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& {\bf Ph.D. Comprehensive Exam, Complex Analysis} & Page 1 of 2 \\
Sept. '94 \hspace{2mm} & Examiners: P. Hewitt, D.A. White &
\end{tabular}
\large{
\begin{enumerate}
\item {\em Laurent Expansions.}
Suppose that $f(z)$ is holomorphic in the annulus:
$\{z \in {\bf C}: 0 \le R_{1} < |z - a| < R_{2} \le \infty\}.$
\begin{enumerate}
\item
Use Cauchy's Integral Formula to show that
\[
f(z) = \sum_{n= - \infty}^{\infty} a_{n} (z-a)^{n}
\]
where
\[
a_{n} = \frac{1}{2 \pi i} \int_{C} \frac{f(t)}{(t-a)^{n+1}} \, dt
\]
and $C= \{t: |t-a| = r\}$ for some $r,$ $R_{1} < r < R_{2}$. Show
also that the series converges absolutely inside the annulus.
\item
Assume further that $R_{1}=0$ and, for some $\alpha >0$, and $M >
0$,
\[
\max_{0 \le \theta \le 2 \pi} |f(a + re^{i\theta})| \le M
r^{-\alpha} \mbox { for all $r$,} \: 0 < r < R_{2}.
\]
Show that $a_{-n} = 0$ if $n > \alpha$.
\end{enumerate}
\item
{\em Argument Principle.} Suppose that $f$ is meromorphic in a
simply connected domain and $C$ is a simple closed rectifiable
curve oriented counterclockwise in that domain. Let $a_{j},$
$1\le j \le J$ be all the zeroes of $f$ and $b_{k},$ $1 \le k \le
K$ all the poles of $f$ inside $C$, with zeroes counted according
to their multiplicity and poles according to their order. Assume
also that no zeroes or poles of $f$ lie on $C$.
% Show that
% \[
% \frac{1}{2 \pi i} \int_{C} \frac{f'(z)}{f(z)} \,dz = J - K.
% \]
Interpret $\displaystyle\int_C \frac{f'(z)}{f(z)} \,dz$ and
$\displaystyle\int_C z\frac{f'(z)}{f(z)} \,dz$ in terms of the
zeroes and poles. Prove all your assertions.
\item
{\em Harmonic Conjugates.}
\begin{enumerate}
\item
Suppose that $u(x,y)$ is an harmonic function defined on a domain
$D$. Show that for every $(a,b) \in D$, there is a function
$v(x,y)$ defined and harmonic in a neighborhood of $(a,b)$ which
is an ``harmonic conjugate'' of $u$ in the sense that $f(x+iy) =
u(x,y) + iv(x,y)$ defines an holomorphic function.
\item
Show, by specific example that the harmonic conjugate need not
be defined on the whole of the original domain $D$. (Find $D$ and
$u$ harmonic on $D$ so that there is no $v$ defined on all $D$ so
that $f$ as defined above is holomorphic.)
\end{enumerate}
\item {\em Montel's Theorem.}
A collection ${\cal F}$ of functions defined on a common domain
$D$ is said to be {\em normal\/} if every sequence in ${\cal F}$
has a subsequence which converges uniformly on every compact
subset of $D$.
% Also ${\cal F}$ is said to
% be {\em locally bounded\/} if for
% every compact subset $K$ of $D$, there is a constant $M$
% so that $|f(z)|