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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 be comprehensive.  The final
exam is at 8:30 am, Monday, June 8, in the regular classroom.
\bigskip

\begin{enumerate}

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

\begin{enumerate}
\item 
\[
\frac{1}{\pi}\int_D |f'(z)|^2dxdy = 1.
\]
\item 
\[
\frac{1}{\pi}\int_D |f'(z)|dxdy =
\frac{1-|a|^2}{|a|^2}\log\left(\frac{1}{1-|a|^2}\right). 
\] 
\end{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

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. 

\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 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

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

\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


\bigskip

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

\end{enumerate}

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