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

\textbf{Quiz 3}\quad \textsl{March 21, 2003}

\bigskip

\begin{problem}{1.}%
Let $f$ be an entire function. Show that $f(z)$ equals the average of
the values of $f$ on any circle with center $z$.  \end{problem}

\begin{problem}{2.}%
Let $f$ be an entire function and let $\alpha$ be a real constant, $0 <
\alpha < 1$. Suppose $|f(z)| \le |z|^\alpha$ for all~$z$. Show that $f$ is
identically zero. [Hint: Use Cauchy's inequality.]
\end{problem}

\begin{problem}{3.}%
Evaluate $\int_{\CC}\Big(\frac{\cos(z)}{z}\Big)^3dz$ where $\CC$ is the 
circle $|z| = 1$.
\end{problem}

\newpage

\begin{problem}{4.}%
Derive the Taylor series for $\sin(z)$ centered at $z = \pi/4$.
\end{problem}

\begin{problem}{5.}%
Derive the Laurent series for $\frac{1}{z^2 +z}$ centered at $z = 0$ in
the following domains.

\emph{a)} $0 < |z| < 1$

\vspace*{2.5in}

\emph{b)} $|z| > 1$

\end{problem}

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