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\textbf{Math 425: Complex Variables}
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\textbf{Quiz 3}\quad \textsl{March 19, 2004}

\bigskip

\begin{problem}{1.}%
Prove that if $f(z)$ is entire and $\Re(f(z)) > 0$ for all $z$ then $f(z)$
is constant.  [Hint: First show that if $\Re(w) > 0$, then
$\Big|\frac{w-1}{w+1}\Big| < 1$.]
\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~$z$. Prove that $f$ is
identically zero. [Hint: Use Cauchy's Inequality.]
\end{problem}

\begin{problem}{3.}%
Evaluate $\int_\CC \frac{e^{z}\sin(z)}{z^3}\,dz$ where $\CC$ is the circle
$|z| = 4$.
\end{problem}

\newpage

\begin{problem}{4.}%
Find the maximum of $\big|e^{-z(z+1)}\big|$  on the rectangle $-1 \le \Re(z)
\le 1$, $-1 \le \Im(z) \le 1$.
\end{problem}


\begin{problem}{5.}%
Derive the Taylor series for $\frac{1}{1+z}$ at the point $z = 1 + i$ and
determine its radius of convergence.
\end{problem}

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