From:           	richardg@math.albany.edu
To:             	ja984@math.albany.edu
Date sent:      	Wed, 6 Jun 2001 11:01:45 -0400
Subject:        	(Fwd) C-Prelim Tex File
Priority:       	normal


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From:           	zhu <kzhu@csc.albany.edu>
Date sent:      	Mon, 4 Jun 2001 13:18:17 -0400 (EDT)
To:             	richardg@csc.albany.edu
Subject:        	C-Prelim Tex File

\magnification=1200
\baselineskip=15pt
\nopagenumbers
\vglue 1cm
\centerline{\bf Prelim in Complex Analysis, June 2001}
\bigskip
\item{1.} Evaluate the following integrals.
$$\int_{|z|=2}\tan z\,dz,\qquad\qquad \int_0^\pi{dt\over 5-4\cos t}.$$
\item{2.} Find the Laurent series of the function
$$f(z)={1\over(z-1)(z-2)}$$ in the region $1<|z-3|<2$. \medskip
\item{3.} Suppose $f(z)$ is an entire function with ${\rm
Re}\,f(z)>10$ for all $z$. Show that $f$ is constant. \medskip
\item{4.} Let $F$ be the family of functions $f$ analytic in $|z|<1$
such that $$\int_{|z|<1}|f(z)|\,dA(z)\le1,$$ where $dA$ is area
measure on $|z|<1$. Show that $F$ is a normal family. \medskip
\item{5.} Does there exist an analytic function $f$ in $|z|<1$ such
that $$0<\left|f\left({1\over n}\right)\right|<e^{-n}$$ for
$n=2,3,4,\cdots$? Justify your answer. \medskip \item{6.}
\itemitem{(a)} Show that $$\left|{1-2z\over2-z}\right|<1$$ for all
$|z|<1$. \itemitem{(b)} Suppose $f$ is analytic in $|z|<1$,
$f(0.5)=0$, and $|f(z)|\le1$ for all $|z|<1$. Show that
$$|f(z)|\le\left|{1-2z\over2-z}\right|$$ for all $|z|<1$. \medskip
\centerline{END} \end
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