Asymptotic Behaviour of the Mapping Function at an Analytic Cusp with Small Perturbation of Angles

Abstract

We say that a simply connected domain in the complex plane has an analytic cusp at the origin if its boundary at the origin is given by two regular analytic curves which form a cusp. We investigate the asymptotic behaviour of a conformal map onto the upper half plane at the origin. Therefore we introduce the notion of an analytic cusp with small perturbation of angles. Assuming this condition we determine the asymptotic behaviour in terms of a holomorphic function and give upper bounds for the derivatives of the mapping function.