722 - Complex Analysis

This is a truly first course on complex analysis, at the pace of a first year graduate
course.

The book, which is rigorous but reader friendly, will be very helpful, but often I will
not follow it. I shall put more emphasis on real variables methods and differential forms.
e.g. residue theory is not something specific about holomorphic functions, it is about
closed non-exact differential forms. And homotopy, in R^2 and possibly R^3, will be
discussed. Still, the important basic theorems and notions of complex analysis will
be covered: analytic continuation, Schwarz Lemma, zero sets, singularities,
sub-harmonicity etc...

Riemann integration (line integrals, Green's Theorem, differentiation with respect to
parameters,..) will be an essential tool, Lebesgue integration will never be used.