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\section{Complex Differentiation}

Goal: study the theory of complex-valued
differentiable functions in one complex variable.
\begin{enumerate}[(1)]
    \item $p(z) = a_d z^d + \cdots + a_1 z + a_0$
        polynomial, coefficients in $\RR$, $\QQ$,
        $\ZZ$, $\CC$.
    \item Recall computing the convergence of
        \[ \sum_{n = 1}^\infty \frac{1}{n^2},
        \quad \sum_{n = 1}^\infty \frac{1}{n^s},
        \quad s > 1 \]
        We could also consider this as a complex
        function in complex variable $s$.
    \item These functions are related to harmonic
        functions $u(x, y) \colon \RR^2 \to \RR$,
        $u_{xx} + u_{yy} = 0$.
\end{enumerate}

\begin{notation*}
    $z \in \CC$, $z = x + iy$, real, imaginary
    parts. \par\noindent
    $\ol{z}$ complex conjugate $\ol{z} = x - iy$.
    \\
    $|z|$, $\arg(z)$ or $\Arg(z)$.
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/9f89197698ab11ed.png}
    \end{center}
    $\theta$ with positive real axis, length of
    vector is $|z|$. $\theta = \arg(z)$, $\theta
    \in [0, 2\pi)$ then $\Arg(z)$.
\end{notation*}

\subsubsection*{Basic Notions}

\begin{itemize}
    \item 
        \begin{flashcard}[open-set-in-C]
            $\mathcal{U} \subset \CC$ is \emph{open} if
            \cloze{$\forall u \in \mathcal{U}$, $\exists \eps
            > 0$ such that
            \[ D(x, \eps) \defeq \{z \in \CC \colon |z -
            u| < \eps\} \subset \mathcal{U} .\]}
        \end{flashcard}
        (This is sometimes also written as
        $\mathbb{D}(x, \eps)$ or $B(x, \eps)$).
    \item
        \begin{flashcard}[path-in-C]
            a \emph{path} in $\mathcal{U} \subset
            \CC$ is \cloze{a continuous map $\gamma : [a, b]
            \to \mathcal{U}$, $\mathcal{C}'$ if
            $\gamma'$ exists and is
            \fcemph{continuous}.
            (one-sided derivatives at endpoints). \\
            $\gamma$ is \fcemph{\emph{simple}} if it is
            \fcemph{injective}.}
        \end{flashcard}
    \item
        \begin{flashcard}[path-connected-in-U]
            $\mathcal{U} \subset \CC$ is
            \emph{path-connected} if \cloze{$\forall z, w \in
            \mathcal{U}$ there exists path in
            $\mathcal{U}$ with endpoints at $z$, $w$.}
        \end{flashcard}
        \begin{remark*}
            If $\mathcal{U}$ is open, $z, w \in
            \mathcal{U}$ connected by a path
            $\gamma$ in $\mathcal{U}$, then
            $\exists$ path $\tilde{\gamma}$ in
            $\mathcal{U}$ connecting $z$ and $w$
            consisting of finitely many horizontal
            and vertical segments.
        \end{remark*}
\end{itemize}

\begin{flashcard}[domain]
\begin{definition*}[Domain]
    A \emph{domain} is \cloze{a
    \fcemph{non-empty}, \fcemph{open},
    \fcemph{path-connected} subset of $\CC$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[differentiable-holomorphic-entire]
\begin{definition*}
    \begin{enumerate}[(1)]
        \item $f \colon \mathcal{U} \to \CC$ is
            \emph{differentiable} \cloze{at $u \in
            \mathcal{U}$ if
            \[ f'(u) \defeq \lim_{z \to u}
            \frac{f(z) - f(u)}{z - u} \]
            exists.}
        \item $f : \mathcal{U} \to \CC$ is
            \emph{holomorphic} \cloze{at $u \in
            \mathcal{U}$ at $u \in \mathcal{U}$ if
            $\exists \eps > 0$ such that $f$ is
            differentiable at $z$ for all $z \in
            D(u, \eps)$ (``analytic'').}
        \item $f \colon \CC \to \CC$ is
            \emph{entire} \cloze{if it is holomorphic
            everywhere.}
    \end{enumerate}
\end{definition*}
\end{flashcard}

\begin{remark*}
    All differentiation rules (sum, product,
    quotient, inverse, chain, \ldots) hold, by the
    same proofs.
\end{remark*}

\noindent
Identifying $\CC$ with $\RR^2$ we may write $f
\colon \mathcal{U} \to \CC$ as $f(x + iy) = u(x,
y) + iv(x, y)$ where $u$ and $v$ are real and
imaginary parts of $f$.

