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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 up to section 2.5 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 Let $\ds f(z)=u(x,y)+iv(x,y)$.  Suppose that $f$ is analytic.  Prove that

\[ \det \begin{pmatrix} u_x &u_y\\ v_x &v_y \end{pmatrix} = |f'(z)|^2.
\]

\bigskip

\item  Let $\ds f(z)=u(x,y)+iv(x,y)$ be 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.

\item Suppose $\ds \sum_0^{\infty} |a_n|^2$ converges.  Prove that
  $f(z)=\sum_0^{\infty} a_nz^n$ is analytic for  
$\ds |z|<1$.  Prove $\ds \lim_{r\to 1} \int_0^{2\pi}|f(re^{it})|^2dt$
exists and compute it. 

\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 statments.

\begin{enumerate}

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

\end{enumerate}


\bigskip

\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 $z^n-1$.

\bigskip

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

\bigskip

\item Find the Laurent series of the function $\frac{1}{z}$ in the
annulus $D=\{z: 2<|z-1|<\infty\}$.  

\bigskip
 


\item  Let $\ds f(z)=\frac{p'(z)}{zp(z)}$, where $\ds
  p(z)=\prod_{j=1}^n(z-z_j)$ and the $\ds z_j$ are distinct and
  different from $0$.  Find all the poles of $\ds f$ and compute the
  residues of $\ds f$ at these poles.
\bigskip

\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 Absolute Value (Modulus) and Argument of a complex number
\item Complex Derivative
\item Cauchy-Riemann equations
\item Harmonic Conjugate
\item Complex Analytic
\item Power Series
\item Complex Exponential Function
\item Complex Logarithm
\item Cauchy's Integral Theorem and Formula
\item Morera's Theorem
\item Liouville's Theorem
\item Isolated Singularities (types)
\item Residues
\item Laurent Series

\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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