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Trance-0
2025-02-18 13:10:08 -06:00
parent 271424a25b
commit 2090e12d7f
3 changed files with 9922 additions and 9779 deletions
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9
next.config.mjs
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@@ -5,15 +5,20 @@ const withNextra = nextra({
themeConfig: './theme.config.jsx',
latex: {
renderer: 'katex',
// options: {
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,

90
package-lock.json generated
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@@ -9772,6 +9772,96 @@
"type": "github",
"url": "https://github.com/sponsors/wooorm"
}
},
"node_modules/@next/swc-darwin-arm64": {
"version": "15.1.4",
"resolved": "https://registry.npmjs.org/@next/swc-darwin-arm64/-/swc-darwin-arm64-15.1.4.tgz",
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"cpu": [
"arm64"
],
"optional": true,
"os": [
"darwin"
],
"engines": {
"node": ">= 10"
}
},
"node_modules/@next/swc-darwin-x64": {
"version": "15.1.4",
"resolved": "https://registry.npmjs.org/@next/swc-darwin-x64/-/swc-darwin-x64-15.1.4.tgz",
"integrity": "sha512-7sgf5rM7Z81V9w48F02Zz6DgEJulavC0jadab4ZsJ+K2sxMNK0/BtF8J8J3CxnsJN3DGcIdC260wEKssKTukUw==",
"cpu": [
"x64"
],
"optional": true,
"os": [
"darwin"
],
"engines": {
"node": ">= 10"
}
},
"node_modules/@next/swc-linux-arm64-gnu": {
"version": "15.1.4",
"resolved": "https://registry.npmjs.org/@next/swc-linux-arm64-gnu/-/swc-linux-arm64-gnu-15.1.4.tgz",
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"cpu": [
"arm64"
],
"optional": true,
"os": [
"linux"
],
"engines": {
"node": ">= 10"
}
},
"node_modules/@next/swc-linux-arm64-musl": {
"version": "15.1.4",
"resolved": "https://registry.npmjs.org/@next/swc-linux-arm64-musl/-/swc-linux-arm64-musl-15.1.4.tgz",
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"cpu": [
"arm64"
],
"optional": true,
"os": [
"linux"
],
"engines": {
"node": ">= 10"
}
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"node_modules/@next/swc-win32-arm64-msvc": {
"version": "15.1.4",
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"cpu": [
"arm64"
],
"optional": true,
"os": [
"win32"
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"engines": {
"node": ">= 10"
}
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"version": "15.1.4",
"resolved": "https://registry.npmjs.org/@next/swc-win32-x64-msvc/-/swc-win32-x64-msvc-15.1.4.tgz",
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"cpu": [
"x64"
],
"optional": true,
"os": [
"win32"
],
"engines": {
"node": ">= 10"
}
}
}
}

50
pages/Math416/Math416_L11.md
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@@ -109,7 +109,7 @@ Cauchy's theorem is true if $\gamma$ is a triangle.
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