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\centerline{\bf Complex Analysis, Examination 1, Math 505}
\smallskip
\centerline{\bf October14, 2002}
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1. Let $(S,d)$ be a metric space.
\roster
(a) Define, in terms of $d$, what is meant by saying that a
subset $U$ of $S$ is an open set.
(b) Define what is meant by saying that $f:S\longrightarrow
\Bbb R$ is a continuous mapping.
(c) Fix a point $p\in S$ and define
$g:S\longrightarrow \Bbb R$ by
$$g(x)=\sin \{(d(x,p))^2\}.$$
Show that $g$ is continuous on $S$.
(You may assume that the sine function is continuous on $\Bbb
R$)
\endroster
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2. Let $(S,d)$ be a metric space and $K$ a subset of $S$.
\roster
(a) Define what is meant by saying $K$ is compact
(b) Show, from your definition of compactness, that a closed
subset of a compact set is compact.
(c) Show that the intersection of a nested decreasing sequence
of compact sets is a nonempty compact set. Must it be connected?
\endroster
\newpage
3. Let $\gamma:[\alpha,\beta]\longrightarrow\Omega$ be a
piecewise smooth curve in a region $\Omega$ on which $p$ and
$q$ are continuous real or complex valued functions.
\roster
(a) Write down precisely, as a definite integral, what is meant
by
$\int_{\gamma} pdx+qdy.$
(b) If $F(x,y)=\int_{\gamma} pdx+qdy$ is independent of the
piecewise smooth path $\gamma$ chosen within $\Omega$ from some
fixed point $(x_0,y_0)\in\Omega$ to any $(x,y)\in\Omega$, then
compute
$\frac{\partial F}{\partial z}.$
(c) If $f(z)$ is an analytic function on $\Bbb C$ with real
part $u=x^2-y^2,$ then determine an expression for $f(z)$. Give
reasons and write your answer in terms of the variable $z$.
\endroster
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4. Let $f$ be an analytic function on the open unit disk $D$
centred at the origin. Explaining any notation introduced,
\roster
(a) state Cauchy's Theorem for $f$,
(b) state the Cauchy integral formula for $f$,
(c) evaluate $\int_{\gamma} \frac{f(z)}{z-1}dz$ where $\gamma$
is any piecewise smooth closed curve in $D$.
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