Math 6420 • PDE

Consider the transformation on the extended complex plane \(\overline{\mathbb{C}}\)
\[
  f(z)=w=z+\frac{1}{z}.
\]
\begin{enumerate}
\item[(a)] Find all possible fixed points.
\item[(b)] Show that the image of the points in the upper half \(z\)-plane that are exterior
to the circle \(\lvert z\rvert=1\) corresponds to the entire upper half \(w\)-plane.
\item[(c)] Find the image of the unit circle.
\item[(d)] Show that \(f(z)\) is a two-to-one function.
\end{enumerate}