% LaTeX file -- PM 352, Winter 2010, Assignment 3

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

\pagestyle{empty}
\topmargin 1cm
\textheight 9.2in
\textwidth 6in
\oddsidemargin .25in
\evensidemargin .25in
\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 3 \hfill Due Friday February 12.}} 

\vskip2em 

\begin{Ex}

\item Suppose that $T$ is a M\"obius map which takes $\bR$ to itself and sends $\infty$ to 0.
\begin{parts}
\item What is the image of the family of lines parallel to $\bR$?
\item What is the image of the family of lines perpendicular to $\bR$?
\end{parts}


\item 
\begin{parts}
\item Show that a M\"obius map takes $\bD = \{ z : |z|<1 \}$ onto itself if and only if
it has the form $Tz = e^{i\theta} \frac{z-a}{1-\ol{a}z}$ for some $a \in \bD$ and $\theta\in\bR$.
\item Suppose that $\C_1$ and $\C_2$ are two disjoint circles in $\bC$.  Show that there
is a M\"obius map $T$ so that $T\C_1$ and $T\C_2$ are concentric.\\
\hint first map $\C_1$ onto the unit circle so that $T\C_2$ is inside the unit disk
with its centre on the real axis.  
Then use a M\"obius map from (a) which also maps $\bR$ to $\bR$.
Remember that the centre of a circle is not preserved by these maps, so you need 
another way to determine whether the circles are concentric.
\end{parts}


\item \begin{parts}
\item If the Riemann sphere $S = \{(x,y,z) : x^2+y^2+z^2=1\}$ is rotated 
one quarter turn about the $y$-axis so that the north pole goes to $(1,0,0)$, 
what is the corresponding M\"obius map $T$ on $\bC$ induced by the 
sterographic projection of this motion of the sphere?
\item Let $Q$ be the square with vertices $\pm1\pm i$.  
Find the image of $Q$ under the map $T$.
\end{parts}


\item  Let $a,b$ be positive real numbers.
Let $\gamma(t) = a \cos t + i b \sin t$ for $0 \le t \le 2\pi$ be an ellipse.
Evaluate $\displaystyle\oint_\gamma \frac{dz}{z}$ in two ways, and deduce that
$\dint_0^{2\pi} \frac{dt}{a^2\cos^2 t + b^2 \sin^2 t} = \frac{2\pi}{ab}$.


\item Suppose that $f(z)$ is analytic in an open set containing the closed ball $\ol{B_r(p)}$.
Let $\gamma$ denote the boundary circle of $\ol{B_r(p)}$ oriented anticlockwise.
Show that for $a \in B_r(p)$ that
\[ f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^{n+1}}\,dz .\]
\hint use the power series expansion to split $\frac{f(z)}{(z-a)^{n+1}}$ into
the first few terms plus an analytic piece.

\item Let $f(z)$ be an entire function.  Suppose that there are real constants $A$ and $B$
so that $|f(z)| \le A + B |z|^n$ for all $z \in \bC$.
Prove that $f$ is a polynomial of degree at most $n$.


%\item Let $R>1$. Suppose that $f(z)$ is continuous on $B_R(0)$
%and analytic on $B_R(0)\setminus [-1,1]$.  Prove that $f$ is analytic on $B_R(0)$. 
%\hint use Morera's theorem.  Estimate the integral around a rectangle crossing the real axis by
%splitting the curve into three rectangles, where one is very narrow with both long edges 
%very close to the real axis.


\item[A.\ \  \textbf{Bonus Problem. \ Please hand in separately.}]  \strut \newline
Let $U$ be an open subset of the upper half plane such that $\ol{U} \cap \bR = [a,b]$.
Suppose that $f$ is analytic on $U$ and extends to be a continuous function on
$U\cup (a,b)$ taking \textit{real} values on $(a,b)$.
Prove that there is an analytic function $\tilde f$ defined on $U\cup (a,b) \cup V$, 
where $V = \{ \bar z : z \in U \}$, such that $\tilde f|_U = f$.\\


\end{Ex}
\end{document}