exterior of circle onto exterior of ellipse mapping
Left image: color=Abs(z)
Right image: w=z+ 1/z

Left image: color=Abs(z) for z: Abs(z)>2
Right image: w=z+ 1/z
One can see that it is conformal mapping from exterior of circle centered at the origin of complex z-plane onto exterior of ellipse in complex w-plane

so :
w = Psi(z) = z + 1/z
z = Phi(w) = (w + sqrt(sqr(w)-4))/2