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\title{Math 336 Midterm, April 27, 2009}
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Name:\hrulefill

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One notebook sized page of notes is allowed on the test. 

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\begin{enumerate}


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\item  Let $u$ and $v$ be harmonic on an open set.  
\begin{enumerate}
\item Is it necessarily true that $u^2$ is harmonic? What about $uv$?
\item Answer the  same question in the case that $v$ is the harmonic conjugate
  of $u$.
\end{enumerate}


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\item  Let $\ds f(z)=u(x,y)+iv(x,y)$ where $u$ and $v$ are real, twice-differentiable on an open
  set.   Suppose
  that the real and imaginary parts of $\ds f(z)$ and $zf(z)$ are
  harmonic.  Prove that $f$ is analytic.


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\item Compute 
\[
\int_{|z|=1}\frac{dz}{2z^2-5z+2}.
\]

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\item Let $f$ be analytic on $\ds D=\{z: |z| < 1\}$.  Let $\ds
  d=\sup_{z,w\in D}\{|f(z)-f(w)|\}$.  Prove that 
\[
2|f'(0)| \le d.
\]
Hint: For $0<r<1$, prove
\[
2f'(0)=\frac{1}{2\pi i}\int_{|z|=r}\frac{f(z)-f(-z)}{z^2}dz,
\]
And let $\ds r\to 1$. 
\end{enumerate}

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