Conformal transformations

\(1.\hspace{0,5cm}\) Show that if two circles don't intersect then there is a pair of points \(z_{1},z_{2}\) that is symmetric with respect to both circles.

\(2.\hspace{0,5cm}\) Show that any two circles that don't intersect can be mapped conformally to concentric circle.

\(1.\hspace{0,5cm}\) Using a fractional linear transformation that maps one of the circles to a line (and the other to a circle) we can reduce the problem to that in the previous exercise.

\(2.\hspace{0,5cm}\) Call the circles \(S_{1}\) and \(S_{2}\). Using the previous example, start with a pair of points \(z_{1},z_{2}\) which are symmetric with respect to both circles. Next pick a fractional linear transformation \(T\) that maps \(z_{1}\) to \(0\) and \(z_{2}\) to infinity. For example,

\(\hspace{4cm}T(z)=\frac{z-z_{1}}{z-z_{2}}.\)

Since \(T\) preserves symmetry \(0\) and \(∞\) are symmetric with respect to the circle \(T(S_{1})\). This implies that \(0\) is the center of \(T(S_{1})\). Likewise \(0\) is the center of \(T(S_{2})\). Thus, \(T(S_{1})\) and \(T(S_{2})\) are concentric.