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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 4.4 in the text (excluding those sections for
which there was no homework). 
\bigskip

\begin{enumerate}

\item p. 195, \# 11, \# 14

\bigskip

\item p. 253, \# 15, \# 16

\bigskip

\item Let $u(z)$ be harmonic on $\ds D=\{|z|<1\}$, continuous on the
  closure of $D$, and non-negative.  Use the Poisson integral formula to prove
\begin{equation*}
\frac{1-r}{1+r}u(0)\le u(z) \le \frac{1+r}{1-r}u(0),
\end{equation*}
where $\ds |z|=r$.
\bigskip

\item Let $\ds C$ be the open arc of the unit circle in the first
  quadrant. Find a bounded function $\ds u$, which is harmonic on $\ds 
  \{z: |z|<1\}$, with 
\begin{equation*}
 \lim_{z\to \zeta}u(z)=
\begin{cases}
1 \text{ if $\zeta \in C$} \\
0 \text{ if $\zeta \not \in \overline{C}, \ |\zeta|=1$}
\end{cases}
\end{equation*}

Find an unbounded harmonic function with the same properties.

\item Is there an analytic function $f$ such that $\ds |f(z)|<1$ for
  $\ds |z|<1$, with $\ds f(0)=\frac{1}{2}$ and $\ds f'(0)=1$?  Hint:
  Use Schwarz's Lemma.

\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 $u$ 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 Let $u(z)$ be harmonic in all of $\mathbf{C}$.  Suppose $\ds
  |u(z)|\le c|z|^n$ for some positive constant $c$.   Prove that $u$ is
  the real part of a complex polynomial of degree at most $n$.  Hint:
  Use 
\begin{equation*}
u(z)=\text{Re} \left\{\left(\frac{1}{2\pi
  i}\right)\int_{|\zeta|=r}u(\zeta)\left(\frac{\zeta+z}{\zeta-z}\right)
\frac{d\zeta}{\zeta}\right\}, 
\end{equation*}
if $\ds |z|<r$.
\bigskip 


\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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