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\nopagenumbers
\def\Log{{\rm Log}\,}
\centerline{\bf Prelim in Complex Analysis, August 2003}
\bigskip
\noindent Let $C$ be the complex plane and $D=\{z\in C:|z|<1\}$
be the open unit disk.
\bigskip
\item{1.} Evaluate the following integrals.
$$\int_{|z|=2}\csc z\,dz,\qquad
\int_{|z|=1}{1-\cos z\over z^2}\,dz,\qquad
\int_0^\pi{d\theta\over2+\cos\theta}.$$
\item{2.} If $f(z)$ is entire and $|f(z)|\le|z|^{3/2}$ for all
$z$, show that $f$ is identically zero.
\medskip
\item{3.} Show that a function $f:D\to C$ is constant if and only
if both $f$ and $\overline f$ are analytic in $D$.
\medskip
\item{4.} Show that the class $X$ of analytic functions $f$ in $D$ with
$$\int_D|f(z)|\,dx\,dy\le1$$
is a normal family.
\medskip
\item{5.} Find the real and imaginary parts of the complex number
$$z=\Log(1+i)+\cos(1+i),$$
where $\Log$ is the branch of the logarithm on $C-\{x:x\le0\}$
with $\Log(1)=2\pi i$.
\medskip
\item{6.} If $f:D\to C$ is a bounded analytic function, show that
$$\sup_{z\in D}(1-|z|^2)|f'(z)|\le\sup_{z\in D}|f(z)|.$$
\item{7.} Show that if $f:D\to C$ is analytic and one-to-one, then
$f'(z)\not=0$ for every $z\in D$.
\medskip
\item{8.} If $f:D\to D$ is analytic and $f(0)=f'(0)=0$, show
that $|f(z)|\le|z|^2$ for all $z\in D$.
\medskip
\item{9.} Find the Laurent series of $f(z)=1/[z(1-z)]$ at $z=0$,
at $z=1$, at $z=2$, and at $z=\infty$.
\medskip
\item{10.} Let
$$B(z)=\prod_{k=1}^n{a_k-z\over1-\overline a_kz},$$
where $a_1,\cdots,a_n$ are distinct points in $D-\{0\}$. Show that
$$B(z)=\prod_{k=1}^n{1\over\overline a_k}+\sum_{k=1}^n{1\over
\overline a_k\overline{B'(a_k)}(1-\overline a_kz)}.$$
\end