% vim: tw=50
% 13/02/2023 10AM

\begin{flashcard}[global-maximum-principle]
\begin{corollary*}[Global Maximum principle]
    \cloze{
    Let $\mathcal{U} \subseteq \CC$ be a bounded
    domain, and let $\ol{\mathcal{U}}$ be its
    closure (the closure of $\mathcal{U}$ is the
    intersection of all closed supersets of
    $\mathcal{U}$). If $f : \ol{\mathcal{U}} \to \CC$
    is continuous and $f$ is holomorphic on
    $\mathcal{U}$, then $|f|$ attains its maximum
    on $\ol{\mathcal{U}} \setminus \mathcal{U}$.
}
\end{corollary*}

\begin{proof}
    \cloze{
    $\mathcal{U}$ bounded implies
    $\ol{\mathcal{U}}$ is bounded, hence $|f|$ has
    a maximum on $\mathcal{U}$, call it $M$. If
    $|f(z_0)| = M$ for $z_0 \in \mathcal{U}$, then
    local maximum principle implies $f \equiv
    f(z_0)$ on any disk  $D(z_1, r) \subseteq
    \mathcal{U}$. By identity theorem, $f \equiv
    f(z_0)$ on $D(z_0, r)$ hence $f \equiv f(z_0)$
    on $\mathcal{U}$, hence $f \equiv f(z_0)$ on
    $\ol{\mathcal{U}}$. So $M$ is achieved by
    $|f|$ on $\ol{\mathcal{U}} \setminus
    \mathcal{U}$.
}
\end{proof}
\end{flashcard}

\subsubsection*{Generalise Cauchy's Integral Formula}

Goal: generalise CIF by allowing more general
closed curves for the integration. We have an
issue:
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/bed54580ab8811ed.png}
\end{center}
Then
\[ \int_{\gamma_2} f = 2 \int_{\gamma_1} f \]
We need to deal with the issue of ``winding
around'' a point more than once; however, once we
correctly quantify this notion, we'll see it is
the \emph{only} issue to generalising CIF.

\myskip
Na\"ive hope: ``counting'' crossings of a slit in
the plane:
\begin{center}
    \includegraphics[width=0.3\linewidth]
    {images/1a56896eab8911ed.png}
\end{center}
can happen infinitely often!

\begin{flashcard}[existence-of-r-theta-decomposition]
\begin{theorem*}
    Let $\gamma : [a, b] \to \CC \setminus \{w\}$
    be a continuous curve. Then there exists
    continuous function $\theta : [a, b] \to \RR$
    with
    \[ \gamma(t) = w + r(t)e^{i\theta t} \]
    with $r(t) = |\gamma(t) - w|$.
\end{theorem*}

\begin{proof}
    \cloze{
    WLOG translate to assume $w = 0$. Since $\arg
    \gamma(t) = \arg
    \frac{\gamma(t)}{|\gamma(t)|}$, we can replace
    $\gamma$ with $\frac{\gamma}{|\gamma|}$ to
    assume $|\gamma(t)| = 1$ for all $t \in [a,
    b]$.

    \myskip
    Notice that if $\gamma \subseteq \CC \setminus
    \RR_{\le 0}$, then $t \mapsto \Arg(\gamma(t))$
    gives a continuous choice of $\theta$. More
    generally, if $\gamma$ lies in any slit plane
    $\CC \setminus \left\{ z :
    \frac{z}{e^{-\alpha}} \in \RR_{\le 0}
    \right\}$, then $\theta(t) \defeq \alpha + \Arg
    \left( \frac{z}{e^{i\alpha}} \right)$ will do.

    \myskip
    Our strategy is to subdivide $\gamma$ so that
    its pieces lie in slit-planes, and so $\theta$
    may be continuously defined on the pieces.
    \begin{center}
        \includegraphics[width=0.3\linewidth]
        {images/0d3a9f9eab8a11ed.png}
    \end{center}
    $\gamma$ is continuous on $[a, b]$, so uniform
    continuous, and $\exists \eps > 0$ such that
    $|s - t| < \eps \implies |\gamma(s) -
    \gamma(t)| < 2$. Subdividing $a = a_0 < a_2 <
    \cdots < a_{n - 1} < a_n = b$ with $a_{j + 1}
    - a_j < 2\eps$, then
    \[ \left| \gamma(t) - \gamma \left( \frac{a_{j
    + 1} - a_j}{2} \right) \right| < 2 \qquad
    \forall t \in [a_j, a_{j + 1}] \]
    So $\gamma([a_{j - 1}, a_j])$ lies in a slit
    plane, and we can define $\theta_j$ a
    continuous choice of argument for $\gamma
    |_{[a_{j - 1}, a_j]}$ for $j \in \{1, \ldots,
    n\}$. We have
    \[ \gamma(a_j) = e^{i\theta_j(a_i)} =
    e^{i\theta_{j + 1}(a_j)} \]
    for $j \in \{1, \ldots, n - 1\}$. So
    \[ \theta_{j + !}(a_j) = \theta_j(a_j) + 2\pi
    n_j \]
    for some $n _j \in \ZZ$. Modifying each
    $\theta_j$, $j \ge 2$, by a suitable integer
    multiple of $2\pi$ ensures the $\theta_j$ fit
    together to a continuous choice of $\theta$ on
    $[a, b]$.
}
\end{proof}
\end{flashcard}

