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

[jaydfox]

However did we silently switch from base change \( \eta\to e \) (\( b=\eta,a=e \)) to the base change \( e\to\eta \) (\( b=e,a=\eta \))?
I think we are more interested in the first one!

Well, they are both important, as singularities in either will be an issue, but I suppose that the change from base eta back to base e should be my more immediate concern.

In theory, most of the principles should be the same. The trivial singularities would be at all branched locations of log^[n-2](1) instead of of log^[n-2](-1), if I'm thinking this through correctly (which I might not be, it's 4 AM here and I just got up to give someone a lift to the airport).