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\title{Math 336 Final Exam, 8:30 am, June 8, 2009}
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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 $\ds \{z:|z-2|<2\}$,\newline
  with pole at $1$.

 
\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 Suppose $f$ is analytic on $H=\{z=x+iy: y>0\}$ and suppose $\ds |f(z)|
\le 1$ on $H$ and $f(i)=0$. Prove

\[
|f(z)|\le \left|\frac{z-i}{z+i}\right|.
\]





\newpage




\item   
Let $\ds \zeta(s)$ be the Riemann zeta function.  Let $N$ be the
  set of positive integers and $P$ be the set of primes.  Prove that
\[
\log (\zeta(s))=\sum_{m\in N, p\in P}\frac{1}{m}p^{-ms}
\]
where $\log(z)$ denotes the branch of the logarithm that is purely real
on the positive real axis and the
convergence is absolute and uniform on compact subsets of
$\ds\{Re(s)>1\}$. You should prove that $\log (\zeta(s))$ makes sense.
Prove the result for $s>1$ and go from there.

\newpage

\item Suppose $a, b, c, d$ are complex numbers, not negative
integers, and $a+b=c+d$. 
Prove that

\[
\prod_{n=1}^{\infty} \frac{(a+n)(b+n)}{(c+n)(d+n)}
\]

converges absolutely and express the answer using $\ds \Gamma$
functions. 

\newpage

\item Name five mathematicians prominently associated with the Prime
  Number Theorem and give approximate dates for their work on the PNT.

\end{enumerate}

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