On the Schwarz-Christoffel transformation and 𝑝-valent functions

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