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


One notebook sized page of notes (both sides) will be allowed on the test.
The test 
will cover up to \S 5.1 in the text and will be comprehensive.  The final
exam is at 8:30 am, Monday, June 9, in the regular classroom.
\bigskip

\begin{enumerate}

\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 $\Omega$.

\bigskip

\item  Let $\ds D_2 = \{z: |z|<2\}$ and $\ds I= \{x\in \mathbf{R}:
  -1\le x \le 1\}$. Find a bounded harmonic function $u$, defined in $\ds
  D_2-I$ such that $u$ does not extend to a harmonic function defined in
  all of $\ds D_2$.

\bigskip

\item Find a conformal map from the region between the
two lines $y=x$ and $y=x+2$ to the upper half plane, which sends $0$ to $0$.

\bigskip

\item Find a function, $h(x,y)$, harmonic in $\{x>0,\ y>0\}$, such that
\begin{equation*}
h(x,y)=
\begin{cases}
0& \text{if $0<x<2, y=0$,}\\
1& \text{if $x>2, y=0$,}\\
2& \text{if $x=0, y>0$}
\end{cases}
\end{equation*}

\item Suppose that $u$ is harmonic on all of $\ds \mathbb{C}$ and $u\ge
  0$.  Prove that $u$ is constant.

\bigskip

\item The function 
\begin{equation*}
f(z)=\frac{1}{(z+1)(z+3)}
\end{equation*}
has three different Laurent series expansions in powers of $z$.  Find them
all and specify the annulus for each series.

\bigskip

\item Let 
\begin{equation*}
f(z)=\exp(z+\frac{1}{z})
\end{equation*}
Prove that 
\begin{equation*}
\text{Res}[f,0]=\sum_{k=0}^{k=\infty}\frac{1}{k!(k+1)!}
\end{equation*}
Carefully justify all steps in your computation.

\bigskip

\item Suppose that $u$ is a harmonic function and $v$ is its
  conjugate harmonic function.  Prove that $uv$ is harmonic.

\bigskip

\item Let $\ds f(z)=1-\frac{1}{2^z}+\frac{1}{3^z}-\frac{1}{4^z}+\dots$.
  Prove that the series converges absolutely and uniformly on compact
  sets to an analytic function if $\ds
  \text{Re}(z)>1$. Prove that $\ds f(z)=(1-2^{1-z})\zeta(z)$ if $\ds
  \text{Re}(z)>1$.  Prove that the series converges conditionally when
  $z=x$ is real and
  $\ds 0<x<1 $.  Prove that this statement is still true on the infinite
  strip $\ds 0<\text{Re}(z)<1$.

\bigskip


\item Let $f$ be analytic on a connected open set $\Omega$ and suppose
  $\ds \overline{\mathbb{D}}= \{ z: |z|\le 1\} \subset \Omega$.  Suppose
  that $f$ is real 
  on $\ds \{z: |z|=1\}$. Prove that $f$ is constant on $\Omega$. 

\bigskip

\item Let $u(z)$ be harmonic in $\ds \{z:0<|z|<1\}$.  Let $\ds
  P=\int_{|z|=r}\frac{\partial u}{\partial n}ds$, where $0<r<1$.  Show
  that $P$ does not depend on $r$.  Prove that 
\begin{equation*}
u(z)=\frac{P}{2\pi}\log |z| +\text{Re}(f(z))
\end{equation*}
where $f$ is analytic in $\ds \{z:0<|z|<1\}$.

\bigskip

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

\bigskip

\item  Find the Green's function for the region $\ds \{z:|z-2|<3\}$
  with pole at $4$.

\bigskip

\item The characteristic function, $\ds \chi_S$  of a set is defined by 
\begin{equation*}
\chi_S(x)=
\begin{cases}
1, \text{ if $x\in S$}\\
0,  \text{ if $x\not\in S$}
\end{cases}
\end{equation*}
\begin{enumerate}
\item Let $\ds I=[a,b] \subset \mathbf{R}$ be a compact interval.  Prove
  that \newline $\ds 
\lim_{x \to \pm \infty} \widehat{\chi}_{_I}(x) =0$ by directly computing
$\ds \widehat{\chi}_{_I}(x)$ and then computing the limit. 
\item Let $\ds \{I_j\}_{j=1}^n$ be a finite collection of compact intervals
  in $\ds \mathbf{R}$ and let $\ds \{c_j\}_{j=1}^n$ be a finite collection of
  constants.  Let $\ds g=\sum_{j=1}^n c_j\chi_{I_j}$.  Using part (a), prove
  that $\ds \lim_{x \to \pm \infty}\widehat{g}(x) = 0$.  (This can be
  used to give an alternate proof of the Riemann-Lebesgue lemma.
\end{enumerate}

\bigskip

\item  There may be homework problems or example problems from the text
on the final.  Don't forget previous sample problems. The final will be
comprehensive. 

\end{enumerate}

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