A One-Parametric Family of Conformal Mappings from the Half-Plane onto a Family of Polygons

Abstract

We consider a one-parametric family of conformal mappings from the upper half-planeonto the family of the polygons obtained by an angle-preserving shift of one or more vertices of someinitial polygon. We obtain some differential equationfor the family of mappings and some system of ordinary second-orderdifferential equations for the preimages of the vertices of the family of polygonswith Cauchy initial-value conditions.