\documentclass{article}

\setlength{\textheight}{9in}
\setlength{\voffset}{-.75in}
\setlength{\textwidth}{6.5in}
\setlength{\hoffset}{-.75in}
\setlength{\parindent}{0pt}
\everymath={\displaystyle}
\renewcommand{\Re}{\ensuremath{\mathrm{Re}}}
\pagestyle{empty}

\newcommand{\problemspace}{\vspace*{18pt}}

\begin{document}
\thispagestyle{empty}

\textbf{Math 425: Complex Variables}

\textbf{Final Exam}\quad \textsl{May 6, 2002}

\problemspace

\emph{Part I:} Carefully state the following Theorems.

\begin{itemize}

\item \textsc{Cauchy's Theorem}

\item \textsc{Liouville's Theorem}

\item \textsc{The Residue Theorem}

\item \textsc{The Maximum modulus Principle}

\item \textsc{The Argument Principle}

\end{itemize}

\bigskip

\emph{Part II:} Solve the following problems.

\bigskip

\begin{enumerate}

\item Find the real and imaginary parts of $(1+i)^{100}$.

\problemspace

\item Calculate the sum $\sum_{n=1}^\infty n\frac{i^n}{2^{n+1}}$.

\problemspace

\item Show that $u(x,y) = x^2 - y^2 + 2x + y$ is harmonic and then find an
analytic function $f(z)$, $z = x + iy$, such that $\Re(f(z)) = u(x,y)$.

\problemspace

\item Suppose $f(z) = \frac{1}{z^2 + 4}$ is developed in a power series
around $z = 3$. Find its radius of convergence.

\problemspace

\item Let $\gamma(t) = t + i\sin(t)$, $0 \le t \le \pi$. Compute
$\int_\gamma \frac{z}{z^2+1}\,dz$.

\problemspace

\item  Let $\gamma$ be the circle of radius 2 centered at~$-1$. Compute
$\int_\gamma \frac{e^z}{z+1}\,dz$.

\problemspace

\item Find the Laurent series for $f(z) = \frac{\log(1+z)}{z^3}$ around
$z=0$.

\problemspace

\item Calculate $\int_0^\infty \frac{x^2}{(x^2+1)^2}\,dx$.

\problemspace

\item Find the smallest positive integer $n$ so that all zeros of
$p(z) = 3z^3 + 2z^2 + z + 7$ lie in the disk $|z| \le n$.

\problemspace

\item Find a fractional linear transformation that maps the half plane
$\Re(z) > 1$ into the unit disk $|z| < 1$.

\problemspace

\end{enumerate}

\end{document}
