On the mapping functions of domains extended by sector reflections

Abstract

A plane domain G w , symmetric with respect to the origin O, is extended by “sector reflections” to a symmetric domain Ğ w . A “mapping function” [Wbreve](z) of Ğ w (mapping the unit disk G z onto Ğ w ) is constructed in an elementary way in terms of a mapping function w(z) of G w using essentially the reflection principle. In the case of polygonal domains considered here, the results can be interpreted as relations between Schwarz-Christoffel integrals. This method applies to most domains considered in [2]. where only the ratio of conformal radii Ro(Ğ w )/R o (G w ) was determined. The study of the mapping functions themselves was suggested by Max Schiffer.