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…)