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\textbf{Math 425: Complex Variables}
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\textbf{Final Exam}\quad \textsl{May 6, 2003}

\bigskip

There are 10 problems worth 10 points each.

\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.}

Whenever an answer is a complex number, please give its rectangular
coordinates.
 
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\begin{center} \begin{minipage}{2in} \begin{itemize}
\setlength{\itemsep}{10pt} \item[\ ] \textbf{Score} \item[ 1.]
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\item[ 9.] \makebox[.5in][r]{\hrulefill} \item[10.]
\makebox[.5in][r]{\hrulefill} \item[\textbf{Total:}]
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\begin{problem}{1.}%
Find all solutions of the equation $\Big(z + \frac{1}{z}\Big)^2 = 1$.
\end{problem}

\begin{problem}{2.}%
Define $e^z$ and show that it is entire.
\end{problem}

\begin{problem}{3.}%
Show that $u(x,y) = x^3 - (3y^2+1)x$ is harmonic and find its harmonic
conjugate.
\end{problem}

\begin{problem}{4.}%
Let $\CC$ be the line segment from 1 to $-1+i$. Evaluate $\int_\CC
\overline{z}\,dz$.
\end{problem}

\begin{problem}{5.}%
Let $\CC$ be a contour from $-1$ to $1$ that lies below the $x$-axis,
except for its endpoints. Evaluate $\int_\CC\sqrt[3]{z}\,dz$.
\end{problem}

\begin{problem}{6.}%
Find an analytic function whose Taylor series is $\sum_{n=1}^\infty n^2z^n$.  
\end{problem}

\begin{problem}{7.}%
Derive the Laurent series expansion of $f(z) = \frac{1}{z^2-3z+2}$ in
powers of $z$ that is valid in the annulus $1<|z|<2$.
\end{problem}

\begin{problem}{8.}%
Evaluate $\int_\CC \frac{e^z}{z^3+z^4}\,dz$ where $\CC$ is the circle $|z| =
2$ oriented counterclockwise.
\end{problem}

\begin{problem}{9.}%
Use residues to calculate $\int_0^\infty \frac{\cos(x)}{x^2+1}\,dx$.
\end{problem}

\begin{problem}{10.}%
Determine the number of roots, counting multiplicities, of the equation
$z^5 - 3z^3 + 1 = 0$ in the annulus $1<|z|<2$.
\end{problem}

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