Math416 Lecture 21

Chapter 9: Generalized Cauchy’s Theorem

Simple connectedness

Proposition 9.1

Let $\phi$ be a continuous nowhere vanishing function from $[a,b]\subset\mathbb{R}$ to $\mathbb{C}\setminus\{0\}$. Then there exists a continuous function $\psi:[a,b]\to\mathbb{C}$ such that $e^{\psi(t)}=\phi(t)$ for all $t\in[a,b]$.

Moreover, $\psi$ is uniquely determined up to an additive integer multiple of $2\pi i \mathbb{Z}$.

Proof:

Uniqueness:

Suppose $\phi_1$ and $\phi_2$ are both continuous functions so that $e^{\phi_1(t)}=\phi(t)=e^{\phi_2(t)}$ for all $t\in[a,b]$.

Then $e^{\phi_1(t)-\phi_2(t)}=1$ for all $t\in[a,b]$. So $\phi_1(t)-\phi_2(t)=2k\pi i$ for some $k\in\mathbb{Z}$.

Existence:

Case 1: Assume $range(\phi)\subset H$ where $H$ is an open half-plane with the origin $0\in \partial H$.

We know there is a branch $l(z)$ of $\log z$ defined on $H$ with $Log(z)=\log|z|+i\theta(z)$ for some $\arg(z)\in(\alpha,\alpha+\pi)$.

Let $\psi(t)=l(\phi(t))$.

Then $e^{\psi(t)}=e^{l(\phi(t))}=\phi(t)$. and $\psi$ is continuous.

Case 2: By compactness of $[a,b]$, there exists a partition $a=t_0<t_1<\cdots<t_n=b$ such that, for each $0\leq j\leq n-1$, $\phi([t_j,t_{j+1}])$ is contained in some open half plane $H_j$ with the origin $0\in \partial H_j$.

Recall:

Compactness: A set is compact if and only if every open cover has a finite subcover.

Let $s\in [a,b]$ and there exists $\epsilon(s)>0$ such that $\phi((s-\epsilon(s),s+\epsilon(s)))$ is contained in some open half plane.

We choose $t_j\in[s_j-\epsilon(s_j),s_j+\epsilon(s_j)]\cup[s_{j+1}-\epsilon(s_{j+1}),s_{j+1}+\epsilon(s_{j+1})]$ for each $j=1,\cdots,n-1$.

On each interval $[t_j,t_{j+1}]$, we can find a $\psi_j(t)$ such that $e^{\psi_j(t)}=\phi(t)$, $\psi_j(t)$ is continuous on $[t_j,t_{j+1}]$. And we can choose $\psi_{j+1}(t_{j+1})=\psi_j(t_{j+1})$.

Defined $\psi(t)=\{\psi_j(t), t\in[t_j,t_{j+1}]\}$ for $j=1,\cdots,n-1$.

QED

Increment of a log and argument

If $f\circ\gamma:[a,b]\to\mathbb{C}\setminus\{0\}$ is continuous, then $\exists \psi:[a,b]\to\mathbb{C}$ such that $e^{\psi(t)}=f(\gamma(t))$ for all $t\in[a,b]$.

We defined the increment in $\log f$ on $\gamma$ as $\Delta(\log f,\gamma)=\psi(b)-\psi(a)$.

The increment in $\arg f$ on $\gamma$ is defined as $\Delta(\arg f,\gamma)=Im[\psi(b)]-Im[\psi(a)]$.

If $\gamma$ is a closed curve, then $f\circ\gamma(a)=f\circ\gamma(b)$. Then $\Delta(\log f,\gamma)\in 2\pi i\mathbb{Z}$, $\Delta(\arg f,\gamma)\in 2\pi\mathbb{Z}$.

Assume $\gamma$ is piecewise continuous and $f$ is continuous and $f(z)\neq 0$ for all $z\in\gamma$.

If $\gamma$ is closed, then $\Delta(\log f,\gamma)=\int_\gamma\frac{f'(z)}{f(z)}dz=0$, $\Delta(\arg f,\gamma)=\frac{1}{i}\int_\gamma\frac{f'(z)}{f(z)}dz=0$.

Special case:

When $f(z)=z-z_0$, $z_0\notin range(\gamma)$, then $\Delta(\arg (z-z_0),\gamma)\in 2\pi\mathbb{Z}$.

The winding number of $\gamma$ around $z_0$ is defined as $n(\gamma,z_0)=\frac{1}{2\pi i}\Delta(\arg (z-z_0),\gamma)$.

also the same as the number of times $\gamma$ winds around $z_0$ counterclockwise.

Winding number is always zero outside the curve.

Contour

A contour is a formed piecewise combination of piecewise continuous closed curves with integer coefficients.

where $\gamma_j$ are piecewise continuous closed curves and $n_j\in\mathbb{Z}$.

A contour is called a simple if the winding number of $\Gamma$ is zero or one.