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 , 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

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 , 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