% LaTeX file -- PM 352, Winter 2010, Assignment 1 \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{\dlim}{\displaystyle\lim\limits} \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} \newcounter{exerci}[section] \renewcommand{\theexerci}{\arabic{exerci}.} \newenvironment{Ex}{\begin{list}% {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt% \labelwidth=12pt\labelsep=6pt\itemsep=2ex}}{\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 1 \hfill Due Monday January 18.}} \vskip1em \begin{Ex} %\item Let $U$ be a non-empty open subset of $\bC$, and let $f$ and $g$ be analytic functions %on $U$. Prove that $h(z) = f(z)g(z)$ is analytic and derive the product rule. \item Let $U$ be a non-empty open subset of $\bC$, and let $f$ be an analytic function on $U$. Define $g$ on $\tilde U = \{ z : \ol{z} \in U \}$ by $g(z) = \ol{f(\ol{z})}$. Prove that $g$ is analytic.\\ \hint check the Cauchy-Riemann equations. \item Let $U$ be the region below. Suppose that $g$ is an analytic function on $U$ such that $g(z)^6 = z$ and $g(-1) = -i$. Find $g(i)$ and $g(3i/4)$ in Cartesian form. \begin{figure}[h] \centering \includegraphics[height = 2in]{fig1} \end{figure} \item \begin{parts} \item Compute $\frac{\partial}{\partial z} z$, $\frac{\partial}{\partial \bar z} z$, $\frac{\partial}{\partial z} \bar z$ and $\frac{\partial}{\partial \bar z} \bar z$. Hence find a formula for $\frac{\partial}{\partial z} (z^j \bar z^k)$ and $\frac{\partial}{\partial \bar z} (z^j \bar z^k)$. \item Show by induction that for $m,n \ge 0$, \[ \frac{\partial^m}{\partial z^m} \frac{\partial^n}{\partial \bar z^n} (z^j \bar z^k) = \begin{cases} \frac{j!k!}{(j-m)!(k-n)!} z^{j-m} \bar z^{k-n} &\IF m \le j \AND n \le k\\ 0 &\IF m > j \OR n > k. \end{cases} \] \item Let $p(z,\bar z) = \sum_{j,k = 0}^N a_{jk} z^j \bar z^k$ be a polynomial in $z$ and $\bar z$. Show that $\frac{\partial p}{\partial \bar z} \equiv 0$ (i.e. $p$ is analytic) if and only if $p = \sum_{j=0}^N a_{j0} z^j$ (i.e.\ no $\bar z$ terms at all). \end{parts} \item Define $\phi(z) =\dfrac{1-z}{1+z}$ for $z \ne -1$. \begin{parts} \item Show that $\phi$ is analytic on $\bC \setminus\{-1\}$ and that $\phi'(z)$ is never $0$. \item Solve $\phi(z) = w$. Deduce that $\phi(\phi(z)) = z$, and that $\phi$ maps $\bC \setminus\{-1\}$ one-to-one and onto itself. \item Show that $\phi$ maps the unit disk $\bD = \{ z \in \bC : |z|<1\}$ one-to-one and onto the right half plane $\bH = \{ z : \re z > 0 \}$. \hint take $z$ in polar form and compute $\re \phi(z)$. \end{parts} \item Let $\bA = \{ z \in \bC : \frac12 < |z| < 2 \}$, and let $f(z) = z + \frac1z$. \begin{parts} \item Show that $f$ is 2-to-1 except at the zeroes of $f'$. \item Show that the image of the circle of radius $r$ is an ellipse except for $r=1$. What is the image in this case? \hint An ellipse is the locus of $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ or, setting $c^2 = a^2-b^2$, the set $\{ z=x+iy \in \bC : |z-c| + |z+c| = a \}$. \item Show that $f(A)$ is the interior of an ellipse. \end{parts} \end{Ex} \end{document}