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\def\trace{\mathop{trace}}
\def\Res{\mathop{Res}}
\def\U{\mathop{U}}

\begin{document}

\raggedleft{August, 1998}
\bigskip

\noindent{\bf Comprehensive Examination \hfill 
        Department of Mathematics}
\vskip .5in

\centering{{\bf COMPLEX ANALYSIS}\\
\vskip .7in PART I: Do three of the following problems.}
\bigskip

\begin{enumerate}

\item
        \begin{enumerate}
        \item   Find 
$$\Res [{e^{iz}\over (z^{2}+1)^{5}}; i].$$
        \item Evaluate
              $$\int_{0}^{\infty}{cos x\over (x^{2}+1)^{5}}dx.$$
        \end{enumerate}

\item Let $u(x,y)$ be an everywhere positive harmonic function on $\bf C$.
Prove that $u(x,y)$ is constant.

\item Show that if $f(z)$ is analytic at $\alpha$ and
      $$g(z)={f(z)+\alpha f'(\alpha) -zf'(\alpha)-f(\alpha)\over
(z-\alpha)^{2}},$$
then $g(z)$ has a removable singularity at $z=\alpha$.

\item Find an entire function having a zero of order $n$ at $z=n$,
$n=1,2,3,...$, and no other zeros.


\end{enumerate}


\newpage

\centering{PART II: Do two of the following problems.}
\bigskip

\begin{enumerate}

\item Suppose $f(z)$ is an entire function with the property that for
every $w\in {\bf C}$ the equation $f(z)=w$ has precisely $k$ solutions.
Show that $f(z)$ is a polynomial of degree $k$.

\item Suppose $\{ f_{n}\}$ is a sequence of analytic functions on a region
$D$ such that there exists a positive constant $M$ with the property that
$$\int\int_{D} \vert f_{n}(z)\vert^{2} dxdy\leq M\hbox{ for all n.}$$
Show that $\{ f_{n}\}$ has subsequence that converges uniformly on compact
subsets of $D$.\par\noindent
Hint: If $f$ is analytic in a neighborhood of a closed ball
$\overline{B(a;R)}$, show that
$$\vert f(a)\vert^{2}\leq {1\over\pi
R^{2}}\int_{0}^{2\pi}\int_{0}^{R}\vert f(a+re^{i\theta})\vert^{2}
rdrd\theta.$$

\item Suppose $f(z)$ is analytic on $\vert z\vert <1$ and continuous on
$\vert z\vert\leq 1$. Assume $f(z)=0$ on an arc of the circle $\vert
z\vert =1$. Prove that $f(z)\equiv 0$.
        \end{enumerate}



\end{document}