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

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

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 $z_0\in K$,

So $\int_{\Gamma} g(z)dz=0$

Thus

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

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, $\forall z\in K, f(z)=\frac{1}{2\pi i}\int_{\Gamma_1}\frac{f(w)}{w-z}dw$

By Fubini’s theorem (In graduate course for analysis),

Since the winding number for $\Gamma$ on $w\in \Gamma_1$ is 0. ($w$ is outside of $\Gamma$)

QED

Homotopy

Suppose $\gamma_0, \gamma_1$ are two curves from $[0,1]$ to $\Omega$ with same end points $P,Q$.

A homotopy is a continuous function of curves $\gamma_t, 0\leq t\leq 1$, deforming $\gamma_0$ to $\gamma_1$, keeping the end points fixed.

Formally, if $H:[0,1]\times [0,1]\to \Omega$ is a continuous function satsifying

1. $H(s,0)=\gamma_0(s)$, $\forall s\in [0,1]$
2. $H(s,1)=\gamma_1(s)$, $\forall s\in [0,1]$
3. $H(0,t)=P$, $\forall t\in [0,1]$
4. $H(1,t)=Q$, $\forall t\in [0,1]$

Then we say $H$ is a homotopy between $\gamma_0$ and $\gamma_1$. (If $\gamma_0$ and $\gamma_1$ are closed curves, $Q=P$)

Lemma 9.12 Technical Lemma

Let $\phi:[0,1]\times [0,1]\to \mathbb{C}\setminus \{0\}$ is continuous. Then there exists a continuous map $\psi:[0,1]\times [0,1]\to \mathbb{C}$ such that $e^\phi=\psi$. Moreover, $\psi$ is unique up to an additive constant in $2\pi i\mathbb{Z}$.

Proof:

Let $\phi_t(s)=\phi(s,t)$, $0\leq t\leq 1$.

Then $\exists \psi_{00}$ such that $e^{\psi_{00}(s)}=\phi(0,t)$.

$\exists \psi_{t}(s)$ such that $e^{\psi_{t}(s)}=\phi_t(s)$.

We want to show $\psi_t(s)$ is continuous in $t$.

Since $\exists \epsilon>0$ such that $\phi(s,t)$ is at least $\epsilon$ away from $0$ for all $s\in [0,1]$ and $t\in [0,1]$.

Moreover, $\phi(s,t)$ is uniformly continuous.

So $\exists \delta>0$ such that $|\phi(s,t)-\phi(s,t_0)|<\epsilon$ if $|t-t_0|<\delta$.

Therefore,

So $\text {Re} \frac{\phi(s,t)}{\phi(s,t_0)}>0$.

Therefore, $\text{Log} \frac{\phi(s,t)}{\phi(s,t_0)}=\chi(s,t)$ is continuous on $s\in [0,1], t\in [t_0-\delta, t_0+\delta]$.

So $e^{\chi(s,t)}=\frac{\phi(s,t)}{\phi(s,t_0)}$, $\chi(s,t_0)=0,\forall s\in [0,1]$

Define $\tilde{\psi}(s,t)=\chi(s,t)+\chi(s,t_0)$. So this function is continuous.

And $e^{\tilde{\psi}(s,t)}=e^{\chi(s,t)+\chi(s,t_0)}=e^{\chi(s,t)}\cdot e^{\chi(s,t_0)}=\phi(s,t)$.

$\tilde{\psi}(s,0)$ and $\psi(t,0)$ on $t\in[t_0-\delta, t_0+\delta]$ are both logs of the same function, and agree to each other on $t_0$.

Therefore, $\tilde{\psi}(s,0)=\psi(s,0)+\text{const}$

QED

Theorem 9.13 Cauchy’s Theorem for Homotopic Curves