A Generalization of the Schwarz-Christoffel Transformation.

The classical Schwarz-Christoffel transformation provides a formula for the conformal mapping of the half-plane onto a plane polygon. A generalization enlarging the class of image domains to include many-sheeted plane polygons with interior winding points was known already to at least Schwarz and Schlifli.' This note is concerned with a further simple extension of the formula which delivers a useful class of mappings defined by multiple-valued functions. These mappings have been applied elsewhere by the author to certain hydrodynamical problems.2 Let w(z) be a multiple-valued analytic function defined on the halfplane, Im z > 0. It is assumed that any regular function element of w(z) can be continued along arbitrary paths to all but a finite number of points. It will be convenient in what follows to consider the image of a singlevalued branch of w(z) obtained by making a suitable cut or cuts in the halfplane which connect all the branch points of w(z) with the boundary. If, on the image Riemann surface thus defined over the w-plane, the images of opposite points on the two edges of the cut are identified, we obtain an ideal Riemann surface or Riemannian manifold in the sense of Koebe.8 (The image of a branch point, if it exists, is included as interior point of the surface.) This domain is a single-valued image of the half-plane under the mapping defined by (a branch of) w(z). In the following we consider the conformal mapping of the half-plane onto certain Riemannian manifolds with polygonal boundaries. We have first, THEOREM 1. Let w(z) be a possibly multiple-valued analytic function defined on the half-plane, Im z > 0. Let the function elements for w(z) be such that in the neighborhood ofany point z = a,