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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\i{{\mathrm i}}
\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\D{{\mathbb D}}
\def\Re{\operatorname{Re}}
\def\Im{\operatorname{Im}}
\def\interior{\operatorname{int}}

\begin{document}

\title{Complex Analysis}
\author{Homework 7}
\date{Due in class Thursday, November 11, 2021}
\maketitle

\begin{enumerate}

\item (Cf.\ Stein \& Shakarchi, Chapter 2 Exercise 12.)  

Suppose that $u \colon \D \to \R$ is harmonic, i.e., $u$ is twice
continuously differentiable as a function of two real variables and
satisfies
\[
  \Delta u
  = \frac{\partial^2 u}{\partial x^2}
  + \frac{\partial^2 u}{\partial y^2} = 0 \,.
\]
Show that there exists a holomorphic function $f$ on $\D$ such that $u
= \Re f$ and $\Im f$ is uniquely defined up on an additive real
constant.

\emph{Hint:} Show that if $f=u+\i v$ were holomorphic, then
$f'(z) = 2 \, \partial u /\partial z$.  Thus, if you can show that
$\partial u /\partial z$ is holomorphic, then you can recover $f$ via
its primitive (explain in detail!).

\item (Stein \& Shakarchi, Chapter 8 Exercise 1.)  A holomorphic
mapping $f \colon U \to V$ is a local bijection on $U$ if for every
$z \in U$ there exists an open disc $D \subset U$ centered at $z$, so
that $f \colon D \to f (D)$ is a bijection.  Prove that a holomorphic
map $f \colon U \to V$ is a local bijection on $U$ if and only if
$f'(z) = 0$ for all $z \in U$.


\emph{Hint:} Modify the Rouch\'e theorem argument shown in class.

\item (Stein \& Shakarchi, Chapter 8 Exercise 2.)  Supppose $f (z)$ is
holomorphic near $z = 0$ and $f (0 ) = f' (0 ) = 0$, while
$f'' (0 ) \neq 0$. Show that there are two curves $\gamma_1$ and
$\gamma_2$ that pass through the origin, are orthogonal at the origin,
and so that $f$ restricted to $\gamma_1$ is real and has a minimum at
$0$, while $f$ restricted to $\gamma_2$ is also real but has a maximum
at $0$.

\emph{Hint:} Write $f (z) = (g(z))^2$ for $z$ near $0$, and consider
the mapping $z \mapsto g(z)$ and its inverse.

\item (Stein \& Shakarchi, Chapter 8 Exercise 5.)  Prove that $f (z) = -\tfrac12\, (z + 1/z)$ is a conformal map from the half-disc
$\{z = x + \i y \colon \lvert z \rvert < 1, y > 0\}$ to the upper
half-plane.

\emph{Hint:} The equation $f (z) = w$ reduces to the quadratic
equation $z^2 + 2wz + 1 = 0$, which has two distinct roots in $\C$
whenever $w = \pm 1$. This is certainly the case if $w \in \mathbb H$.

\end{enumerate}



\end{document}
