From ffc9cc277de5abfd6a70bdd6232b6b18270de5e2 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Thu, 17 Apr 2025 12:13:46 -0500 Subject: [PATCH] update about and some meta --- pages/CSE559A/CSE559A_L24.md | 0 pages/CSE559A/_meta.js | 2 + pages/Math4121/_meta.js | 1 + pages/Math416/Math416_L23.md | 9 ---- pages/Math416/Math416_L25.md | 95 ++++++++++++++++++++++++++++++++++++ pages/Math416/_meta.js | 2 + pages/about.md | 6 ++- 7 files changed, 104 insertions(+), 11 deletions(-) create mode 100644 pages/CSE559A/CSE559A_L24.md create mode 100644 pages/Math416/Math416_L25.md diff --git a/pages/CSE559A/CSE559A_L24.md b/pages/CSE559A/CSE559A_L24.md new file mode 100644 index 0000000..e69de29 diff --git a/pages/CSE559A/_meta.js b/pages/CSE559A/_meta.js index 5fb9b3a..82fa202 100644 --- a/pages/CSE559A/_meta.js +++ b/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)" } diff --git a/pages/Math4121/_meta.js b/pages/Math4121/_meta.js index 22742ac..bacbe55 100644 --- a/pages/Math4121/_meta.js +++ b/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)", } diff --git a/pages/Math416/Math416_L23.md b/pages/Math416/Math416_L23.md index 08ef4e1..2e3b804 100644 --- a/pages/Math416/Math416_L23.md +++ b/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 diff --git a/pages/Math416/Math416_L25.md b/pages/Math416/Math416_L25.md new file mode 100644 index 0000000..937fc1c --- /dev/null +++ b/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 diff --git a/pages/Math416/_meta.js b/pages/Math416/_meta.js index 9997e64..690c30d 100644 --- a/pages/Math416/_meta.js +++ b/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)", } diff --git a/pages/about.md b/pages/about.md index 57fcf47..35e3234 100644 --- a/pages/about.md +++ b/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)