3.8 KiB
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.
\begin{aligned}
[a,b]&=\bigcup_{s\in[a,b]}(s-\epsilon(s),s+\epsilon(s))\cup[a,a+\epsilon(a))\cup(b-\epsilon(b),b] \\
&=\bigcup_{j=1}^n(s_j-\epsilon(s_j),s_j+\epsilon(s_j))\cup[a,a+\epsilon(a))\cup(b-\epsilon(b),b]
\end{aligned}
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.
\begin{aligned}
\Delta(\log f,\gamma)&=\psi(b)-\psi(a) \\
&=\int_a^b\frac{d}{dt}\log f(\gamma(t))dt \\
&=\int_a^b\frac{f'(\gamma(t))\gamma'(t)}{f(\gamma(t))}dt \\
&=\int_\gamma\frac{f'(z)}{f(z)}dz
\end{aligned}
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.
\Gamma=\sum_{j=1}^p n_j\gamma_j
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.