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 Infinite tetration of the imaginary unit sheldonison Long Time Fellow Posts: 640 Threads: 22 Joined: Oct 2008 06/24/2011, 04:36 PM (This post was last modified: 06/24/2011, 08:03 PM by sheldonison.) (06/24/2011, 12:25 PM)tommy1729 Wrote: sheldon , we dont know if the period is a brjuno number , which is why we are not certain if its a siegel disk. .... although you are correct with probability 1 , we know how " tricky and weird " tetration can be. i would also like to point out that 1.7129i^z is periodic with 2pi i /ln(1.7129 i) = 3.57977339 + 1.22650561 i ...Hmmm, we don't even know if the period is irrational either, although a random base on the Shell-Thron is going to be irrational, with probability 1. I wasn't able to follow the other part of your post. 3.57977339 + 1.22650561i would be a period of 1.7129^z, but I'm not sure how that effects the superfunction. The superfunction has a period of 2.9883, and 2.9883 is not a period of 1.7129^z. Presumably, the coefficients of the series for the superfunction can be calculated with some sort of formula from the series for B^(z-L). The Fourier series coefficients are equivalent to the Taylor Series coefficients of the superfunction function wrapped around the Siegel disc. The algorithm I used seems to works, but it isn't very elegant, and it only works a little bit inside the Siegel disc, where convergence is much better. Update, one Thesis paper I started to read, written by Edgar Arturo Saenz Maldonado on the Brjuno number seems to have the formulas. $\lambda=\exp(2\pi i \alpha)$ $f(z)=\lambda z + \sum_{n>=2}a_n z^n$. And .... the formal power series of h {the Seigel disc function} is given by $h(z)=\sum_{i>=1}h_i z^n$ If h is the solution of the functional equation ... $f(h(z))=h(\lambda z)$, the coefficients of the series must satisfy (formally) the following recursive relation: $h_n$=1, for n=1, and for n>=2, $h_n = \frac{1}{\lambda^n-\lambda}\sum_{n=2}^{n}a_m \sum_{n1+...+n_m=n} h_{n1}h_{n2}...h_{n_m}$ where in the second summation, $n_i>=1$ "... By the formulas in question, it is possible to determine the coefficients of the formal power series of $h_f$; the denominators of these coefficients can be written as products of the form $\lambda^n-\lambda$, for n>=2, since $\alpha$ is an irrational number these products could be very small....", which is where the Brjuno number comes from. So this would be a closed form equation for the superfunction for bases on the Shell-Thron boundary, where $\lambda=\exp(2\pi i \alpha)$, and $\alpha$ is an irrational Brjuno number. - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 12:09 AM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 09:52 AM RE: Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 11:35 AM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 01:28 PM RE: Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 08:53 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 09:10 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/14/2008, 08:25 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/15/2008, 05:35 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/19/2011, 09:22 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/20/2011, 05:27 AM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 06:20 AM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 01:36 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/20/2011, 02:46 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/21/2011, 12:13 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/21/2011, 03:02 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/21/2011, 08:00 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/22/2011, 09:28 AM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/22/2011, 03:23 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/23/2011, 08:55 AM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/23/2011, 01:13 PM solved -- they're called Siegel discs - by sheldonison - 06/23/2011, 06:10 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 06:11 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/20/2011, 10:22 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/21/2011, 01:58 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/21/2011, 11:19 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 03:15 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 12:25 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 07:05 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/24/2011, 08:07 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/24/2011, 12:25 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/24/2011, 04:36 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/24/2011, 09:44 PM RE: Infinite tetration of the imaginary unit - by bo198214 - 06/26/2011, 08:06 AM

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