Asymptotic behaviour of zeros of Bieberbach polynomials

Abstract

Let be a simply-connected domain in the complex plane and let Ω n π denote the n th degree Bieberbach polynomial approximation to the conformal map f of Ω onto a disc. In this paper we investigate the asymptotic behaviour ( as n ∞ → ) of the zeros of ' , n n π π and also of the zeros of certain closely related rational approximants to f. Our results 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\ , and Ω we present numerical examples illustrating this.

Convergence of Bieberbach polynomials in domains with interior cusps

“…x > ξ, n ∈ N. by (4.11). Hence the conformal mapping ϕ must have an analytic continuation through ∂G, by (4.15) and Theorem 2.1 of [19], which contradicts our assumption.…”

Bergman polynomials on an archipelago: Estimates, zeros and shape reconstruction

“…The result is given in Walsh [36, pp. 130-131] (see also [18,Thm 2.1]) for a single Jordan region and, as Walsh asserts, is immediately extendable to several Jordan regions. By applying Lemma 3.4 to f = K(•, ζ), and by using the reproducing property (2.2), in conjunction with (2.16) and (3.6), we obtain:…”

“…The result is given in Walsh [36, pp. 130-131] (see also[18, Thm 2.1]) for a single Jordan region and, as Walsh asserts, is immediately extendable to several Jordan regions.…”

Erdős–Turán Type Theorems on Quasiconformal Curves and Arcs

“…n does not have zeros in G except for z 0 . It was shown in [25] that the distributions of the zeros of ? n and ?$ n are governed by the location of the singularities of the mapping function f 0 .…”

“…n ] or [?$ n ]. By [25] the boundary L= G attracts zeros of ? n and ?$ n iff the function f 0 cannot be analytically extended to a neighborhood of G .…”