/2016, 03:58 PM RE: Derivative of exp^[1/2] at the fixed point? - by tommy1729 - 12/30/2015, 01:27 PM

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