Asymptotic behaviour of zeros of Bieberbach polynomials

摘要

Let Ω be a simply-connected domain in the complex plane and let πn denote the nth-degree Bieberbach polynomial approximation to the conformal map f of Ω onto a disc. In this paper we investigate the asymptotic behaviour (as n→σ) of the zeros of πn, πn′ and also of the zeroes of certain closely related rational approximants to f. Our result show that, in each case, the distribution of the zeros is governed by the location of the singularities of the mapping function f in C⧹ω, and we present numerical examples illustrating this.