91Ó°ÊÓ

a. Show that \(w=2 z+1 / z\) maps the exterior of the unit circle conformally onto the exterior of the ellipse: $$ \frac{x^{2}}{9}+y^{2}=1 $$ b. Find a conformal mapping of the exterior of the ellipse \(x^{2} / 9+y^{2}=1\) onto the exterior of a real line segment.

Given a conformal mapping \(f\) of \(R\) onto \(U\) (the unit disc) and \(z_{0} \in R\), find a conformal mapping \(g\) of \(R\) onto \(U\) with \(g\left(z_{0}\right)=0\) and \(g^{\prime}\left(z_{0}\right)>0\).