08/15/2009, 10:40 PM

(08/15/2009, 09:44 PM)bo198214 Wrote:(08/15/2009, 07:13 PM)jaydfox Wrote: Note that this grid of points covers the entire right half of the complex plane, so that when we iteratively perform logarithms, we can always find points close to the real line.

But still I dont get this. Iterated (branches of) logarithms take any point to one of the fixed points of exp. Why do they come arbitrary close to the real axis with increasing n?

I hope the picture will clarify.

I hope so.

The reason, of course, has to do with our choice of starting point. We want log^[n](-1), which as you point out, would seem to go to a fixed point.

But consider the first of the n iterated logarithms. If you choose the log(-1) at pi*i, then yes, you are going to descend into a fixed point as you continue taking iterated logarithms.

But notice that my first choice of log(-1) is way out on the imaginary line, maybe at something like (10^100 + 1)*pi*i. Notice that, even in the primary branch of the logarithm, it will take quite a while for us to get near the fixed point, and indeed, the iterated logarithms will get very close to the real line.

Well, without further ado, I will start showing some pictures (sagenb.org came back online):

This first picture is to show the case of n=4. I find log(log(-1)), in the principal branch, and I get 1.14472988584940 + 1.57079632679490*I. I create a straight line from 0 to this point, which is the blue line in the picture.

The exponentiation of this line gives me a curve with 1 and pi*i as endpoints, which is the green curve in the picture.

Finally, the second exponentiation gives me a curve with endpoints of e and -1. Note that with two iterations remaining, I will get a singularity when I get to the third iterated logarithm in base eta.

~ Jay Daniel Fox