Late, I know, but better than never.
This stuff honestly is straightforward. Having seen complex variable before, the approach is a little weird (partial derivatives of a complex function and chain rule usage are a little suspect, but it works).
Laplace's equation is solved pretty thoroughly in a bunch of classes, so that's pretty much par for the course. It is really cool to see the stuff on p. 197 about using the real vs. complex part as the potential. Didn't know you could look at it that way.
What exactly happens physically at the interface where, mathematically, the pressure becomes negative? That was really my only major question from the chapter.
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