% LaTeX file -- PM 352, Winter 2010, Assignment 6

\documentclass[11pt]{amsart}
\usepackage{amssymb, amstext, amscd, amsmath}
\usepackage{graphicx}  

\pagestyle{empty}
\topmargin 1cm
\textheight 9.2in
\textwidth 6.2in
\oddsidemargin .15in
\evensidemargin .15in
\voffset -1.5cm

%      Blackboard bold letters
\newcommand{\bA}{{\mathbb{A}}}
\newcommand{\bB}{{\mathbb{B}}}
\newcommand{\bC}{{\mathbb{C}}}
\newcommand{\bD}{{\mathbb{D}}}
\newcommand{\bE}{{\mathbb{E}}}
\newcommand{\bF}{{\mathbb{F}}}
\newcommand{\bG}{{\mathbb{G}}}
\newcommand{\bH}{{\mathbb{H}}}
\newcommand{\bI}{{\mathbb{I}}}
\newcommand{\bJ}{{\mathbb{J}}}
\newcommand{\bK}{{\mathbb{K}}}
\newcommand{\bL}{{\mathbb{L}}}
\newcommand{\bM}{{\mathbb{M}}}
\newcommand{\bN}{{\mathbb{N}}}
\newcommand{\bO}{{\mathbb{O}}}
\newcommand{\bP}{{\mathbb{P}}}
\newcommand{\bQ}{{\mathbb{Q}}}
\newcommand{\bR}{{\mathbb{R}}}
\newcommand{\bS}{{\mathbb{S}}}
\newcommand{\bT}{{\mathbb{T}}}
\newcommand{\bU}{{\mathbb{U}}}
\newcommand{\bV}{{\mathbb{V}}}
\newcommand{\bW}{{\mathbb{W}}}
\newcommand{\bX}{{\mathbb{X}}}
\newcommand{\bY}{{\mathbb{Y}}}
\newcommand{\bZ}{{\mathbb{Z}}}

%      Lower case bold letters
\newcommand{\Ba}{{\mathbf{a}}}
\newcommand{\Bb}{{\mathbf{b}}}
\newcommand{\Bc}{{\mathbf{c}}}
\newcommand{\Bd}{{\mathbf{d}}}
\newcommand{\Be}{{\mathbf{e}}}
\newcommand{\Bf}{{\mathbf{f}}}
\newcommand{\Bg}{{\mathbf{g}}}
\newcommand{\Bh}{{\mathbf{h}}}
\newcommand{\Bi}{{\mathbf{i}}}
\newcommand{\Bj}{{\mathbf{j}}}
\newcommand{\Bk}{{\mathbf{k}}}
\newcommand{\Bl}{{\mathbf{l}}}
\newcommand{\Bm}{{\mathbf{m}}}
\newcommand{\Bn}{{\mathbf{n}}}
\newcommand{\Bo}{{\mathbf{o}}}
\newcommand{\Bp}{{\mathbf{p}}}
\newcommand{\Bq}{{\mathbf{q}}}
\newcommand{\Br}{{\mathbf{r}}}
\newcommand{\Bs}{{\mathbf{s}}}
\newcommand{\Bt}{{\mathbf{t}}}
\newcommand{\Bu}{{\mathbf{u}}}
\newcommand{\Bv}{{\mathbf{v}}}
\newcommand{\Bw}{{\mathbf{w}}}
\newcommand{\Bx}{{\mathbf{x}}}
\newcommand{\By}{{\mathbf{y}}}
\newcommand{\Bz}{{\mathbf{z}}}

%      Capital script letters
  \newcommand{\A}{{\mathcal{A}}}
  \newcommand{\B}{{\mathcal{B}}}
  \newcommand{\C}{{\mathcal{C}}}
  \newcommand{\D}{{\mathcal{D}}}
  \newcommand{\E}{{\mathcal{E}}}
  \newcommand{\F}{{\mathcal{F}}}
  \newcommand{\G}{{\mathcal{G}}}
\renewcommand{\H}{{\mathcal{H}}}
  \newcommand{\I}{{\mathcal{I}}}
  \newcommand{\J}{{\mathcal{J}}}
  \newcommand{\K}{{\mathcal{K}}}  
\renewcommand{\L}{{\mathcal{L}}}
  \newcommand{\M}{{\mathcal{M}}}
  \newcommand{\N}{{\mathcal{N}}}
\renewcommand{\O}{{\mathcal{O}}}
\renewcommand{\P}{{\mathcal{P}}}
  \newcommand{\Q}{{\mathcal{Q}}}
  \newcommand{\R}{{\mathcal{R}}}
\renewcommand{\S}{{\mathcal{S}}}
  \newcommand{\T}{{\mathcal{T}}}
  \newcommand{\U}{{\mathcal{U}}}
  \newcommand{\V}{{\mathcal{V}}}
  \newcommand{\W}{{\mathcal{W}}}
  \newcommand{\X}{{\mathcal{X}}}
  \newcommand{\Y}{{\mathcal{Y}}}
  \newcommand{\Z}{{\mathcal{Z}}}

