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\noindent{\Large{\textsc{Complex Variables \hfill Assignment 6.1  \\ Fall 2024 \hfill Due October 9}}}

\large

\vspace{1cm}

\begin{exercise}
Let $P$ and $Q$ be polynomials in $z$ with complex coefficients.  
\begin{itemize}

\item[\bf a.] Use Exercise 5.3.3 to show that 
\[
\lim_{z \to \infty} \frac{P(z)}{Q(z)} = \infty
\]
if and only if $\deg P > \deg Q$.

\item[\bf b.]  Conclude that if $P$ is nonconstant, then
\[
\lim_{z \to \infty} P(z) = \infty.
\]
\end{itemize}


\end{exercise}

\begin{exercise}
Let $f$ and $g$ be complex-valued and defined in a neighborhood of $\alpha \in \C \cup \{ \infty \}$.  Show that if $\displaystyle \lim_{z \to \alpha} f(z) = a \in \C^{\times}$ and $\displaystyle\lim_{z \to \alpha} g(z) = \infty$, then
\[
\lim_{z \to \alpha} f(z) g(z) = \infty.
\]
[{\em Suggestion.}  Use Exercise 4.3.3.]
\end{exercise}

\begin{exercise}
If we represent $\R^3$ by $\C \times \R$, the standard equation for the unit sphere $S^2$ becomes $|z|^2 + v^2 = 1$, where $(z,v) \in \C \times \R$.  The complex plane itself is represented by points of the form $(z,0)$, or the equation $v=0$.  Let $N = (0,1)$ represent the north pole of $S^2$

\begin{itemize}

\item[\bf a.]  Define $\pi : S^2 \setminus \{ N \} \to \C$ as follows.  Given a point $p \in S^2 \setminus \{ N \}$, draw the line through $N$ and $p$, and define $\pi(p)$ to be the point where this line meets the complex plane $v=0$.  Show that if $p=(z,v) \in S^2$, then
\[
\pi(z,v) = \frac{z}{1-v}.
\]

\item[\bf b.]  Conversely, use the same construction to show that
\[
\pi^{-1}(z) = \left( \frac{2z}{|z|^2 + 1}, \frac{|z|^2-1}{|z|^2+1}  \right)
\]
for $z \in \C$.

\item[\bf c.]  Show that $\displaystyle \lim_{z \to \infty }\pi^{-1}(z) = N$.

\end{itemize}

\end{exercise}



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