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

[jaydfox]

Jay, you confused me converting between base e to eta, instead of eta to e. I agree that the principles are the same, and the singularities seem to be fatal.

Sorry, in thinking back, I think I started with going from e to eta, because it seemed that we needed to be able to get both ways, and at heart we start by going from base e to eta, then using cheta, then going back to base e.

At any rate, there is value in both, which is easy to summarize: in order to graph contour lines for the conversion from eta to e, it is easier to look simply at the conversion from e to eta. This is, for example, the easiest way to graph contour lines for ln(z): just draw lines with exp(z), using constant real or imaginary parts as needed.

Likewise, working with the conversion from eta to e would be a simple and effective way to draw contour lines in the reverse conversion, from e to eta. Since I'm still working on graphs, and I need both directions anyway, it made sense to start with going from e to eta.