\begin{remark*}
    $\theta$ is not unique, since $\theta(t) +
    2\pi n$ is also valid for all $n \in \ZZ$. If
    $\theta_1, \theta_2$ are two functions as in
    the theorem, then $\theta_1 - \theta_2$ is
    continuous, takes values in (discrete) $2\pi
    \ZZ$, so constant.
\end{remark*}

\begin{flashcard}[winding-number]
\begin{definition*}[Winding Number]
    \cloze{
    Let $\gamma : [a, b] \to \CC$ be a closed
    curve, $w \not\in \gamma$. The \emph{winding
    number} or \emph{index} of $\gamma$ about $w$
    is
    \[ I(\gamma; w) \defeq \frac{\theta(b) -
    \theta(a)}{2\pi} ,\]
    where $\gamma(t) = w + r(t)e^{i\theta(t)}$
    with $\theta$ continuous.
}
\end{definition*}
\end{flashcard}

\begin{flashcard}[winding-number-integral-formula]
\begin{lemma*}[Winding Number Integral Formula]
    \cloze{
    Let $\gamma : [a, b] \to \CC \setminus \{w\}$
    be a closed curve. Then
    \[ I(\gamma; w) = \frac{1}{2\pi i} \int_\gamma
    \frac{\dd z}{z - w} \]
}
\end{lemma*}

\begin{proof}
    \cloze{
    $\gamma$ piecewise-$C^1$ implies $r(t)$ and
    $\theta(t)$ are piecewise-$C^1$ as well, where
    $\gamma(t) = w + R(t) e^{i\theta(t)}$.
    \begin{align*}
        \int_\gamma \frac{\dd z}{z - w}
        &= \int_a^b \frac{\gamma'(t)}{\gamma(t) -
        w} \dd t \\
        &= \int_a^b \frac{r'(t)}{r(t)} +
        i\theta'(t) \dd t \\
        &= [\ln r(t) + i\theta(t)]_{t = a}^{t = b}
        \\
        &= 2\pi i I(\gamma; w)
    \end{align*}
    since $\gamma$ is closed and $\theta(b) -
    \theta(a) = 2\pi I (\gamma; w)$.
}
\end{proof}
\end{flashcard}

\begin{proposition*}
    If $\gamma : [0, 1] \to D(a, R)$ is a closed
    curve, then $\forall w \not\in D(a, R)$,
    $I(\gamma; w) = 0$.
\end{proposition*}

% .image

\begin{proof}
    Consider the M\"obius map $z \mapsto \frac{z -
    w}{a - w}$, This takes $a \mapsto 1$, $w
    \mapsto 0$, so $D(a, R) \mapsto D(1, r)$ for
    some $r < 1$. So then $D(a, R)$ is contained
    in the slit plane $\CC \setminus \left\{ z :
    \frac{z - w}{a - w} \in \RR_{\le 0} \right\}$.
    So there is a branch of $\arg(z - w)$ defined
    on $D(a, r)$. And so
    \[ I(\gamma; w) = \frac{\arg(\gamma(1) - w) -
    \arg(\gamma(0) - w)}{2\pi} = 0 \qedhere \]
\end{proof}

\begin{flashcard}[homologous-to-zero]
\begin{definition*}[Homologous to zero]
    \cloze{
    Let $\mathcal{U} \subseteq \CC$ be open. Then
    a closed curve $\gamma$ in $\mathcal{U}$ is
    \emph{homologous to zero in $\mathcal{U}$} if
    $\forall w \not\in \mathcal{U}$, $I(\gamma; w) =
    0$.
}
\end{definition*}
\end{flashcard}

\begin{flashcard}[simply-connected]
\begin{definition*}[Simply Connected]
    \cloze{
    $\mathcal{U}$ is \emph{simply connected}
    if every closed curve in $\mathcal{U}$ is
    homologous to zero.
}
\end{definition*}
\end{flashcard}

\begin{remark*}
    For $\mathcal{U}$ open this is equivalent to
    the homotopy definition of simply connected.
\end{remark*}

\begin{enumerate}[(1)]
    \item Any disk is simply connected by previous
        proposition.
    \item Any punctured disk $D(a, R) \setminus
        \{a\}$ is not simply connected, since
        curves can wind around $a$:
        \begin{center}
            \includegraphics[width=0.4\linewidth]
            {images/ae399af4ab8e11ed.png}
        \end{center}
\end{enumerate}

\begin{flashcard}[general-CIF]
\begin{theorem*}[General CIF]
    \cloze{
    Let $f : \mathcal{U} \to \CC$ be holomorphic
    on a domain $\mathcal{U}$, and $\gamma$ is a
    closed curve homologous to zero in
    $\mathcal{U}$. Then $\forall w \in \mathcal{U}
    \setminus \gamma$,
    \[ I(\gamma; w) f(w) = \frac{1}{2\pi i}
    \int_\gamma \frac{f(z)}{z - w} \dd z ,\]
    and $\int_\gamma f(z) \dd z = 0$
}
\end{theorem*}
\end{flashcard}