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Observations on power series involving logarithmic singularities
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:

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 , but right now this is all I can tell. Now what I wonder is if all the fixed points are close to ? 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 fixed point of exp(x)):
where the n in is approximately how many 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 . 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

Messages In This Thread
RE: Observations on power series involving logarithmic singularities - by andydude - 10/29/2007, 11:50 PM

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