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

[jaydfox]

The values are only equal if you continue the function along two paths that you can deform into each other without crossing a singularity.)

Or, interestingly enough, if you wind around non-trivial singularities in such a manner that the windings cancel out.

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. This is not necessarily true for the non-trivial singularities, and I suspect it is false.