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+# Math416 Lecture 23
+
+## Chapter 9: Generalized Cauchy Theorem
+
+### Separation lemma
+
+Let $\Omega$ be an open subset in $\mathbb{C}$, let $K\subset \Omega$ be compact. Then There exists a simple contour $\Gamma$ such that
+
+$$
+K\subset \text{int}(\Gamma)\subset \Omega
+$$
+
+#### Corollary 9.9 for separation lemma
+
+Let $\Gamma$ be the contour constructed in the separation lemma. Let $f\in O(\Omega)$ be holomorphic on $\Omega$. Then $\forall z_0\in K$ such that
+
+$$
+f(z_0)=\frac{1}{2\pi i}\int_{\Gamma}\frac{f(z)}{z-z_0}dz
+$$
+
+Proof:
+
+Suppose $h\in O(G)$, then $\int_{\partial S} h(z)dz=0$, by Cauchy's theorem for square, followed from the triangle case.
+
+So $\int_{\Gamma} h(z)dz=0=\sum_{j=1}^n \int_{\partial S_j} h(z)dz$
+
+Fix $z_0\in K$,
+
+$$
+g(z_0)=\begin{cases}
+\frac{f(z)-f(z_0)}{z-z_0} & z\neq z_0 \\
+f'(z_0) & z=z_0
+\end{cases}
+$$
+
+So $\int_{\Gamma} g(z)dz=0$
+
+Thus
+
+$$
+\begin{aligned}
+\int_{\Gamma}\frac{f(z)}{z-z_0}dz-\int_{\Gamma}\frac{f(z_0)}{z-z_0}dz&=0 \\
+\int_{\Gamma}\frac{f(z)}{z-z_0}dz&=f(z_0)\int_{\Gamma}\frac{1}{z-z_0}dz \\
+&=f(z_0)\cdot 2\pi i
+\end{aligned}
+$$
+
+QED
+
+#### Theorem 9.10 Cauchy's Theorem
+
+Let $\Omega$ be an open subset in $\mathbb{C}$, let $\Gamma$ be a contour with $int(\Gamma)\subset \Omega$. Let $f\in O(\Omega)$ be holomorphic on $\Omega$. Then
+
+$$
+\int_{\Gamma} f(z)dz=0
+$$
+
+Proof:
+
+Let $K\subset \mathbb{C}\setminus \text{ext}(\Gamma)$.
+
+By separation lemma, $\exists \Gamma_1$ s.t. $K\subset \text{int}(\Gamma_1)\subset \Omega$.
+
+Notice that Separation lemma ensured that $w\neq z$ for all $w\in \Gamma_1, z\in \Gamma$.
+
+By Corollary 9.9, $\forall z\in K, f(z)=\frac{1}{2\pi i}\int_{\Gamma_1}\frac{f(w)}{w-z}dw$
+
+$$
+\int_{\Gamma} f(z)dz=\frac{1}{2\pi i}\int_{\Gamma}\left[\int_{\Gamma_1}\frac{f(w)}{w-z}dw\right]dz
+$$
+
+By Fubini's theorem (In graduate course for analysis),
+
+$$
+\begin{aligned}
+\int_{\Gamma} f(z)dz&=\frac{1}{2\pi i}\int_{\Gamma_1}\left[\int_{\Gamma}\frac{f(w)}{w-z}dz\right]dw \\
+&=\frac{1}{2\pi i}\int_{\Gamma_1}f(w)\left[\int_{\Gamma}\frac{1}{w-z}dz\right]dw \\
+&=\frac{1}{2\pi i}\int_{\Gamma_1}f(w)\cdot 2\pi i \ \text{ind}_{\Gamma}(w)dw \\
+&=0
+\end{aligned}
+$$
+
+Since the winding number for $\Gamma$ on $w\in \Gamma_1$ is 0. ($w$ is outside of $\Gamma$)
+
+QED
+
+### Homotopy
+
+
diff --git a/pages/Math416/_meta.js b/pages/Math416/_meta.js
index dfff81c..9997e64 100644
--- a/pages/Math416/_meta.js
+++ b/pages/Math416/_meta.js
@@ -26,4 +26,5 @@ export default {
     Math416_L20: "Complex Variables (Lecture 20)",
     Math416_L21: "Complex Variables (Lecture 21)",
     Math416_L22: "Complex Variables (Lecture 22)",
+    Math416_L23: "Complex Variables (Lecture 23)",
 }