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\title{Math 336 Final Exam, 8:30 am, June 9, 2008}
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\markboth{Final}{Final}

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Name:\hrulefill

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One notebook-size page of notes is allowed (each side may be used).
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\begin{enumerate}
 
\item Let $f$ be analytic on a connected open set $\Omega$ and suppose
  \newline 
  $\ds \overline{\mathbb{D}}= \{ z: |z|\le 1\} \subset \Omega$.  Suppose
  that $f$ is real 
  on $\ds \{z: |z|=1\}$. Prove that $f$ is constant on $\Omega$. 



\newpage

\item   Find the Green's function for the region $\ds \{z:|z-1|<3\}$,\newline
  with pole at $2$.

 
\newpage


\item  Let $u(x,y),\ v(x,y)$ be continuously differentiable as functions
of $(x,y)$ in a domain
$\Omega$.  Let $f(z)=u(z)+iv(z)$.  Suppose that for every $z_0 \in \Omega$
there is an $r_0$ (depending on $z_0$) such that 
\begin{equation*}
\int_{|z-z_0|=r} f(z)dz =0,
\end{equation*}
for all $r$ with $r<r_0$.  Prove that $f$ is analytic in $\Omega$.  Hint:
Show that $f$ satisfies the Cauchy-Riemann equations in $\Omega$.



\newpage

\item    Suppose that $u$ is harmonic on all of $\ds \mathbb{C}$ and $u(z)\ge
  0$ for all $\ds z \in \mathbb{C}$.  Prove that $u$ is constant.  Is it
  true that a positive harmonic 
  function on $\ds \{ (x,y): x+y > 0\}$ is constant?


\newpage

\item Let 
\[ g(x)= \begin{cases}
&1\text{, if } |x|<1\\
&0\text{, if } |x|>1;
\end{cases}  
\]

\[ h(x)=
\begin{cases}
&|x| \text{, if } |x|<1\\
& 0\text{, if } |x|>1;
\end{cases}
\]
\[
f(x)=g(x)- h(x).
\]
Compute the Fourier transform (my definition in class) of $\ds g, h$ and
hence of $\ds f$.   
\newpage




\item  Suppose $\ds f_n(z)$ is a sequence of analytic functions on an
  open set $\ds \Omega$ and that $\ds |f_n(z)| <1$ for all $z \in
  \Omega$. Suppose $\ds \sum f_n(z)$ converges uniformly and absolutely on
  compact subsets of $\ds \Omega$. Prove that
\[f(z)= \prod(1+f_n(z))\]
defines an analytic function and $\ds f(z) \ne 0$ ($\ds f$ is never zero.)



\newpage

\item State what you consider to be the most important result of your
  term paper.  Why do you consider it to be interesting?

\end{enumerate}

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