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Fall, 2001
\begin{center}{\large \bf{M.S. (Applied Mathematics)\\
Comprehensive Examination in Analysis}}
\end{center}
\vspace{10mm}
\hspace{1.5cm}{\em Do five (5) questions from {\bf each} of Parts A and B.\
\hspace{1.5cm}Indicate on the front of the blue book which problems you wish to
have graded.}\\
\hspace{1.5cm}${\bf R}$ denotes the set of real numbers and ${\bf C}$ the set of
complex numbers.
\vspace{10mm}
\begin{center}{\bf Part A. Real Analysis}
\end{center}
\begin{enumerate}
\item Suppose $\lim\limits_{n\to\infty}a_n =A$ and $\lim\limits_{n\to\infty}b_n
=B.$ Prove that $\lim\limits_{n\to\infty}a_nb_n =AB$.
\item (a) Let $(M,d)$ be a metric space and suppose $f_n:M \to {\bf R}$ are
continuous functions. Define what it means for $f_n \Rightarrow f$
uniformly on $M$.
(b) Prove: If $f_n \Rightarrow f$ uniformly on $M$ and each $f_n$ is
continuous on $M$, then so is $f$.
\item {\em Abel's Theorem}: Suppose $\sum\limits_{k=0}^{\infty}a_k$
converges and let $f_n(x) = \sum\limits_{k=0}^n a_k x^k$. Show
that $f_n(x) \Rightarrow f(x)$ uniformly on $[0,1]$, where $f(x) =
\sum\limits_{k=0}^{\infty} a_kx^k$.
\item (a) Give two different but equivalent definitions of
compactness for a metric space $(M,d)$.
(b) Prove that a compact metric space is closed and bounded and
give an example to show that the converse statement is not true.
\item Let $f(x)$ be a continuous real-valued function on $[a,b]$
with $f(x) \ge 0$.\\
Prove: If there is one point $c \in [a,b]$
with $f(c) > 0$ then \[\int_a^bf(x)dx > 0.\]
\item Let $f_n(x) = x^n(1-x)$, $g_n(x) = x^n(1-x^n)$, $0 \le x \le
1$. Show that $f_n(x) \Rightarrow 0$ uniformly on $[0,1]$ while
$g(x) \to 0$ pointwise on $[0,1]$ but not uniformly.
\item Let $f:M \to {\bf R}$ be a continuous function on the
compact metric space $(M,d)$.\\
Prove that $f$ is uniformly continuous.
\item (a) Define what it means for a subset $O$ of $M$ to be {\em
open}, where $(M,d)$ is a metric space.
(b) Prove that for any $p \in M$ and $r >0$, the ball $B(p,r)=
\{q \in M: d(p,q) < r\}$ is an open subset of $M$.
\end{enumerate}
\vspace{10mm}
\begin{center}{\bf Part B. Complex Analysis}
\end{center}
\begin{enumerate}
\item (a) Prove that if $f(z)$ and $\overline{f(z)}$ are analytic in a domain
$D \subseteq {\bf C}$ then $f(z)$ is constant in $D$. ({\em Hint:} Cauchy-Riemann
equations)
(b) Prove that if $f(z)$ is analytic in a domain $D \subseteq {\bf C}$ and
$|f(z)|$ is constant in $D$, then $f(z)$ is constant in $D$.
\item (a) Verify that the function $u(x,y) = 2x(1-y)$ is harmonic in ${\bf
R}^2$ and find a harmonic conjugate $v(x,y)$.
(b) Suppose $u(x,y)$ and $v(x,y)$ are conjugate harmonic functions on a domain $D
\subseteq {\bf R}^2$. If $U(x,y) = e^{u(x,y)}\cos v(x,y)$ and $V(x,y) =
e^{u(x,y)}\sin v(x,y)$, show that $U(x,y)$ and $V(x,y)$ are also conjugate
harmonic.
\item (a) Evaluate the contour integral \[\int_C \frac{dz}{\bar{z}}\] where $C$
is the upper half of the circle $|z|=1$ from $z=1$ to
$z=-1$.
(b) Let $C$ be the line segment from $z=i$ to $z=1$. {\em Without} evaluating
the integral directly show that \[\left|\int_C \frac{dz}{\bar{z}}\right| \le
2\]
\item Evaluate \[\oint_C \frac{\sin^2 z}{(z-\pi/6)^3}dz\] if $C$ is the circle
$|z| = 1$ traced once counterclockwise.
\item (a) State Liouville's Theorem.
(b) Suppose a non-constant function $f(z)$ is such that, for two constants $a > 0$ and
$b>0$, $f(z) = f(z+a)$ and $f(z) = f(z+bi)$ for all $z \in {\bf C}$.
(Such a function is said to be {\em doubly periodic}.) Prove that $f(z)$ is
not analytic in the rectangle $0 \le x \le a$, $0 \le y \le b$.
\item Compute {\em all} possible Laurent series for $f(z) =
\frac{1}{z^2-4z+3}$ at $z=1$.
State explicitly the domain of convergence of each series.
\item Use residues to evaluate the contour integral \[\oint_C z(3z+1)e^{2/z}dz
\] where $C = \{z: |z| = 1\}$ traced once counterclockwise.
\item Use residues to evaluate the improper integral
\[\int_0^{\infty}\frac{1}{x^4+1}dz.\]
\end{enumerate}
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