update about and some meta

pages/CSE559A/_meta.js

@@ -25,4 +25,6 @@ export default {
     CSE559A_L20: "Computer Vision (Lecture 20)",
     CSE559A_L21: "Computer Vision (Lecture 21)",
     CSE559A_L22: "Computer Vision (Lecture 22)",
+    CSE559A_L23: "Computer Vision (Lecture 23)",
+    CSE559A_L24: "Computer Vision (Lecture 24)"
 }

pages/Math4121/_meta.js

@@ -38,4 +38,5 @@ export default {
     Math4121_L32: "Introduction to Lebesgue Integration (Lecture 32)",
     Math4121_L33: "Introduction to Lebesgue Integration (Lecture 33)",
     Math4121_L34: "Introduction to Lebesgue Integration (Lecture 34)",
+    Math4121_L35: "Introduction to Lebesgue Integration (Lecture 35)",
 }

pages/Math416/Math416_L23.md

@@ -151,15 +151,6 @@ $\tilde{\psi}(s,0)$ and $\psi(t,0)$ on $t\in[t_0-\delta, t_0+\delta]$ are both l
 
 Therefore, $\tilde{\psi}(s,0)=\psi(s,0)+\text{const}$
 
-
-
-
-
-
-
-
-
-
 QED
 
 #### Theorem 9.13 Cauchy's Theorem for Homotopic Curves

pages/Math416/Math416_L25.md

@@ -0,0 +1,95 @@
+# Math416 Lecture 25
+
+## Continue on Residue Theorem
+
+### Review the definition of simply connected domain
+
+A domain $\Omega$ is called simply connected if $\overline{C}\setminus \Omega$ is connected if and only if every closed curve in $\Omega$ is null-homotopic in $\Omega$.
+
+Proof:
+
+Last time we proved $\impliedby$ part.
+
+If every closed curve in $\Omega$ is null-homotopic in $\Omega$, then $\operatorname{ind}_\Gamma(z)=0$ for all $z\in\mathbb{C}\setminus\Omega$ for all contour in $\Omega$.
+
+$\implies$ $\mathbb{C}\setminus\Omega$ is connected.
+
+$\impliedby$ part:
+
+....
+
+#### Theorem 10.4-6
+
+The following condition are equivalent:
+
+1. $\Omega$ is simply connected.
+2. every holomorphic function on $\Omega$ has a primitive $g$, i.e. $g'(z)=f(z)$ for all $z\in \Omega$.
+3. every non-vanishing holomorphic function on $\Omega$ has a holomorphic logarithm.
+4. every harmonic function on $\Omega$ has a harmonic conjugate.
+
+### Residue Theorem
+
+#### Theorem 10.8 The Residue Theorem
+
+Let $\Omega$ be a domain, $\Gamma$ be a contour such that $\Gamma\cup \operatorname{int}(\Gamma)\subset \Omega$
+
+Let $f$ be holomorphic on $\Omega\setminus \{z_1, z_2, \cdots, z_n\}$ where $z_1, z_2, \cdots, z_n$ are finitely many points in $\Omega$, where $z_1, z_2, \cdots, z_n\notin \Gamma$.
+
+Then
+
+$$
+\int_\Gamma f(z) dz = 2\pi i \sum_{j=1}^n\operatorname{ind}_{\Gamma}(z_j) \operatorname{res}_{z_j}(f)
+$$
+
+Proof:
+
+For each $i\leq j\leq n$, let $C_j$ be a circle centered at $z_j\in \Gamma\setminus \Omega$ such that $\operatorname{int}(C_j)\subset \Omega$, counterclockwise and pairwise disjoint.
+
+Let $\Gamma_1=\Gamma\setminus\{z_1, z_2, \cdots, z_n\}$, $\Gamma_1=\Gamma-\sum_{j=1}^n \operatorname{ind}_{\Gamma}(z_j)C_j$ (This excludes the singularities outside $\Gamma$)
+
