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\title{Math 336 Midterm, May 18, 2009}
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Name:\hrulefill

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One notebook sized page of notes is allowed.
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\item Let $\ds P(z)$ be a polynomial of degree $n$. Prove that 
\[
\int_{|z|=2}\frac{P(z)}{(z-1)^{n+2}}dz = 0.
\]


\newpage



\item  Let $\ds D=\{z:|z|<1\}$. Suppose $f$ is analytic on an open set that
  includes the closure of $D$ and suppose $\ds |f(z)|<1$ if $\ds
  |z|=1$.  Prove that there is a unique $\ds \zeta \in D$ such that $\ds
  f(\zeta)=\zeta$. 



\newpage


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

\[
|f(z)|\le |z|^4/\log|z|, \text{ for } |z|>1.
\]

Prove that $f$ is a polynomial of degree at most $3$.

\newpage



\item Use the residue theorem to compute

\[
\int_{|z|=1}\frac{|dz|}{|z-a|^4}, \text{ where } a>1.
\]


 

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