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\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Complex Analysis Section %Complex Analysis Qualifying Exam
\endtitle
\author
\bf Fall 1995
\endauthor
\endtopmatter
\document

%\s{NOTE}
%{Please state on the outside of your examination book which
%problems  you want graded.
%All reasoning should be justified.}
\vglue\baselineskip

\t{Part I}{Do three (3) of these problems.}

\s{I.1}
{Show that if $u(x,y) + iv(x,y)$ is an analytic function with non-vanishing
derivative in a region $R$, then, for any constants $c_1$ and $c_2$, the 
curves $u(x,y) = c_1$ and $v(x,y) = c_2$ are orthogonal in $R$ (at the points
of their intersection).}

\s{I.2}
{If $-1 < a < 1$, compute
 $$
 \int_0^\infty \frac {x^a} {1+x^2} dx
 $$
 using residues.}

\s{I.3}
{Give a conformal (i.e., biholomorphic) map of $\Bbb C \setminus [1,\infty)$ onto
the open unit disc.}

\s{I.4}
{Suppose $f(z)$ is holomorphic in $\Bbb C \setminus \{0\}$ and satisfies 
 $$
 |f(z)| \leq |z|^2 + \frac 1 {|z|^2} \quad \text{ for } z \ne 0.
 $$
If $f(z)$ is an odd function, what form must it have?}

\vglue\baselineskip
\t{Part II}{Do two (2) of these problems.}

\s{II.1}
{Suppose $f(z)$ is meromorphic in all of $\Bbb C$ and bounded on $\{z: |z| >
R\}$ for some $R>0$. Prove that $f(z)$ is rational. }

\s{II.2}
{Suppose $f$ is analytic on a neighborhood of the closed unit disc $\overline
D$ and one-to-one on the unit circle $\partial D$. Show that $f$ is
one-to-one on $\overline D$.}


\s{II.3}
{Show that there is no one-to-one analytic function which maps $A = \{z: 0 < |z|
< 1\}$ onto $B = \{z:1< |z| < 2\}$.}

\enddocument