Date sent:      	Thu, 01 Jun 2000 08:11:40 -0400 (EDT)
From:           	zhu <kzhu@csc.albany.edu>
To:             	ja984@cnsvax.albany.edu

\magnification=1200
\baselineskip=15pt
\nopagenumbers

\def\C{{\bf C}}
\def\D{{\bf D}}
\def\R{{\bf R}}

\centerline{\bf Preliminary Exam in Complex Analysis}
\medskip
\centerline{June 2000}
\bigskip
\noindent Notation: $\C$ denotes the complex plane; 
$\R^-$ denotes the negative real axis (including the origin);
and $\D$ denotes the unit disk.
\medskip
\item{1.} Suppose $f(z)$ is analytic in $\C-\{0\}$ and satisfies
$$|f(z)|\le{1\over\sqrt{|z|}},\qquad z\in\C-\{0\}.$$
Show that $f$ is identically zero.
\smallskip
\item{2.} Let $\Omega=\C-\R^-$.
\itemitem{1)} Define the principal branch of the logarithm, $Log(z)$, in
the region $\Omega$.
\itemitem{2)} Show that $Log(zw)=Log(z)+Log(w)$ for all $z$ and $w$
in the (open) right half-plane.
\itemitem{3)} Show that the identity in 2) does not hold for all
$z$ and $w$ in $\Omega$.
\smallskip
\item{3.} For each of the following functions find the radius of
convergence for its Taylor series at the specified point.
\itemitem{1)} $f(z)=(\cos z)/(3z+4)$ at $z_0=10$.
\itemitem{2)} $g(z)=1/(z^2+z+1)$ at $z_0=0$.
\smallskip
\item{4.} Suppose $f(z)$ and $\overline{f(z)}$ are both analytic
in $\D$. Show that $f(z)$ is constant.
\smallskip
\item{5.} Evaluate the following integrals.
\smallskip
\itemitem{1)} $\displaystyle\int_C\left(\sinh z+{z\over 2z+1}\right)\,dz$,
where $C$ is the unit circle traversed once clockwise.
\smallskip
\itemitem{2)} $\displaystyle\int_\Gamma(z+3\bar z)\,dz$, where $\Gamma$
is the path from $-1$ to $1$ along the upper semi-circle $|z|=1$.
\smallskip
\itemitem{3)} $\displaystyle\int_0^\pi{d\theta\over2+\cos\theta}$.
\end