From 2090e12d7fbbe5dbaf5b6c6f7fcf56d82be63a0a Mon Sep 17 00:00:00 2001 From: Trance-0 <60459821+Trance-0@users.noreply.github.com> Date: Tue, 18 Feb 2025 13:10:08 -0600 Subject: [PATCH] update --- next.config.mjs | 15 +- package-lock.json | 19636 +++++++++++++++++---------------- pages/Math416/Math416_L11.md | 50 +- 3 files changed, 9922 insertions(+), 9779 deletions(-) diff --git a/next.config.mjs b/next.config.mjs index b8fab84..bfaab67 100644 --- a/next.config.mjs +++ b/next.config.mjs @@ -5,15 +5,20 @@ const withNextra = nextra({ themeConfig: './theme.config.jsx', latex: { renderer: 'katex', - // options: { - // macros: { - // '\\RR': '\\mathbb{R}' - // } - // } + options: { + // suppress warnings from katex for `\\` + strict: false, + // macros: { + // '\\RR': '\\mathbb{R}' + // } + } } }) export default withNextra({ + eslint: { + ignoreDuringBuilds: true, + }, experimental: { // optimize memory usage: https://nextjs.org/docs/app/building-your-application/optimizing/memory-usage webpackMemoryOptimizations: true, diff --git a/package-lock.json b/package-lock.json index 337e5b7..176bc5c 100644 --- a/package-lock.json +++ b/package-lock.json @@ -1,9777 +1,9867 @@ { - 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Proof: -We plan to keep shrinking the triangle until $f(\zeta+h)=f(\zeta)+hf'(\zeta)+\mu(h)$ where $\mu(h)$ is a function of $h$ that goes to $0$ as $h\to 0$. +We plan to keep shrinking the triangle until $f(\zeta+h)=f(\zeta)+hf'(\zeta)+\epsilon(h)$ where $\epsilon(h)$ is a function of $h$ that goes to $0$ as $h\to 0$. Let's start with a triangle $T$ with vertices $z_1,z_2,z_3$. @@ -117,5 +117,53 @@ Let's start with a triangle $T$ with vertices $z_1,z_2,z_3$. We divide $T$ into four smaller triangles by drawing lines from the midpoints of the sides to the opposite vertices. +Let $R_1,\ldots,R_4$ be the four smaller triangles. +For one $R_j$, $\left|\int_{R_j}f(z)dz\right|\geq\frac{1}{4}|I|$, we choose it then call it $T_1$. +There exists $T_1$ such that $\left|\int_{T_1}f(z)dz\right|\geq\frac{1}{4}|I|$. + +Since $L(T_1)=\frac{1}{2}L(T)$, we iterate after $n$ steps, get a triangle $T_n$ such that $L(T_n)=\frac{L(T)}{2^n}$ and $\left|\int_{T_n}f(z)dz\right|\geq\frac{1}{4^n}|I|$. + +Since $K_n=T_n\cup \text{interior}(T_n)$ is compact, we can find $K_n+1\subset K_n$ and $diam(K_n+1)<\frac{1}{2}diam(K_n)$. $diam(K_n)\to 0$ as $n\to\infty$. (Using completeness theorem) + +Since $f$ is holomorphic on $u$, $\lim_{\zeta\to z_0}\frac{f(\zeta)-f(z_0)}{\zeta-z_0}=f'(z_0)$ exists. + +So $f(\zeta)=f(z_0)+f'(z_0)(\zeta-z_0)+R(\zeta)$, we have + +$$ +\int_{T_n}f(\zeta)d\zeta=\int_{T_n}f(z_0)d\zeta+\int_{T_n}f'(z_0)(\zeta-z_0)d\zeta+\int_{T_n}R(\zeta)d\zeta +$$ + +since $f(z_0)d\zeta+\int_{T_n}f'(z_0)(\zeta-z_0)$ is in form of Cauchy integral formula, we have + +$$ +\int_{T_n}f(z_0)d\zeta+\int_{T_n}f'(z_0)(\zeta-z_0)d\zeta=0 +$$ + +Let $e_n=\max\{\frac{R(\zeta)}{\zeta-z_0}:z_0\in T_n\}$ + +Since $diam(K_n)\to 0$ as $n\to\infty$, we have $e_n\to 0$ as $n\to\infty$. + +So + +$$ +\begin{aligned} +|I|&\leq 4^n\left|\int_{T_n}f(\zeta)d\zeta\right|\\ +&\leq 4^n\left|\int_{T_n}R_n(\zeta)d\zeta\right|\\ +&\leq 4^n\cdot L(T_n)\cdot \max_{\zeta\in T_n}|R_n(\zeta)|\\ +&\leq 4^n\cdot \frac{L(T_0)}{2^n}\cdot e_n L(T_n)\\ +&\leq 4^n\cdot \frac{L(T_0)}{2^n}\cdot e_n\cdot \frac{L(T_0)}{2^n}\\ +&\leq e_n\cdot L(T_0)^2 +\end{aligned} +$$ + +Since $e_n\to 0$ as $n\to\infty$, we have $I\to 0$ as $n\to\infty$. + +So + +$$ +\int_{T_n}f(\zeta)d\zeta\to 0 +$$ + +EOP