Math 6720 • Applied Complex Variables and Asymptotic Methods

Bilinear transformations

Consider the transformation on the extended complex plane \(\mathbb{C}_\infty\)
\[f(z)=w=z+\frac{1}{z}.\]
\begin{enumerate}
\item
(7 points) Find all possible fixed points.
\item
(9 points) Show that the image of the points in the upper half \(z\)-plane that are exterior
to the circle \(|z|=1\) corresponds to the entire upper half \(w\)-plane.
\item
(9 points) Find the image of the unit circle.
\end{enumerate}

\item