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\title{Math 336 Midterm, May 17, 2010}
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Name:\hrulefill

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One notebook sized page of notes is allowed.
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\item Let $a$ be a complex number with $|a|>e$.  Prove that $\ds aze^z=1$ has
exactly one solution in $\ds \{z:|z|<1\}$.

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\item Let $\ds D=\{z:|z|<1\}$.  Let $f$ be analytic and non-constant on the 
open connected set $W$, 
and suppose $\ds \overline{D} \subset W$.  Suppose $|f|$ is constant on 
$\ds \partial D$. Prove that $f$ has at least one zero in D.




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\item Suppose $f$ is an entire function so that

\[
|f(z)|\le a+b|z|, \text{  where $a>0$, $b>0$}.
\]

Prove that $f$ is a polynomial of degree at most $1$.

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\item Compute

\[
\int_{-\infty}^{\infty} \frac{dx}{x^4+16} 
\]


 

\end{enumerate}

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