A friend of mine is taking a course on analytic number theory in the spring and needs to learn basic complex analysis in a couple of weeks. I decided to do a post (self-contained, except for Stokes’ formula) on deducing the Cauchy theorems and their applications from Stokes’ theorem now instead of later–when I’ll talk about several complex variables. It might be objected that Stokes’ theorem is just Green’s theorem for , commonly used in undergraduate treatments, but my goal was to take an expository challenge: write something rigorous on complex variables in as short a space as possible without sacrificing readability. So Stokes’ theorem for manifolds is preferable to Green’s theorem as stated in a vague way about “insides of a curve” (before, say, the Jordan curve theorem is proved) and the traditional proof of Green’s theorem via rectangular decompositions.
So, let’s consider an open set , and a function . We can consider the differential
which is a complex-valued 1-form on . It is also convenient to write the differential using the and -derivatives I talked about earlier, i.e.
The reason these are important is that if , we can choose with
by differentiability, and it is easy to check that . So we can define a function to be holomorphic if it satisfies the differential equation
which is equivalent to being able to write
for each and a suitable . In particular, it is equivalent to a difference quotient definition. The derivative of a holomorphic function thus satisfies all the usual algebraic rules, under which holomorphic functions are closed. (more…)