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\textbf{Math 425: Complex Variables}
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\textbf{Final Exam}\quad \textsl{May 5, 2004}

\bigskip

There are 10 problems, each worth 10 points.

In definitions and theorems, all terms must be clearly defined and all
relevant hypotheses and conclusions clearly stated.

In calculations, all important steps and explanations must be included.
 
Whenever an answer is a complex number, it should be given in rectangular
coordinates.

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\begin{problem}{1.}State the Cauchy-Riemann equations and explain their
significance.
\end{problem}

\begin{problem}{2.}Give a complete definition of the complex analytic
function $\log(z)$ and show that $\frac{d}{dz}\log(z) = \frac{1}{z}$.
\end{problem}

\begin{problem}{3.}State Cauchy's Integral Formula.
\end{problem}

\begin{problem}{4.}State the Maximum Modulus Principle.
\end{problem}

\begin{problem}{5.}State the Argument Principle.
\end{problem}

\begin{problem}{6.}Calculate the sum $\sum_{n=1}^{\infty}\frac{n(n-1)}{2^n}$.
\end{problem}

\begin{problem}{7.}Let $\CC$ be the contour $z = t(\pi-t) + i\cos(t)$, $0
\le t \le \pi$. Compute $\int_\CC z e^z\,dz$.
\end{problem}

\begin{problem}{8.}Find the Laurent Series of $f(z) = \frac{1}{z^4+3z^2-4}$
centered at $z=0$ and valid on the annulus $1< |z| < 2$.
\end{problem}

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

\begin{problem}{10.}Find the smallest positive integer $n$ so that all the
zeros of $p(z) = 2z^3 + z^2 + z + 5$ lie in the disk $|z| \le n$.
\end{problem}

\end{document}
