\documentclass[12pt,reqno,oneside]{amsart}

\pagestyle{myheadings}
\markright{\hfill\protect\makebox[2.5in][l]{Name:\protect\rule{2in}{.5pt}}}

\setlength{\parindent}{0pt}
\setlength{\parskip}{10pt}
\setlength{\textheight}{9in}
\setlength{\voffset}{-.75in}
\setlength{\textwidth}{6.5in}
\setlength{\hoffset}{-.75in}
\everymath{\displaystyle}

\newenvironment{problem}[1]{\leavevmode\llap{{#1}~}}{\vfil}
%\newenvironment{problem}[1]{\leavevmode\llap{{#1}~}}{\newpage}
\newcommand{\C}{\ensuremath{\mathbb{C}}}
\newcommand{\CC}{\ensuremath{\mathcal{C}}}
\newcommand{\R}{\ensuremath{\mathbb{R}}}

\begin{document}

\textbf{Math 425: Complex Variables}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Midterm Exam}\quad \textsl{February 27, 2004}

\bigskip

There are 10 problems on pages 2--11. Each problem is worth 10 points.
You must work on the exam entirely on your own. You are not allowed to
discuss the exam with anyone until after they are turned in. You may
use your textbook and a calculator. \emph{To receive credit, you must
show all your work and include all important steps and explanations.}
 
Please work out the problems on separate sheets and then copy your work
and answers neatly into the space below the appropriate problem. You will
lose points if your answers are difficult to read or your work is poorly
organized.

Whenever an answer is a complex number, please give its rectangular
coordinates.

\bigskip
 
\begin{center}
\begin{minipage}{2in}
\begin{itemize}
\setlength{\itemsep}{5pt}
\item[\ ] \textbf{Score}
\item[ 1.] \makebox[.5in][r]{\hrulefill}
\item[ 2.] \makebox[.5in][r]{\hrulefill}
\item[ 3.] \makebox[.5in][r]{\hrulefill}
\item[ 4.] \makebox[.5in][r]{\hrulefill}
\item[ 5.] \makebox[.5in][r]{\hrulefill}
\item[ 6.] \makebox[.5in][r]{\hrulefill}
\item[ 7.] \makebox[.5in][r]{\hrulefill}
\item[ 8.] \makebox[.5in][r]{\hrulefill}
\item[ 9.] \makebox[.5in][r]{\hrulefill}
\item[10.] \makebox[.5in][r]{\hrulefill}
\item[\textbf{Total:}] \makebox[.5in][r]{\hrulefill}
\end{itemize}
\end{minipage}
 
\end{center}
 
\newpage

\begin{problem}{1.}Find all roots of the polynomial $p(z) = z^8 + z^4 + 1$
and plot their locations in the complex plane.
\end{problem}

\begin{problem}{2.}Show that the intersection of two domains in the complex
plane is an open set, but not necessarily a domain.
\end{problem}

\begin{problem}{3.}Define the analytic function $\tan(z)$, describe its
domain, determine its real and imaginary parts, and derive a formula for
its complex derivative.
\end{problem}

\begin{problem}{4.}Find all solutions of the equation $z^i = 1+i$.
\end{problem}

\begin{problem}{5.}Suppose that $w = f(z)$ is analytic in a domain $D$ and
that the range of $f$ is contained in a line in the $w$-plane. Prove that
$f$ is constant.
\end{problem}

\begin{problem}{6.}Suppose that $f(z) = u(x,y) + iv(x,y)$ is analytic in
$\C\setminus\{0\}$ with $u(x,y) = \frac{xy}{(x^2+y^2)^2}$. Find $f'(z)$.
\end{problem}

\begin{problem}{7.}Evaluate the contour integral $\int_\CC
(z+1)\overline{z}\,dz$, where $\CC$ is defined by $z(t) = t + it^2$, $0 \le
t \le 1$.
\end{problem}

\begin{problem}{8.}Show that $\left|\int_\CC \frac{\cos(\overline{z})}{z^2
- 3z + 2}\,dz\right| < 115$, where $\CC$ is the circle $|z| = 4$.
\end{problem}

\begin{problem}{9.}Evaluate $\int_1^i z^{2/3}\,dz$ where the path is any
contour from 1 to $i$ that, except for its endpoints, lies \emph{outside}
the first quadrant.
\end{problem}

\begin{problem}{10.}Use the Cauchy Integral Formula to evaluate $\int_\CC
\frac{1}{z^4-16}\,dz$ where $\CC$ is the circle $|z-i|=2$.

\end{problem}

\end{document}
