Zeros of Faber polynomials for Joukowski airfoils

2018/09/27 by N. Levenberg, Levenberg, N., F. Wielonsky +1
Mathematics · #30C15 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1809.10439

openalex publication_date 2018/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for K are not understood in this general setting. Joukowski airfoils provide a particular class of such sets. We determine the (unique) weak-* limit of the full sequence of normalized counting measures of the Faber polynomials for Joukowski airfoils; it is never equal to the potential-theoretic equilibrium measure of K. This implies that many of these airfoils admit an electrostatic skeleton and also explains an interesting class of examples of Ullman related to Chebyshev quadrature.

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