This is just something I've been working on and I need some help.
For any number, n, taking iterations of f(n) = ln (n) will eventually produce a complex numbers with imaginary components, the first of which being some number + pi*i (which seems relatively expectable given e^(pi*i) = -1).
But with enough iterations, both the real and imaginary components approach a limit. The real component, I've discovered, is 1/pi, but I can't figure out if there is a significance to the imaginary component. It is equal to approximately 1.337235701. Can anyone help in identification?
Also, because of the way this complex number, let's call it R, arose, R = e^R = ln R. I believe this is the only number with these properties, is that correct?
Any other base of logarithm will too produce a limit, though it will be different from the one produced by base e. This knowledge may be helpful if a relationship between the real and imaginary components of different log based can be found.
Just some things to think over. Any help is appreciated. I can't seem to find much about this online.