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\title{Math 336 Final Exam, 8:30 am, June , 2010}
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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 $\ds m \ge 0$ be an integer.  Compute the residue of $\ds \Gamma(z)$
at $-m$.

\newpage

\item  Let $\ds g(z,t)$ be a continuous function of $z,t$ where $\ds z
\in D$, $\ds t \in K$, $D$ is an open subset of $\mathbb{C}$ and $K$ is a
compact subset of $\mathbb{R}^n$. Suppose that $g$ is harmonic as a function
$z$.  Prove that $\ds G(z) = \int_K g(z,t)dt$ is harmonic.

\newpage



\item   Let $\ds f(z)=\frac{z-a}{1-\bar{a}z}$, where $|a|<1$. Let $\ds
  D=\{z:|z|<1\}$. Prove that


\[
\frac{1}{\pi}\int_D |f'(z)|^2dxdy = 1.
\]

\newpage 

\item Decide whether or not the following products converge.

\begin{enumerate}

\item $\ds\prod_{j=1}^{\infty} e^{(-1)^j/\sqrt{j}}$
\item $\ds\prod_{j=1}^{\infty} \left(1+i\frac{(-1)^j}{\sqrt{j}}\right)$

\end{enumerate}




\newpage




\item Prove that all conformal maps from the upper half plane to the unit 
disk have the form

\[
\alpha \frac {z-\beta}{z-\bar{\beta}}, 
\]

where $\ds |\alpha|=1,\ Im (\beta) >0$.

  

\newpage

\item 

\begin{enumerate}
\item Find a bounded harmonic function $u$ defined on $\ds D=\{z: |z|<1\}$ so 
that 
\[
 \lim_{z\to e^{it}}u(z) =
\begin{cases}
1, &\text{if $0<t<\pi/2$}\\
0, &\text{if $\pi/2<t<2\pi$}
\end{cases}
\]
\item Find an unbounded harmonic function with the same property.
\end{enumerate} 

\newpage

\item  Prove that if $\ds |z|<1$
\begin{equation*}
\prod_{k=0}^{k=\infty} (1+z^{2^k})=\frac{1}{1-z}
\end{equation*}



\end{enumerate}

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