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

[jaydfox]

As you deform the paths to determine that they are the same, you may have to push one or more trivial singularities through a path. Each time you push a trivial singularity "through" a path, you keep of track of whether it crossed from the right or left side (relative to the direction of travel along the path from the starting point), and if the number of right and left singularities (corresponding to windings in opposite directions) are the same, then you can still manage to arrive at the same value.

Oh thats indeed interesting. So is the branch (say on the real axis to have a closed path) determined by the sum of the (oriented) winding numbers around each trivial singularity (assuming no windings around other singularities)?

Effectively, yes. I'm trying to test different types of paths to determine if there are exceptions, but as far as I can tell, it works approximately like this.

Given a starting and ending point, it is possible to create a path between them that starts and ends in the principal branch, even if it must sometimes leave the principal branch. I'm working on pictures of what I mean, so in the meantime, hopefully you can picture what I mean.

If this path is used to close the path we are interested in (the one that goes "through" the thicket of singularities), then we can simply count the number of windings around each singularity (most will be one winding), to determine which branch we are in for the first of the iterated logarithms. Care must be taken not to enclose a non-trivial singularity, but other than that, I'm not sure if there are any exceptions.

Here's the weird part: Unless we choose a path that "unwinds" the windings of the enclosed region, we will find that the image of the first of the iterated logarithms will simply go out along the imaginary axis, even as the real part sort of oscillates between very large and small values. The next logarithm will then give us some oscillating path that remains in the strip between 0*i and pi*i, such that the remaining iterated logarithms will converge on the real axis.

This has me concerned that the base-change formula is truly undefined for non-real values, because does it even make any sense to say that all complex values are somehow mapped back to a real number (which would be the result as n goes to infinity)? Worse yet, this mapping is determined by the path taken.