+$f\in O(\Omega_1)$, $\Gamma_1\in \Omega_1$
+
+and $\operatorname{ind}_{\Gamma_1}(z)=0$ for all $z\in \mathbb{C}\setminus \Omega_1$, either $z\notin \Gamma$ or $z\in\{z_1, z_2, \cdots, z_n\}$.
+
+$\operatorname{ind}_{\Gamma_1}(z_j)=\operatorname{ind}_{\Gamma}(z_j)-1\cdot\operatorname{ind}_{C_j}(z_j)=0$ for all $j=1, 2, \cdots, n$.
+
+By Cauchy's theorem, $\int_{\Gamma_1}f(z)dz=0$.
+
+So, since $f(z)=\sum_{k=-\infty}^\infty a_k(z-z_0)^k$, and $\gamma(t)=z_k+Re^{it}$ for $t\in[0, 2\pi]$,$\gamma'(t)=iRe^{it}$,
+
+$$
+\begin{aligned}  
+\int_\Gamma f(z)dz&=\int_{\Gamma_1}f(z)dz+\sum_{j=1}^n\int_{C_j}f(z)dz\\
+&=0+\sum_{j=1}^n \operatorname{ind}_{\Gamma}(z_j) \int_{C_j}f(z)dz\\
+&=0+\sum_{j=1}^n \operatorname{ind}_{\Gamma}(z_j)  \int_{0}^{2\pi}f(z_j+Re^{it})ie^{i\theta}dt\\
+&=0+\sum_{j=1}^n \operatorname{ind}_{\Gamma}(z_j)  \int_{0}^{2\pi}\left(\sum_{k=-\infty}^\infty a_k (z_j-z_0)^k e^{int}\right)  iRe^{i\theta}dt\\
+&=0+\sum_{j=1}^n \operatorname{ind}_{\Gamma}(z_j)  i\sum_{k=-\infty}^\infty a_k R^{k+1}\left(\int_{0}^{2\pi} e^{i(k+1)t}dt\right)\\
+&=\sum_{j=1}^n 2\pi i \operatorname{ind}_{\Gamma}(z_j)  \operatorname{res}_{z_j}(f)\\
+\end{aligned}
+$$
+
+QED
+
+#### Corollary 10.9 Cauchy's Integral Formula
+
+If $\Gamma$ is a simple contour, $z_0\in \operatorname{int}(\Gamma)$, $g\in O(\Omega)$, then
+
+$$
+g(z_0)=\frac{1}{2\pi i}\int_\Gamma \frac{g(z)}{z-z_0}dz
+$$
+
+Proof:
+
+The right hand side is the residue of $g(z)/(z-z_0)$ at $z_0$.
+
+By the residue theorem,
+
+Notice that $g(z)=a_0+a_1(z-z_0)+a_2(z-z_0)^2+\cdots$, and $\frac{1}{z-z_0}=a_0\sum_{k=0}^\infty (z-z_0)^k$.
+
+So $a_0=g(z_0)$, and $a_k=\frac{g^{(k)}(z_0)}{k!}$ for $k\geq 1$.
+
+$$
+\int_\Gamma \frac{g(z)}{z-z_0}dz=2\pi i \operatorname{res}_{z_0}\left(\frac{g(z)}{z-z_0}\right)=2\pi i g(z_0)
+$$
+
+QED

pages/Math416/_meta.js

@@ -27,4 +27,6 @@ export default {
     Math416_L21: "Complex Variables (Lecture 21)",
     Math416_L22: "Complex Variables (Lecture 22)",
     Math416_L23: "Complex Variables (Lecture 23)",
+    Math416_L24: "Complex Variables (Lecture 24)",
+    Math416_L25: "Complex Variables (Lecture 25)",
 }

pages/about.md

@@ -8,11 +8,13 @@ This page is built with [Nextra](https://nextra.site/).
 
 With the front end framework [Next.js](https://nextjs.org/)
 
-Deployed on [Vercel](https://vercel.com/)
+CI-CD is maintained via [Jenkins](https://www.jenkins.io/) with [Docker](https://www.docker.com/).
+
+With deployment support from [Vercel](https://vercel.com/) and [Cloudflare](https://www.cloudflare.com/)
 
 Data is stored on [GitHub](https://github.com/)
 
-Course materials are collected from [Washingtong University in St. Louis](https://wustl.edu/)
+Course materials are collected from [Washington University in St. Louis](https://wustl.edu/)
 
 Also the procrastination of project [Notechondria](https://github.com/Trance-0/Notechondria)