Math 6720 • Applied Complex Variables and Asymptotic Methods

Determine the conformal map \(f\) that takes the region \(D\) exterior to two circular cylinders
of radius \(R\) to the region outside a circular disc of unit radius centered at the origin (see
figure 1). Use the following steps:
\begin{enumerate}
\item
Show how the map \(\omega=i-2R/z\) takes the region \(D\) to the infinite strip \(0<\operatorname{Im}\omega<2\).
\item
Show how the map \(\xi=e^{\pi\omega/2}\) takes the infinite strip to the upper-half complex \(\xi\)-plane.
\item
Show how the map \(\eta=(\xi+i)/(\xi-i)\) takes the upper-half complex \(\xi\)-plane to the
region outside the unit disc, that is, \(|\eta|>1\).
\item
Find \(f\).
\end{enumerate}

\emph{Figure 1 description: The map \(f\) sends the \(z\)-plane region exterior to two vertically tangent circular cylinders of radius \(R\) to the \(\eta\)-plane region exterior to the unit disc. The original figure shows the two tangent circles on the left, an arrow labeled \(f\), and a unit circle in the \(\eta\)-plane on the right.}

Figure 1: Find the conformal map \(f\)