Lecture Notes

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====== Review for Midterm 1 ======

$\gdef\D{\mathbb{D}}$

In the past month, we have covered first two chapters of Stein's book. 

==== Preliminaries ====
First, we talked about complex numbers and reviewed some concepts of topologies. For complex numbers, you should be familiar with the following terms
  * real part, imaginary part. 
  * absolute value, argument, 
  * complex conjugate. 
  * addition, multiplication. 
  * Stereographical projection formula will not be on the test, but you should be able to give a geometrical definition of it. 

For topological notions, one should know
  * open set, closed set. Arbitrary union of open set is still open; arbitrary intersection of closed set is closed. 
  * compact set, a subset $K \In \R^n$ is compact if and only if any of the following equivalent conditions is true
    *  $K$ is bounded and closed. 
    *  any sequence of points in $K$ has convergent subsequence
    *  any open cover of $K$ has finite sub cover. 
  * Continuous function. If $f$ is continuous, then 
    * pre-image of open is open, pre-image of closed is closed. 
    * image of compact set is compact. 

==== Holomorphic function ====

We give the definition of holomorphic function (existence of complex derivative) and several examples. At this moment, we don't know if the derivative is continuous yet. The definition is useful in that, it is easy to check. 

Absolute convergent power series give rises to holomorphic functions. The radius of convergence. Hadamard's formula. 

==== Integral along curves ====
Here is just basic definition of line integrals with complex integrand $f$, and complex line element $dz$. 

1. What do we mean when we say a 'curve'? Orientation of a curve? Piecewise smooth curves. 

2. How do we define the complex integral along a curve? 

3. How to estimate an integral's absolute value? $$ | \int_\gamma f(z) dz | \leq \int_\gamma |f(z)| |dz| $$

4. 'Fundamental theorem of calculus': if $F$ is holomorphic in $\Omega$, and $F' = f$ is continuous, $\gamma$ is a curve in $\Omega$ from $a$ to $b$,  then $\int_\gamma f dz = F(b) - F(a)$. 

===== Chapter 2 =====
In this chapter, we study integrals where the integrand is a //holomorphic// function. Main useful result is the Cauchy theorem: if $f$ is holomorphic in $\D$, then $\int_\gamma f dz=0$ for any **closed** cur $\gamma$. This follows from Goursat theorem and the construction of primitives. The proof of these theorems uses typical techniques and should be familiarized. 


Then, we have the Cauchy integral formula: if $C$ is a simple closed curve, $f$ is holomorphic on $C$ and in the open set $\Omega$ in the interior of $C$, then for any $z \in \Omega$, we have 
$$ f(z) = \frac{1}{2\pi i} \int_C \frac{f(w)}{w-z} dw. $$
The book state the theorem for $C$ the unit circle, $\Omega$ the open disk. The proof of the theorem uses key-hole contour, but conceptually, one can think of shrinking $C$ down to a small $\epsilon$-radius circle around $z$, and evaluate that integral explicitly, where one replaces $f(w)$ by constant $f(z)$ and show that the error is negligible. 

From the main theorem, we derived many corollaries. One is the presentation of higher derivatives using Cauchy integrals. And one can estimate $$f^{(n)}(z)$$ the sup norm of $f(z)$ and distance of $z$ to the boundary of the domain (the closer to the boundary, the worse the estimation get). 

Finally, we derives several useful applications. One of them is the Morera theorem, which gives a converse to Goursat theorem, and uses integral to test the holomorphicity. 

===== Suggestion to Review =====
It might be useful to read other textbooks to see how the materials are presented, like Ahlfors (Ch 1, 2, 3.1 and 4.1, 4.2). It is also useful to find a partner to debate, "why is this true"? "how to construct..." "what is the simplest example or counter-example for ..." 

In the steps of the proofs, there are many details to be filled in. Try to question yourself or your friends about theses and fill in the details, so that you are fully convinced. It is better to be verbose and sure, than to be vague and feel shaky (inside).

Again, I like to hear and answer questions on piazza and zoom chat channel (chat is recommended for quick response). 

===== Sample problems =====
1. Some of the midterm questions will conceptual, for example like this
  * Why we cannot switch the order of limits? If we define the double sum as
$$ \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} := \lim_{n \to \infty} \lim_{m \to \infty} \sum_{i=0}^n \sum_{j=0}^m a_{ij} $$
Can you give an example where $$\sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} \neq \sum_{j=0}^\infty \sum_{i=0}^\infty a_{ij}? $$
Can you modify that to given an example where
$$ \int_{x=0}^1 \int_{y=0}^1 f(x,y) dy dx \neq \int_{y=0}^1 \int_{x=0}^1 f(x,y) dx dy? $$

Why absolute convergence is a sufficient condition for switching the order of summation (or integration)? 

2. Some questions will be calculations, like computing
$$ \int_{|z|=1} \frac{e^z}{z} dz $$
$$\int_{|z|=2} \frac{1}{z^2+1} dz $$
$$\int_{|z|=R} \frac{|dz|}{|z-a|^2} , |a| < R $$
Hint for the last one, write it as an integral with holomorphic integrand and $dz$, $|dz| = R d \theta = R dz / iz $. 

Some questions will be definitional, or easy to look up in the book (assuming you know where to look), 
  * what is the definition of $\d/\d z$ and $\d/\d \bar z$? 
  * Why does $\d f / \d z = 2 \d u / d z$? 
  * Is composition of holomorphic functions holomorphic? Is there a chain rule for $\d / \d z$? 




3. Let $g(x)$ be a continuous function on $[-1,1]$. For $z \in  \C \RM [-1,1]$, define
$$ f(z) = \frac{1}{2\pi i} \int_{x=-1}^1 g(x) \frac{1}{x-z} dx $$
  * Show that $f(z)$ is holomorphic in $\C \RM [-1,1]$. 
  * (hard) Show that for any $x \in (-1,1)$, 
$$ \lim_{\epsilon \to 0^+} f(x + i \epsilon) - f(x - i \epsilon) = g(x) $$
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