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\title{Math 336 Midterm, April 25, 2008}
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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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\item  Let $\ds f(z)$ be analytic and $\ds z(t)=u(t)+iv(t)$ be
  differentiable as a function of the real variable $t$.  Let $\ds
  g(t)=f(z(t))$.  Prove  that $\ds g'(t)=f'(z(t))z'(t)$, where $f'$ is
  the complex derivative of $f$ and the multiplication of $f'$ and $z'$ is
  multiplication of complex numbers.  Carefully explain your reasons.  You
  may quote results about real functions.  


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\item  Let $\ds f(z)=u(x,y)+iv(x,y)$ be 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 Let $\ds f_n$ be a sequence of analytic functions defined on an open
  set $\Omega$.  Suppose $\ds f_n$ converges to a function $\ds f$,
  uniformly on compact subsets of $\Omega$.  Prove that $f$ is analytic
  and $\ds \lim_{n \to \infty} f'_n = f'$. 


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\item Compute the following contour integral

\[\int_{|z|=1} \frac{1}{(3z-1)(2z-1)}\] 




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