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\begin{document}

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

\textbf{Midterm Exam}\quad \textsl{February 28, 2003}

\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.
A correct answer without supporting work will receive no credit.}
 
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 credit if your answers are difficult to read or your work is poorly
organized.

\bigskip
 
\begin{center}
\begin{minipage}{2in}
\begin{itemize}
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\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.}
Describe the images of the lines $x = c$ and $y = k$ under the mapping $w =
\frac{1}{z}$ and graph several of each.
\end{problem}

\begin{problem}{2.}
Show that if $D_1$ and $D_2$ are domains with a non-empty intersection,
then $D_1 \cup D_2$ is a domain.
\end{problem}

\begin{problem}{3.}
Solve the equation $z^4 +(1-2\sqrt{3}i)z^2 - (3 +\sqrt{3}i) = 0$. (Find
rectangular coordinates.)
\end{problem}

\begin{problem}{4.}
Show that
$\left| \int_\CC \frac{\overline{z}^2+1}{z^3 - 3z^2 - z + 3}\,dz \right| < 21$
where $\CC$ is the circle $|z| = 2$.
\end{problem}

\begin{problem}{5.}
Suppose that $f(z)$ is analytic in a domain $D$. Prove that if
$\overline{f(z)}$ is also analytic in $D$ then $f(z)$ is constant.
\end{problem}

\begin{problem}{6.}
Show that $u(x,y) = e^x(x\cos y - y\sin y)$ is harmonic and find its
harmonic conjugate.
\end{problem}

\begin{problem}{7.}
Prove the following
\begin{itemize}
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\item[a)] $\frac{d}{dz}e^z = e^z$
\item[b)] $\frac{d}{dz}\sin(z) = \cos(z)$
\end{itemize}
\end{problem}

\begin{problem}{8.}
Let $\CC$ be the contour given by the line segments from $0$ to $1+i$
and from $1+i$ to $1$. Evaluate $\int_\CC z^2\overline{z}\, dz$. (Find
rectangular coordinates.)
\end{problem}

\begin{problem}{9.}
Let $a$ and $b$ be non-zero real numbers. Evaluate $\int_{-b}^bz^{a-1}\,dz$
where the path is any contour from $-b$ to $b$ that, except for its
endpoints, lies above the real axis. (Find rectangular coordinates.)
\end{problem}

\begin{problem}{10.}
Evaluate $\int_\CC \frac{e^{\pi z}}{z^2+1}\,dz$ where $\CC$ is the circle
$|z+i| = 1$ oriented counterclockwise. (Find rectangular coordinates.)
\end{problem}

\end{document}
