  Given a bounded n-connected domain in the plane bounded by non-intersecting
Jordan curves, and given one point on each boundary curve, L. Bieberbach proved
that there exists a proper holomorphic mapping of the domain onto the unit disc
that is an n-to-one branched covering with the properties that it extends
continuously to the boundary and maps each boundary curve one-to-one onto the
unit circle, and it maps each given point on the boundary to the point 1 in the
unit circle. We modify a proof by H. Grunsky of Bieberbach's result to show
that there is a rational function of 2n+2 complex variables that generates all
of these maps. We also show how to generate all the proper holomorphic mappings
to the unit disc via the rational function.
