From 1165614cab70c417087583ee25ca772fc6a1f27a Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Thu, 27 Mar 2025 11:57:16 -0500
Subject: [PATCH] update

---
 pages/Math416/Math416_L19.md | 74 ++++++++++++++++++++++++++++++++++++
 pages/Math416/_meta.js       |  1 +
 pages/Math416/index.md       |  4 +-
 3 files changed, 77 insertions(+), 2 deletions(-)
 create mode 100644 pages/Math416/Math416_L19.md

diff --git a/pages/Math416/Math416_L19.md b/pages/Math416/Math416_L19.md
new file mode 100644
index 0000000..45b4936
--- /dev/null
+++ b/pages/Math416/Math416_L19.md
@@ -0,0 +1,74 @@
+# Math416 Lecture 19
+
+## Continue on the Laurent series
+
+### Laurent series
+
+If $f$ is holomorphic in $A(z_0;R_1,R_2)$ then $f=\sum_{n=-\infty}^{\infty} a_n (z-z_0)^n$ where the Laurent series converges on the annulus $A(z_0;R_1,R_2)$
+
+$$
+\int_{C(z_0,r)} f(z)(z-z_0)^{-k-1} dz = \sum_{n=-\infty}^{\infty} a_n \int_{C(z_0,r)} (z-z_0)^{n-k-1} dz=a_k 2\pi i
+$$
+
+> $C(z_0,r)$ is a circle centered at $z_0$ with radius $r$
+
+### Isolated singularities
+
+A punctured disk at $z_0$ is $A(z_0;0,R)=\{z:0<|z-z_0|<R\}$
+
+Say a function $f$ has an isolated singularity at $z_0$ if it is holomorphic in a punctured disk $A(z_0;0,R)$
+
+$f$ has a Laurent series in $A(z_0;0,R)$
+
+$$
+f(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n
+$$
+
+that converges in $A(z_0;0,R)$
+
+#### Principal part of a Laurent series
+
+The principal part of a Laurent series is the sum of the terms with negative powers of $(z-z_0)$
+
+$$
+\sum_{n=-\infty}^{-1} a_n (z-z_0)^n
+$$
+
+Say the isolated singularity is
+
+- removable if $a_n=0$ for all $n<0$
+  - If $f(z)$ has a removable singularity at $z_0$, then extend $f$ to $\mathbb{D}_{z_0,R}$ by defining $f(z_0)=a_0$. This extended $f$ is holomorphic on $\mathbb{D}_{z_0,R}$ and $f(z)=\sum_{n=0}^{\infty} a_n (z-z_0)^n$ for $z\in \mathbb{D}_{z_0,R}$
+- pole if $a_{-k}\neq 0$ and $a_n=0$ for all $n<-k$
+  - A pole with order $1$ is a simple pole
+- essential if the cases above are not true
+
+Example:
+
+1. $f(z)=\frac{\sin z}{z}$ has a removable singularity at $z=0$.
+
+the power series is
+
+$$
+\sin z = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots
+$$
+
+So the Laurent series is
+
+$$
+\frac{\sin z}{z} = 1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \cdots
+$$
+
+The singularity is removable by defining $f(0)=1$
+
+2. $f(z)=\frac{z^2-1}{(z-1)(z-3)}=\frac{(z-1)(z+1)}{(z-1)(z-3)}$
+
+There are two poles at $z=1$ and $z=3$
+
+the singularity at $z=1$ is removable by defining $f(1)=1$
+
+the singularity at $z=3$ is a simple pole with order 1 $f(z)=\frac{z+1}{z-3}=\frac{(z-3)+4}{z-3}=4(z-3)^{-1}+1$
+
+3. $f(z)=\frac{(z+1)^2(z+2)^3}{(z-1)^2(z-5)^6(z-8)}$
+
+there are three poles at $z=1,5,8$, the order of the poles are 2, 6, 1 respectively.
+
diff --git a/pages/Math416/_meta.js b/pages/Math416/_meta.js
index 285ebed..0aeecca 100644
--- a/pages/Math416/_meta.js
+++ b/pages/Math416/_meta.js
@@ -22,4 +22,5 @@ export default {
     Math416_L16: "Complex Variables (Lecture 16)",
     Math416_L17: "Complex Variables (Lecture 17)",
     Math416_L18: "Complex Variables (Lecture 18)",
+    Math416_L19: "Complex Variables (Lecture 19)",
 }
diff --git a/pages/Math416/index.md b/pages/Math416/index.md
index 26ac691..b50446f 100644
--- a/pages/Math416/index.md
+++ b/pages/Math416/index.md
@@ -3,13 +3,13 @@
 Complex variables. This is a course that explores the theory and applications of complex analysis as extension of Real analysis.
 
 The course is taught by Professor.
-John E. McCarthy <mccarthy@math.wustl.edu>
+John E. McCarthy <mailto:mccarthy@math.wustl.edu>
 
 Some interesting fact is that he cover the lecture terribly quick. At least for me. I need to preview and review the lecture after the course ended. The only thing that I can take granted of is that many theorem in real analysis still holds in the complex. By elegant definition designing, we build a wonderful math with complex variables and extended theorems, which is more helpful when solving questions that cannot be solved in real numbers.
 
 McCarthy like to write $\zeta$ for $z$ and his writing for $\zeta$ is almost identical with $z$, I decided to use the traditional notation system I've learned to avoid confusion in my notes.
 
-I will use $B_r(z_0)$ to denote a disk in $\mathbb{C}$ such that $B_r(z_0) = \{ z \in \mathbb{C} : |z - z_0| < r \}$
+I will use $B_r(z_0)$ to denote a disk in $\mathbb{C}$ such that $B_r(z_0) = \{ z \in \mathbb{C} : |z - z_0| < r \}$. In the lecture, he use $\mathbb{D}(z_0,r)$ to denote the disk centered at $z_0$ with radius $r$. If $\mathbb{D}$ is used, then it means the unit disk $\mathbb{D}=\{z:|z|<1\}$. You may also see the closure of the disk $\overline{B_r(z_0)}$ and $\overline{\mathbb{D}}$, these are equivalent definition.
 
 I will use $z$ to replace the strange notation of $\zeta$. If that makes sense.