On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary

Abstract

Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial Fn(z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)]n. Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of ∂Ω, we obtain asymptotic formulas for Fn that yield fine results on the limiting distribution of the zeros of Faber polynomials.