Review of holomorphic functions of one variable - 1

We will focus on complex geometry for quite a long time from now on. Before we step into the world of complex manifolds and related structure, a review of holomorphic functions of one variable is of great help. The first part will focus on some conclusions which are useful in the theory of holomorphic functions of several variables, and the second part will devote to the Riemann mapping theorem.

Suppose is a connected open set in and is a function. Then is called holomorphic if for each , the limit exists. Viewing as a subset of , we have the partial derivatives of with respect to and . It can be seen that is holomorphic on if and only if the Cauchy-Riemann equation holds on , with the derivative Let then the above conditions become

An important part of the complex analysis is concerned with the integral of a function along a contour (piecewise differentiable path).

Theorem (Cauchy-Goursat). Suppose is a connected open set in and is a holomorphic function. Then for each contour that is contractible in ,

Using Cauchy-Goursat theorem, we can consider contours of simpler shapes without changing the value of the integral. An important fact about the holomorphic functions is that they are actually smooth and analytic, shown by the following theorems.

Theorem (Cauchy integral formula). Suppose is a connected open set in and is holomorphic. Then exists and is holomorphic on for each , with where is a simple contour that can be contracted to in .

Theorem (Taylor series). Suppose is a connected open set in and is holomorphic. Then for each , the Taylor expasion holds in a neighborhood .

Corollary. Suppose is a connected open set in and is holomorphic. If is a zero of with order , i.e., then there is a holomorphic function such that and

By this corollary, we may consider a contour contracted to such that there are no other zeros of inside , since is nozero in a neighborhood of . Noting that we have implying that where the last equality holds from the Cauchy integral formula and the Cauchy-Goursat theorem as is holomorphic inside . This extends to the argument principle.

Theorem (Argument principle for zeros). Suppose is a connected open set in and is holomorphic. If are the zeros of inside a simple contour , with orders , then

Theorem (Extension theorem). Suppose is a connected open set in , and is bounded and holomorphic. Then can be extended to a holomorphic function .

Proof: Define by for and where is a sufficiently small contour aorund contained in . The bounded of can imply the holomorphic property of at .

Another important corollary of the Cauchy integral formula is the maximal module principle.

Theorem (Maximal module principle). Suppose is a connected open set in and is holomorphic. If attains a maximal value , then is constant on .

Proof: Consider the inequality which holds for sufficiently small and implies that is constant aroud . The Cauchy-Riemann equations then implies that is constant near , and an application of the Taylor series and the path-connectedness of shows that is constant on the whole .

Next we come across the theorem which decribes the topological property of a holomorphic function.

Theorem (Open mapping theorem). Suppose is a connected open set in and is a non-constant holomorphic function. Then is an open mapping.

Proof: It suffices to show that is an open set. Take any and let . There is such that is the unique zero of in . Let and take . We claim that . Consider any and let . We have and . Thus where the last equality holds for the argument principle. We then see that has a zero in .