Math416 Lecture 24

Continue on Generalized Cauchy’s Theorem

Homotopy

A homotopy between two curves $\gamma_0, \gamma_1 : [0, 1] \to \mathbb{C}$ is a continuous map $H : [0, 1] \times [0, 1] \to \mathbb{C}$ such that $H(z, 0) = \gamma_0(z)$ and $H(z, 1) = \gamma_1(z)$ for all $z \in [0, 1]$.

Lemma:

Let $\Omega$ be open in $\mathbb{C}$, Let $\gamma_0, \gamma_1$ be closed contour, homotopic in $\Omega$. Then $\operatorname{ind}_{\gamma_0} (z) = \operatorname{ind}_{\gamma_1} (z)$ for all $z \in \Omega$.

Proof:

Let $H(s,t)$ be a homotopy between $\gamma_0$ and $\gamma_1$. Let $z_0\in \mathbb{C} \setminus \Omega$.

Defined $\phi:[0,1]\times[0,1]\to \mathbb{C}\setminus \{0\}$, $\phi(s,t)=H(s,t)-z_0$.

By Technical Lemma , $\exists$ continuous $\psi:[0,1]\times[0,1]\to \mathbb{C}$ such that $e^{\psi}=\phi$.

For each $t$, $\gamma_t(s)=H(s,t)$ is a closed curve.

$\operatorname{ind}_{\gamma_t}(z_0)=\frac{1}{2\pi i}\left[\psi(1,t)-\psi(0,t)\right]$.

This is continuous (in $t$), integer valued, thus constant.

QED

Theorem 9.14 Homotopy version of Cauchy’s Theorem

Let $\Omega$ be open, $\gamma_0, \gamma_1$ be two piecewise continuous curves in $\Omega$ that are homotopic.

Then $\int_{\gamma_0} f(z) \, dz = \int_{\gamma_1} f(z) \, dz$ for all $f\in O(\Omega)$.

Proof:

$\Gamma=\gamma_0-\gamma_1$, then $\operatorname{ind}_{\Gamma}(z)=0$ for all $z\in \mathbb{C}\setminus \Omega$.

QED

Corollary of Theorem 9.14

If $\gamma_0$ is null-homotopic in $\Omega$ (i.e. $\gamma_0$ is homotopic to a point), then $\int_{\gamma_0} f(z) \, dz = 0$ for all $f\in O(\Omega)$.

Chapter 10: Further development of Complex Function Theory

Simple connectedness

Definition (non-standard) simply connected

Let $\Omega$ be a domain in $\mathbb{C}$. We say $\Omega$ is simply connected if $\overline{\mathbb{C}}\setminus \Omega$ is connected. ($\overline{\mathbb{C}}=\mathbb{C}\cup \{\infty\})$

Example:

disk is simply connected.

annulus is not simply connected.

$\mathbb{C}$ is simply connected.

Any convex domain is simply connected.

Standard definition: $\Omega$ is simply connected if every closed curve in $\Omega$ is null-homotopic in $\Omega$.

Theorem of equivalent definition of simply connected

For open connected subsets of $\mathbb{C}$, the standard definition and the non-standard definition are equivalent.

Proved end of book.

Proposition for simply connected domain

$\Omega$ is simply connected $\iff$ every contour in $\Omega$ has winding number $0$ about every point in $\mathbb{C}\setminus \Omega$.

Proof:

If $\Omega$ is simply connected, let $\gamma$ be a curve in $\Omega$, then $\operatorname{ind}_{\gamma}(z)=0$ for all $z$ in the unbounded component of $\overline{\mathbb{C}}\setminus \Omega$. This contains all of $\mathbb{C}\setminus \Omega$.

Conversely, assume $\Omega$ is not simply connected, then $\exists K\cup L=\overline{\mathbb{C}}\setminus \Omega$, where $K$ and $L$ are disjoint closed, without loss of generality, assume $\infty\in L$.

Let $H=\Omega\cup K=\mathbb{C}\setminus L$.

$H$ is open, $K$ is compact subset of $H$, so by Separation Lemma , $\exists \gamma\in H\setminus K=\Omega$ such that $K\subset \operatorname{int}(\gamma)$.

Theorem 10.3 Cauchy’s Theorem for simply connected domain

corollary of Proposition for simply connected domain

Let $\Omega$ be a simply connected domain, let $\gamma$ be a closed curve in $\Omega$. Then $\int_{\gamma} f(z) \, dz = 0$ for all $f\in O(\Omega)$.

Proof:

Know that is true if $\operatorname{ind}_{\gamma}(z)=0$ for all $z\in \mathbb{C}\setminus \Omega$.

By Proposition, $\Omega$ is simply connected $\iff$ every closed curve in $\Omega$ has winding number $0$ about every point in $\mathbb{C}\setminus \Omega$.

So the result is true.

QED

Theorem 10.4-6

The following condition are equivalent:

1. $\Omega$ is simply connected.
2. every holomorphic function on $\Omega$ has a primitive $g$, i.e. $g'(z)=f(z)$ for all $z\in \Omega$.
3. every non-vanishing holomorphic function on $\Omega$ has a holomorphic logarithm.
4. every harmonic function on $\Omega$ has a harmonic conjugate.

Proof:

$(1)\iff (2)$:

First we show $(1)\implies (2)$.

Assume $\Omega$ is simply connected.

Define $g(z)=\int_{z_0}^{z} f(w) \, dw$ for $z_0\in \Omega$ fixed. Then by Cauchy’s Theorem, this definition does not depend on the path.

$\frac{g(z+h)-g(z)}{h}=\frac{1}{h}\left[\int_{z}^{z+h} f(w) \, dw\right]$

$\frac{1}{h}\left[\int_{z}^{z+h} f(w) \, dw\right]\to f(z)$ as $h\to 0$.

So on $[z,z+h]\subset \Omega$, if $|f(w)-f(z)|<\epsilon$, then $|\frac{g(z+h)-g(z)}{h}-hf(z)|<h\epsilon$.

To show $(2)\implies (1)$, we prove $\neg (1)\implies \neg (2)$.

$(1)\iff (3)$:

If $\Omega$ is not simply connected, there is some closed curve $\gamma$ and some $z_0\not in \Omega$ such that $\operatorname{ind}_{\gamma}(z_0)\neq 0$.

SO $\int_{\gamma} \frac{1}{z-z_0} \, dz\neq 0$.

So $\frac{1}{z-z_0}$ does not have a primitive on $\Omega$. $\frac{1}{z-z_0}$ have no logarithm on $\Omega$.

This shows $(3)\implies (1)$.

Suppose $\Omega$ is simply connected.$f\in O(\Omega)$ and $f$ is non-vanishing. We want to show that $f$ has a logarithm on $\Omega$.

Let $z_0\in \Omega$ be fixed. And $a\in \log f(z_0)$.

$\frac{f'(z)}{f(z)}=g'$ Since $g\in O(\Omega)$, $we can assume$g(z_0)=a$.

$g(z)=a+\int_{z_0}^{z} g'(w) \, dw$

So

So $fe^{-g}=c$ for some $c\in \mathbb{C}$.

So $f=ce^g$

QED

Continue on Residue Theorem on Thursday.