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 $n=2$, 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 ${O \subset \mathbb{C}}$, and a ${C^2}$ function ${f: O \rightarrow \mathbb{C}}$. We can consider the differential

$\displaystyle df := f_x dx + f_y dy$

which is a complex-valued 1-form on ${O}$. It is also convenient to write the differential using the ${z}$ and ${\bar{z}}$-derivatives I talked about earlier, i.e.

$\displaystyle f_z := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) f, \quad f_{\bar{z}} := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) f.$

The reason these are important is that if ${w_0 \in O}$, we can choose ${A,B \in \mathbb{C}}$ with

$\displaystyle f(w_0+h) = f(w_0) + Ah + B \bar{h} + o(|h|), \ h \in \mathbb{C}$

by differentiability, and it is easy to check that ${A=f_z(w_0), B=f_{\bar{z}}(w_0)}$. So we can define a function ${f}$ to be holomorphic if it satisfies the differential equation

$\displaystyle f_{\bar{z}} = 0,$

which is equivalent to being able to write

$\displaystyle f(w_0 + h) =f(w_0) + Ah + o(|h|)$

for each ${w_0 \in O}$ and a suitable ${A \in \mathbb{C}}$. In particular, it is equivalent to a difference quotient definition. The derivative ${f_z}$ of a holomorphic function thus satisfies all the usual algebraic rules, under which holomorphic functions are closed. (more…)

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 ${M}$ which is locally homeomorphic to ${\mathbb{C}}$ via charts (i.e., homeomorphisms) ${\phi_i:U_i \rightarrow V_i}$ for ${U_i \subset M, V_i \subset \mathbb{C}}$ open and such that ${\phi_j \circ \phi_i^{-1}: V_i \cap V_j \rightarrow V_i \cap V_j}$ is holomorphic.

Examples

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of ${\mathbb{C}}$ is a Riemann surface in a natural manner.

The Riemann sphere ${P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}}$ or ${S^2}$ is a Riemann sphere with the open sets ${U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}}$ and the charts

$\displaystyle \phi_1 =z, \ \phi_2 = \frac{1}{z}.$

The transition map is ${\frac{1}{z}}$ and thus holomorphic on ${U_1 \cap U_2 = \mathbb{C}^*}$.

An important example comes from analytic continuation, which I will briefly sketch below. A function element is a pair ${(f,V)}$ where ${f: V \rightarrow \mathbb{C}}$ is holomorphic and ${V \subset \mathbb{C}}$ is an open disk. Two function elements ${(f,V), (g,W)}$ are said to be direct analytic continuations of each other if ${V \cap W \neq \emptyset}$ and ${f \equiv g }$ on ${V \cap W}$. 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 ${\gamma = (f,V)}$, we can consider the totality ${X}$ of all equivalence classes of function elements that can be obtained by continuing ${\gamma}$ along curves in ${\mathbb{C}}$. Then ${X}$ is actually a Riemann surface. (more…)