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\noindent{\Large{\textsc{Complex Variables \hfill Assignment 5.3  \\ Fall 2024 \hfill Due October 2}}}

\large

\vspace{1cm}

\begin{exercise}
Let $X = \C \cup \{ \infty \}$.  For $\epsilon>0$, define $D(\infty; \epsilon) = \{ z \in \C \, | \, |z|>\frac{1}{\epsilon} \} \cup \{ \infty \}$.  We say $U \subseteq X$ is {\em open} if for every $z \in U$ there exists $\epsilon > 0$ so that $D(z;\epsilon) \subseteq U$.

\begin{itemize}

\item[\bf a.]  Show that $X$ and $\varnothing$ are both open.

\item[\bf b.]  If $U_{\alpha}$ ($\alpha \in I$) are open subsets of $X$, show that
\[
\bigcup_{\alpha \in I} U_{\alpha}
\]
is also open.  Here $I$ is an arbitrary indexing set.

\item[\bf c.]  If $U_1, U_2, \ldots , U_n$ are open subsets of $X$, show that 
\[
\bigcap_{j=1}^{n} U_j
\]
is also open.

\end{itemize}

Parts {\bf a}-{\bf c} show that the collection of open subsets of $X$ is a {\em topology} on $X$.  It should be clear that the topology $\C$ inherits as a subspace of $X$ is the usual one.

\end{exercise}

\begin{exercise}
Let $f(z)$ be defined on a neighborhood of $\infty$.  Show that $f(1/w)$ is defined on a deleted neighborhood of $w=0$, and that
\[
\lim_{z \to \infty} f(z) = \lim_{w \to 0} f(1/w).  
\]
\end{exercise}

\begin{exercise}
Let $\C[X]$ denote the ring of all polynomials in $X$ with complex coefficients, and for $P(X) \in \C[X]$ let $\deg P$ denote the degree of $P(X)$.

\begin{itemize}

\item[\bf a.]  Let $P(X) \in \C[X]$ and let $\widetilde{P}(X) = X^{\deg P} P(1/X)$.  Prove that if $P(X)$ is nonzero, then $\widetilde{P}(X) \in \C[X]$ and $\widetilde{P}(0)$ is the leading coefficient of $P(X)$.

\item[\bf b.]  Let $P(X), Q(X) \in \C[X]$ and define $\widetilde{P}(X)$ and $\widetilde{Q}(X)$ as above.  Show that 
\[
\frac{P(1/X)}{Q(1/X)} = X^{\deg Q - \deg P} \frac{\widetilde{P}(X)}{\widetilde{Q}(X)}.
\]

\item[\bf c.]  Let $P(X), Q(X) \in \C[X]$ be nonzero.  Use part {\bf b.} and the preceding exercise to show that $\displaystyle \lim_{z \to \infty} \frac{P(z)}{Q(z)}$ exists (in $\C$) if and only if $\deg Q \ge \deg P$. 

\end{itemize}

\end{exercise}



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