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

\textbf{Quiz 4}\quad \textsl{April 2, 2004}

\bigskip

\begin{problem}{1.}%
Establish Euler's Identity, $e^{i\theta} = \cos(\theta) + i\sin(\theta)$,
using the power series for $e^z$.
\end{problem}

\begin{problem}{2.}%
Compute $\lim_{z\to0}\frac{\sin(z^{10}) -z^{10}}{z^{30}}$ by
first expressing the numerator as a power series.
\end{problem}

\begin{problem}{3.}%
Find an analytic function $f(z)$ that has the power series expansion
$f(z) = \sum_{n=1}^\infty \frac{n}{n+1}z^n$ for $|z|<1$, using
differentiation and/or integration of series.
\end{problem}

\newpage

\begin{problem}{4.}%
Determine the Laurent series of $\frac{z}{z^2+3z+2}$ on $1<|z|<2$.
\end{problem}


\begin{problem}{5.}%
Compute the residue of $f(z) = \frac{1}{z^3+z}$ at each of its isolated
singular points.
\end{problem}

\end{document}
