Real and complex behaviour of the base change function (was: The "cheta" function)

[jaydfox]

Now this approach only got us the "closest" singularity to whatever point on the real line we started at (0 in my example). Instead of a quarter circle, we can do a 3/4 circle, 5/4 circle, etc., and we can pick different k values in the (2k+1)*pi*i formula, to find arbitrarily many singularities, and depending on the exact path being used, we can find these singularities in arbitrarily many different branches.

I will draw up some pictures do demonstrate.

Alas, I'm at home and don't have access to a math library. When I'm at home, I typically go to www.sagenb.org. By the way, I recommend it to anyone who wants quick access to a powerful library!

However, sagenb.org appears to be having problems, as the SAGE libraries aren't loading. Pictures might have to wait until Monday.

Anyway, it occurs to me that for n>=4, we can create a very nice "grid" of singularities, in the exp^[n-4](x) image.

Simply pick all the points whose real part is equal to log((2k+1)*pi), k a non-negative integer, and whose imaginary part is equal to (2m+1)/2*pi, for m an integer.

Exponentiating once will get you to +/- (2k+1)*pi*i, which exponentiating again will get you to -1.

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.

As n goes to 5, 6, etc., with each logarithm, we get a strip of points between -pi*i and pi*i, and if we use other branches, we can always keep the complex plane filled with points that will be singularities.

These are what I would call the "trivial" singularities, because they are singularities, no matter what branch of the logarithm of eta we use.

The non-trivial singularities are the ones Henryk pointed out, where we get eta, eta^eta, etc., but only in the primary branch of the logarithm of eta! This is not so easy a feat to accomplish as n gets very large, because we have to wind in and around singularities to make sure we are in the primary branch of the logarithm of base eta.