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 Observations on power series involving logarithmic singularities andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 10/29/2007, 11:50 PM (This post was last modified: 10/29/2007, 11:51 PM by andydude.) Well, I think I've heard of this connection before, but I'm not sure if there are any counter-examples... I think this may only be true of complex-analytic functions / holomorphic functions (in the domain minus the singularities) or meromorphic functions (in the domain including the singularities) but not true of real-analytic functions, if i recall correctly. But I don't know for sure. Anyways, I did a plot to educate myself as to what it was you were talking about, and I found some interesting things: the Lambert W-function describes an exponential-like curve through the fixed points, and the curve goes through a lattice-like structure of fixed points. The plot is shown below: http://tetration.itgo.com/pdf/SuperLogPoles2.pdf Where the circle would be the radius of convergence of slog centered at z=0 using this connection. I was also thinking that since there is a countably infinite number of singularities, does the natural super-logarithm constitute a meromorphic function? Or does the number of singularities have to be finite? I'm sure theres a better explaination of the grid other than "they're close" to $\pi/2\ (\text{mod}\ 2\pi i)$, but right now this is all I can tell. Now what I wonder is if all the fixed points are close to $\pi/2\ (\text{mod}\ 2\pi i)$? or if I'm seeing this pattern and it actually does not exist? This is definitely interesting. PS. I have also noticed that there is an obvious pattern in the number of fixed points between two fixed points. This can be shown by (with $a_k = -W_k(-1)$ a fixed point of exp(x)): $\begin{tabular}{rl} \pi i + a_{0} - a_{-1} & = 0.467121 i \approx 0 \\ 5\pi i + a_{1} - a_{-2} & = 0.530701 i \approx 0 \\ 9\pi i + a_{2} - a_{-3} & = 0.375917 i \approx 0 \\ 13\pi i + a_{3} - a_{-4} & = 0.295789 i \approx 0 \\ 17\pi i + a_{4} - a_{-5} & = 0.246132 i \approx 0 \end{tabular}$ where the n in $n \pi i$ is approximately how many $\pi$ intervals there are between a Lambert W-function fixed point and its conjugate. Notice that all the fixed points along the two exponential curves are obtained from the Lambert W-function, whereas the other fixed points are obtained from adding or subtracting $2\pi$. It took me a while to realize it was A016813, or (4n + 1), but then it was obvious since you can see that it adds two fixed points on each side every time you go to the right. Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Observations on power series involving logarithmic singularities - by jaydfox - 10/26/2007, 11:09 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/26/2007, 11:32 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/26/2007, 11:41 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/27/2007, 06:32 AM RE: Observations on power series involving logarithmic singularities - by Gottfried - 10/29/2007, 11:30 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/29/2007, 05:37 PM RE: Observations on power series involving logarithmic singularities - by Gottfried - 10/30/2007, 06:29 AM RE: Observations on power series involving logarithmic singularities - by andydude - 10/29/2007, 11:50 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/30/2007, 02:25 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/30/2007, 03:40 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/30/2007, 05:33 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/31/2007, 08:55 PM RE: Observations on power series involving logarithmic singularities - by Gottfried - 11/03/2007, 06:02 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 07:28 AM RE: Observations on power series involving logarithmic singularities - by bo198214 - 11/05/2007, 11:08 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 02:28 PM RE: Observations on power series involving logarithmic singularities - by Gottfried - 11/06/2007, 07:29 AM RE: Observations on power series involving logarithmic singularities - by Gottfried - 11/06/2007, 11:51 AM RE: Observations on power series involving logarithmic singularities - by bo198214 - 11/06/2007, 12:46 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 07:42 AM RE: Observations on power series involving logarithmic singularities - by andydude - 11/05/2007, 08:20 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 02:19 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 02:33 PM RE: Observations on power series involving logarithmic singularities - by andydude - 11/10/2007, 06:19 AM RE: Observations on power series involving logarithmic singularities - by bo198214 - 11/10/2007, 12:46 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/10/2007, 06:02 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/11/2007, 01:01 AM RE: Observations on power series involving logarithmic singularities - by andydude - 11/12/2007, 08:26 AM RE: Observations on power series involving logarithmic singularities - by andydude - 11/12/2007, 08:34 AM RE: Observations on power series involving logarithmic singularities - by bo198214 - 11/12/2007, 10:59 AM

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