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& {\bf M.S. Applied Mathematics} & \hspace{9mm} \\
April 2004 \hspace{1mm} & Comprehensive Exam in Analysis& H.L. Bentley, Denis White \\
& 3 hours
\end{tabular}
%\large
{\bf Instructions:} Do 7 questions including at least 2 from Part B
on Complex Analysis
\begin{center}
{\bf Part A: Real Analysis}
\end{center}
\begin{enumerate}
\item
If $\{ s_{n} \}$ is a complex sequence, define its arithmetic means by
\[
\sigma_{n} = \frac{s_{0}+s_{1}+ \cdots + s_{n}}{n+1}
\hspace{5mm} (n=0,1,2,\ldots).
\]
If $\lim s_{n}=s$, prove that $\lim \sigma_{n} = s$.
\item Suppose that $f$ is a real function defined on $\mathbf{R}^{1}$
which satisfies
\[
\lim_{h \rightarrow 0}[f(x+h)-f(x-h)]=0
\]
for every $x \in \mathbf{R}^{1}$. Does this imply that $f$ is
continuous?
\item
Suppose that $f: X \rightarrow Y$ is a mapping between
metric spaces $(X,d)$ and $(Y, \delta )$.
\begin{enumerate}
\item State the definition of {\em uniform continuity} of $f$ in this
setting.
\item Suppose that $f$ is continuous and that $(X,d)$ is compact.
Show that $f$ is uniformly continuous.
\end{enumerate}
\item
Let $X$ be an infinite set. For $p \in X$ and $q \in X$, define
\[
d(p,q) =\left\{ \begin{array}{lr} 1 & \mbox{ if } p \neq q \\
0 & \mbox{ if } $p=q$
\end{array}
\right.
\]
Prove that this is a metric. What subsets of the resulting metric space are open? Which are closed? Which are compact?
\item
Prove that the series
\[
\sum_{n=1}^{\infty} (-1)^{n} \frac{x^{2}+n}{n^{2}}
\]
converges uniformly in every bounded interval, but does not converge absolutely
for any value of $x$.
\item
Define
\[
f(x) = \left\{ \begin{array}{lrl}e^{-1/x^{2}}& \mbox{ if }& x \neq 0 \\
0 & \mbox{ if }& x = 0
\end{array}
\right.
\]
Prove that $f$ has derivatives of all orders at $x=0$, and that
$f^{(n)}(0)=0$ for $n =1,2,3, \ldots $.
\item
Let $f:[0,1] \rightarrow \mathbf{R}$ be defined by
\[
f(x) = \left\{ \begin{array}{lr} 0 & \mbox{ if $x$ is rational} \\
x & \mbox{ if $x$ is not rational}
\end{array}
\right.
\]
Determine whether or not $f$ is Riemann integrable on [0,1].
If it is then evaluate $\int_{0}^{1} f(x) \, dx$.
\end{enumerate}
\begin{center}
{\bf Part B: Complex Analysis}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% Denis's questions are below %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Instructions: Do at least 2 questions from Part B
\begin{enumerate}
\item %1
Compute all possible Laurent series at $z=0$ for the function.
\[
f(z) = \frac{1}{z^{2}-z-2}
\]
Specify the domain of convergence of each series.
\item %2
Let $u(x,y) = x^{3}+2xy -3xy^{2}$.
\begin{enumerate}
\item
Show that $u$ is harmonic.
\item
Find all harmonic conjugates of $u$.
\item
Find an analytic function $f(z)$ so that $u(x,y) = \Re f(x+iy)$.
\end{enumerate}
\item % 3
Use the residue theorem to evaluate
\(\displaystyle{ \int_{0}^{\infty} \frac{1}{(x^{2}+4)^{2}}} \, dx\)
\end{enumerate}
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