\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\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 8}
\date{Due in class Thursday, November 18, 2021}
\maketitle

\begin{enumerate}

\item Recall from class that $f$ is an automorphism of the unit disc
$\D$ if and only if there exist $\theta \in \R$ and $\alpha \in \D$
such that
\[
  f(z) = \e^{\i \theta} \, \frac{\alpha -z}{1-\overline \alpha z} \,.
\]
Pulling back this automorphism of the disc to the upper half plane via
\[
  \psi = F^{-1} \circ f \circ F \,,
\]
where $F \colon \mathbb H \to \D$ is the conformal map
\[
  F(z) = \frac{\i - z}{\i + z} \,,
  \qquad
  F^{-1} (w) = \i \, \frac{1-w}{1+w} \,,
\]
show that
\[
  \psi(z) = \frac{az+b}{cz+d}
\]
where $a,b,c,d$ are real and satisfy $ad-bc=1$.

\emph{Hint:} The computations are tedious, you may want to use a
computer algebra system.  Alternatively, look at the hints in Stein \&
Shakarchi, Chapter 8, Exercise 16 which provide a structured way of
doing this argument by hand.

\item Show that the exponential $f(z) = \e^z$ is a conformal map from
the strip
\[
  \Omega = \{ 0 < \Im z < b \}
\]
to a sector
\[
  S = \{ 0 < \operatorname{arg} z < b \}
\]
provided $0<b<2\pi$.

\item Show that the Joukowski map
\[
  f(z) = \frac12 \, \biggl( z + \frac1z \biggr)
\]
is a conformal map from the exterior of the unit disk $\{z \colon
\lvert z \rvert > 1\}$ to the exterior of the segment of the real axis
$[-1,1]$.


\item (Stein \& Shakarchi, Chapter 8 Exercise 11.)  
Show that if $f \colon D_R(0) \to \C$ is holomorphic, with $|f (z)|
\leq M$ for some $M > 0$,
then
\[
 \biggl| \frac{f (z) - f (0)}{M^2 - \overline{f(0)} \, f(z)}
 \biggr| \leq \frac{\lvert z \rvert}{MR} \,.
\]
\emph{Hint:} Use the Schwarz lemma.

\end{enumerate}



\end{document}
