A crash course in one complex variable

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.

 

Riemann surfaces

Finally, it’s time to get to the definition of a Riemann surface. A Riemann surface is a connected one-dimensional complex manifold. In other words, it’s a Hausdorff space which is locally homeomorphic to via charts (i.e., homeomorphisms) for open and such that is holomorphic.

Examples

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of is a Riemann surface in a natural manner.

The Riemann sphere or is a Riemann sphere with the open sets and the charts

The transition map is and thus holomorphic on .

An important example comes from analytic continuation, which I will briefly sketch below. A function element is a pair where is holomorphic and is an open disk. Two function elements are said to be direct analytic continuations of each other if and on . By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element , we can consider the totality of all equivalence classes of function elements that can be obtained by continuing along curves in . Then is actually a Riemann surface.