2.2 KiB
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