# Conformally mapping an ellipse into the unit circle

I'm currently studying for a complex analysis final and I don't think I've really developed the intuition for conformal mappings yet. I'm attempting a problem from Ahlfors: map the outside of the ellipse $(x/a)^2+(y/b)^2=1$ onto $|w|<1$ with preservation of symmetries. I believe I should use the inverse of the Joukowski transformation at some point (as it maps ellipses to circles) to get a circle of radius $R$ and then rescale. However, I run into trouble when I try to find an $R$ that will work. Any thoughts?

It is through a special Joukowski transformation $z=\alpha w + \beta/w$ with real constants $\alpha$ and $\beta$ (assuming $z=x+iy$ and $w=u+iv$). The constants are determined using the fact that the boundary of the ellipse is mapped to the boundary of the disk $|w|=1$, or $$z = x+iy = \alpha (u+iv)+\beta/(u+iv)=\alpha (u+iv)+\beta (u-iv).$$ Then the equation $u^2+v^2=1$ becomes $$\frac{x^2}{(\alpha+\beta)^2} + \frac{y^2}{(\alpha-\beta)^2}=1$$, from which you can determine $\alpha$ and $\beta$ by $|\alpha+\beta|=a, |\alpha-\beta|=b$.