\newcommand{\FOR}{\text{ for }}
\newcommand{\FORAL}{\text{ for all }}
\newcommand{\AND}{\text{ and }}
\newcommand{\IF}{\text{ if }}
\newcommand{\OR}{\text{ or }}
\newcommand{\qand}{\quad\text{and}\quad}
\newcommand{\qor}{\quad\text{or}\quad}
\newcommand{\qfor}{\quad\text{for}\quad}
\newcommand{\qforal}{\quad\text{for all}\quad}
\newcommand{\qif}{\quad\text{if}\quad}

\newcommand{\Arg}{\operatorname{Arg}}
\newcommand{\cis}{\operatorname{cis}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\im}{\operatorname{Im}}
\newcommand{\re}{\operatorname{Re}}

\newcommand{\ep}{\varepsilon}


\newcommand{\rC}{\mathrm{C}}
\newcommand{\CT}{\mathrm{C}(\bT)}
\newcommand{\ol}{\overline}
\newcommand{\one}{\boldsymbol{1}}
\renewcommand{\phi}{\varphi}
\newcommand{\upchi}{{\raise.35ex\hbox{$\chi$}}}
\newenvironment{sbmatrix}{\left[\begin{smallmatrix}}{\end{smallmatrix}\right]}
\newcommand{\mt}{\emptyset}
\newcommand{\lip}{\langle}
\newcommand{\rip}{\rangle}
\newcommand{\ip}[1]{\lip #1 \rip}
\newcommand{\wh}{\widehat}

\newcommand{\dint}{\displaystyle\int}
\newcommand{\dlim}{\displaystyle\lim\limits}
\newcommand{\dsum}{\displaystyle\sum\limits}

\newcounter{exerci}[section]
\renewcommand{\theexerci}{\arabic{exerci}.}
\newenvironment{Ex}{\begin{list}%
 {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt%
 \labelwidth=12pt\labelsep=6pt\itemsep=3ex}}{\end{list}\medbreak}
\newenvironment{parts}{\vspace{1ex}\begin{enumerate}\renewcommand{\itemsep}{1ex}
 \renewcommand{\labelenumi}{(\alph{enumi})}}{\end{enumerate}}
\newcommand{\hint}{\textbf{Hint:} }

\begin{document}
\thispagestyle{empty}

%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 352 \hfill Assignment 6 \hfill Due Wednesday March 30.}} 

\vskip2em 

\begin{Ex}
\item 
\begin{parts}
\item Let $\Omega$ be a \textit{convex} region. 
Suppose that $f(z)$ is analytic on $\Omega$ and $\re f'(z) > 0$ for all $z\in\Omega$.
Prove that $f$ is one to one.\\
\hint find an expression for $\frac{f(z_2)-f(z_1)}{z_2-z_1}$ as an line integral.
\item Show by example that (a) may fail for non-convex regions.\\
\hint consider a power of $z$ on $\Omega = \bD \setminus (-1,0]$.
\end{parts}


\item Let $\Omega = \{ z : 0 < \re z < \pi,\ \im z > 0 \}$.
\begin{parts}
\item Find an explicit conformal map of $\Omega$ onto $\bH = \{z : \im z > 0 \}$.\\
\hint use a combination of an exponential, a M\"obius map and a power.
\item Express your map in terms of the cosine function.
 Hence show that $\cos z$ maps $\Omega$ conformally onto $-\bH$.
\end{parts}


\item Let $S = \{ z : |\re z|<1 \AND |\im z| < 1 \}$ be the unit square.
Let $f$ be a conformal map of $\bD = \{ z : |z|<1\}$ onto $S$ such that $f(0)=0$.
\begin{parts}
\item Prove that $f(iz) = if(z)$.  i.e. $f$ has four-fold symmetry.\\
\hint Let $g(z) = f(iz)$ and $h(z) = g^{-1}(if(z))$.  Apply Schwarz's Lemma to $h$.
\item Let the power series expansion of $f$ be $f(z) = \sum_{n=0}^\infty a_n z^n$.
Prove that $a_n = 0$ when $n \not\equiv 1 \pmod 4$.
\end{parts}


\item Show that there is no conformal map of the region $\Omega = \{z : 0 < |z| < 1\}$
onto any annulus $\bA_r = \{ z : 1 < |z| < r\}$ for $1<r<\infty$.\\
\hint such a map would have an isolated singularity at $0$. Remember that $f$ is open.
Where is $f(0)$?  What maps there?

\item  Let $z^{1/2}$ be the principle branch of the square root in 
$\bC\setminus\{iy:y\le 0\}$, the plane slit along the negative imaginary axis.
Define $f(z) = z^{1/2}(z-3)$.
\begin{parts}
\item Show that $f$ is analytic on $\bH=\{z : \im z > 0\}$, continuous on $\ol{\bH}$ and
$\dlim_{|z|\to\infty} |f(z)| = \infty$.\vspace{-1ex}
\item Find the image of the real axis.
\item Find $\Omega = f(\bH)$, and show that $f$ maps $\bH$ conformally onto $\Omega$.
\end{parts}

\item[A.\ \  \textbf{Bonus Problem. \ Please hand in separately.}]  \strut \newline
\begin{parts}
\item Generalize question 3 as much as you can.
\item Suppose that $\Omega$ is a simply connected region which is symmetric 
about the the real line $\bR$ (i.e. $z\in\Omega$ implies $\bar z \in \Omega$).
Prove that there is a conformal map of $\bD$ onto $\Omega$ which takes 
$(-1,1)$ onto $\Omega \cap \bR$.
\end{parts}

\end{Ex}
\end{document}