\myskip
From Analysis and Topology: $u \colon \mathcal{U}
\to \RR$ as a function of two real variables is
($\RR^2$-)differentiable at $(c, d) \in \RR^2$
with $D u|_{(c, d)} = (\lambda, u)$ if
\[ \frac{u(x, y) - u(c, d) - [\lambda(x - c) +
\mu(y - d)]}{\sqrt{(x - c)^2 + (y - d)^2}} \to 0 \]
as $(x, y) \to (c, d)$.

\begin{flashcard}[Cauchy-Riemann]
\begin{proposition*}[Cauchy-Riemann equations]
    Let $f \colon \mathcal{U} \to \CC$ on an open
    set $\mathcal{U} \subset \CC$. Then $f$ is
    differentiable at $w = c + id \in \mathcal{U}$
    if and only if, \cloze{writing $f = u + iv$, we have
    $u$, $v$ ($\RR^2$-)\fcemph{differentiable} at $(c, d)$
    \emph{and}
    \[ \boxed{\begin{aligned}
        u_x
        &= v_y \\
        u_y
        &= -v_x
    \end{aligned}} \]}
    \fcscrap{``Cauchy-Riemann equations''. ($u_x =
    \pfrac{u}{x}$ and so on).}
\end{proposition*}
\end{flashcard}

\begin{proof}
    $f$ is differentiable at $w \iff f'(w) = p +
    iq$ exists
    \[ \iff \lim_{z \to w} \frac{f(z) - f(w) - (z
    - w)(p + iq)}{|z - w|} = 0 .\]
    Writing $f = u + iv$ and considering real,
    imaginary parts in the quotient above, this
    holds iff
    \[ \lim_{(x, y) \to (c, d)} \frac{u(x, y) -
    u(c, d) - [p(x - c) - q(y - d)]}{\sqrt{(x
    - c)^2 + (y - d)^2}} = 0 \]
    and
    \[ \lim_{(x, y) \to (c, d)} \frac{v(x, y) -
    v(c, d) - [q(x - c) + p(y - d)]}{\sqrt{(x
    - c)^2 + (y - d)^2}} = 0 \]
    This holds if and only if $u, v$ are
    ($\RR^2$-)differentiable at $(c, d)$ and $u_x
    = v_y$ and $u_y = -v_x$ holds.
\end{proof}

\begin{hiddenflashcard}[sufficient-condition-for-differentiable]
Sufficient condition for $f : \mathcal{U} \to \CC$
to be differentiable? \\
\cloze{If $u, v$ are \fcemph{continuously}
differentiable on $\mathcal{U}$ and Cauchy-Riemann
equations hold, then $f$ is differentiable on
$\mathcal{U}$.}
\end{hiddenflashcard}

\begin{remark*}[From Analysis and Topology]
    If the partials $u_x, u_y$ exist and are
    continuous on $\mathcal{U}$, then $u, v$ are
    differentiable on $\mathcal{U}$. So it
    suffices to check partials exist and are
    continuous and Cauchy-Riemann equations hold
    to deduce complex differentiability.
\end{remark*}

\subsubsection*{Examples}

\begin{enumerate}[(1)]
    \item $f(z) = \ol{z}$. $f$ has $u(x, y) = x$,
        $v(x, y) = -y$ so $u_x = 1$, $v_y = -1$.
        So $f(z) = \ol{z}$ is \emph{not}
        holomorphic or differentiable anywhere.
    \item Any polynomial $p(z) = a_d z^d + \cdots
        + a_1 z + a_0$ with $a_i \in \CC$ is
        entire (holomorphic everywhere).
    \item \emph{rational} functions: a quotient of
        polynomials $\frac{p(z)}{q(z)}$ is
        holomorphic on $\CC \setminus
        \{\text{zeroes of $q$}\}$.
\end{enumerate}

\begin{warning*}
    $f = u + iv$ satisfying Cauchy-Riemann
    equations at a point does not imply $f$
    differentiable; see Example Sheet 1.
\end{warning*}

\noindent
Exercise: Let $f \colon \mathcal{U} \to \CC$ on a
domain $\mathcal{U}$ with $f'(z) \equiv 0$ on
$\mathcal{U}$, then $f$ is constant on
$\mathcal{U}$. Sketch: use a nice path and the
mean value theorem.

\myskip
Why are we interested??
\begin{itemize}
    \item [structure] Unlike
        $\RR^2$-differentiable functions,
        holomorphic functions are very
        constrained: for example, if $f$ is
        entire and bounded (i.e. $|f(z)| \le M
        ~\forall z \in \CC$) then $f$ must be
        constant. (contrasts with $\sin$ for
        example over reals)
    \item [analycity] We'll see that $f$
        holomorphic on domain $\mathcal{U}$ has
        holomorphic derivative on $\mathcal{U}$.
        Hence $f$ is infinitely differentiable, as
        are $u, v$. Differentiating Cauchy-Riemann
        equations:
        \[ u_x = v_y \implies u_{xx} = v_{yx} =
        v_{xy} = -u_{yy} .\]
        So $u_{xx} + u_{yy} = 0$; similarly
        $v_{xx} + v_{yy} = 0$. The \emph{real} and
        imaginary parts of a holomorphic function
        are harmonic.
\end{itemize}
