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\title{Math 336 Sample Problems}
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\markboth{Sample Problems}{Sample Problems}


One notebook sized page of notes will be allowed on the test.  The test
will cover up to \S VIII.2 in the text (excluding those sections for
which there was no homework). 
\bigskip

\begin{enumerate}

\item p. 158, \# 13.

\bigskip

\item p. 228, \# 5.



\bigskip

\item  Using Rouch\'e's theorem, show that $z^5+5z^3+z-2$ has
three roots in the set $\{z: |z|<1\}$.

\bigskip

\item Let $\ds D=\{z:|z|<1\}$. Suppose $f$ is analytic on an open set that
  includes the closure of $D$ and suppose $\ds |f(z)|<1$ if $\ds
  |z|=1$.  Prove that there is a unique $\ds \zeta \in D$ such that $\ds
  f(\zeta)=\zeta$. 

\item Let $u$ and $v$ be harmonic on an open connected set $W$.  Suppose
  that $u(z)v(z)=0$ on an open subset of $W$.  Prove that either $u$ or
  $v$ is identically $0$ on $W$.

\bigskip

\item Let $\Omega$ be a bounded connected open set.  Suppose $\ds 0\in
  \Omega$ and that $f$ is an analytic function on $\Omega$ such that if
  $\ds z\in 
  \Omega,\ f(z)\in \Omega$.  Suppose also that $\ds f(0)=0,\ f'(0)=1$.
  Prove that $\ds f(z)=z$.  Hint: use Cauchy's inequalities on the
  functions obtained by composing $f$ with itself $k$ times.

\bigskip

\item Let $f$ be analytic on $\ds D=\{|z|<1\}$.  Suppose $f$ is $1-1$ on
  $\ds D-\{0\}$.  Prove that $f$ is $1-1$ on $D$.

\bigskip

\item Suppose $f$ is analytic in $\ds \{ 0<|z|<r\}$ for some
  $r>0$. Suppose also that $\ds |f(z)|<|z|^{-1+\epsilon}$ in $\ds \{
  0<|z|<\delta \}$, where $\ds \epsilon >0$. Prove that $f$ has a removable
  singularity at $0$. 

\bigskip 

\item Prove that 
\[
\sum_{n=1}^{\infty} d(n)z^n = \sum_{n=1}^{\infty} \frac{z^n}{1-z^n}
\text{  for} |z|<1,
\]
where $d(n)$ is the number of divisors of $n$. Carefully consider
convergence issues.

\bigskip

\item Review contour integration computations.


\item  There may be homework problems, example problems from the text,
  or requests for statements of theorems on the midterm.

\end{enumerate}

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