Key points are not available for this paper at this time.
maps the open unit circle |z I < 1 (hereafter denoted by E) onto P the interior of an m-sided convex polygon. The vertices of the polygon are wj =fi(%) and the exterior angle(2) at the vertex wj is yj7r. Conversely if P is given, then z1, Z2 * * * Zm, c1, and c2 can be determined such that (1.1) maps E onto P, and moreover the origin can be carried into any preassigned point of P and the value of argf'(0) can be arbitrarily preassigned. The equation (1.1) subject to the conditions (1.2) and (1.3) is one form of the Schwarz-Christoffel transformation (3). Schwarz(4) stated that the formula (1.1) is easily generalized to the case where P is a multi-sheeted domain bounded by straight lines and containing branch points, and Christoffel(5) considered this generalization in some detail. Study('), Loewner(7), Gronwall(8), Bieberbach(9), Paatero(10), and
A. W. Goodman (Sun,) studied this question.