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\title{Math 336 Sample Problems}
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\markboth{Sample Problems}{Sample Problems}
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One notebook sized page of notes will be allowed on the test.  The test
will cover through chapter IV, section 5 in the text.
\bigskip

\begin{enumerate}

\item Suppose that $v$ is the harmonic conjugate of $u$ and $u$ is the
harmonic conjugate of $v$.  Show that $u$ and $v$ must be constant.

\bigskip

\item Gamelin, IV.5, problem 3.

\bigskip

\item Gamelin, IV.5, problem 4.

\bigskip


\item Compute $\ds \int_{|z|=3} \frac{dz}{(z-1)(z-2)(z-4)}$.

\bigskip

\item Give an example of a function $f$ for which
 $\int _{|z|=r} f(z)dz = 0$ for all $r>0$  although $f$ is not analytic.

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\item  Let $\ds f(z)=u(x,y)+iv(x,y)$ be twice continuously (real)
  differentiable on an open 
  set.   Suppose
  that the real and imaginary parts of $\ds f(z)$ and $zf(z)$ are
  harmonic.  Prove that $f$ is analytic.

\bigskip

\item Let $a$ be a complex number and suppose $\ds |a|<1$.  Let $\ds
  f(z)=\frac{z-a}{1-\overline{a}z}$. Prove the following statements.

\begin{enumerate}

\item $\ds |f(z)|<1$, if $\ds |z|<1$.
\item $\ds |f(z)|=1$, if $\ds |z|=1$. 

\end{enumerate}

\bigskip

\item Suppose $P(z)$ is a polynomial and that all of it's roots have positive
real part.  Prove that the zeros of the derivative of $P$ have positive real 
part.


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\item Let $z_j=e^{\frac{2\pi ij}{n}}$ denote the $n$ roots of unity.
Let $c_j=|1-z_j|$ be the $n-1$ chord lengths from $1$ to the points
$z_j,j=1,\dots. n-1$. Prove that the product $c_1\cdot c_2\cdots
c_{n-1}=n$.  {\it Hint}:  Consider $z^n-1$.

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\item   Let $\ds f(z)=x^2-y^2 + i\log(x^2+y^2)$. Find the points at which 
$f$ is complex differentiable.  Find the points at which $g(z)=x-iy$ is 
complex analytic.

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\item Let $\ds f(z)=u(z)+iv(z), u=Re(f(z)), v=Imf((z))$ be analytic on an open
  connected set $\Omega$. Suppose there are real numbers $a,b,c$ with
  $\ds a^2+b^2 \ne 0$ and $au(z)+bv(z)=c$ for all $\ds z \in \Omega$.
  Prove that $f$ is constant.



\item Let $f$ be analytic within and on a simple closed curve
  $\Gamma$. Prove that $\ds Re\left (\int_{\Gamma}
    \overline{f}(z)f'(z)dz\right )=0$.
\bigskip


\item Compute $\ds \int_{|z|=r}\frac{|dz|}{|z-a|^2}$, where $\ds |a|\ne
  r$.  Use the fact that on \newline $\ds \{|z|=r\},\
  |dz|=-ir\frac{dz}{z}$; and then use the Cauchy integral formula.
\bigskip


\item You will need to know the definitions of the following terms and
statements of the following theorems.

\begin{enumerate}

\item Modulus (absolute value) and argument of a complex number
\item Complex derivative
\item Complex analytic function
\item Cauchy-Riemann equations
\item Harmonic functions and harmonic conjugate
\item Complex exponential function
\item Complex logarithm
\item Cauchy's integral theorem and formula
\item Maximum principle
\item Linear fractional transformations

\end{enumerate}

\bigskip

\item[13.]  There may be homework problems or example problems from the text
on the midterm.

\end{enumerate}

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