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

[bo198214]

In looking at the singularities for small values of k, It seems that the function becomes undefined (or multi-valued?), once passing the neighborhood of the singularity.

The singularities are branching points (like log has a branching point at 0). So properly there must be a cut associated with the singularity. I see whether I make a picture of the cuts associated with these singularities. But each \( f_n \), \( n\ge 3 \) is multivalued, according along which path you continue the function to a particular point. (like the logarithm depends on how often winds the continuing path around the singularity at 0, the value of \( f_n \) depends on how the path winds around any singularity. The values are only equal if you continue the function along two paths that you can deform into each other without crossing a singularity.)