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)):
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 \( \pi/2\ (\text{mod}\ 2\pi i) \), 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.\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} \)
Andrew Robbins
