Potential at infinity of a polynomial image of the disk
type: Item
instance of: scholarly article
title: Potential at infinity of a polynomial image of the disk
zbMATH Open document ID: 0964.31002
DOI: 10.1007/BF02467117
publication date: 2001-07-22T00:00:00Z
review text: Let \(q(t)= a_0t+\cdots+ a_nt^{n+1}\), \(a_0\neq 0\) real, be a polynomial univalent in the unit disc \(\Delta\). Let \(D= q(\Delta)\) and \(p(z)= \frac{1}{\pi} \int_D \frac{d\mu(\xi)}{z-\xi}\) be the potential of the domain \(D\). Then for large \(z\), \(p(z)= \frac{c_0}{z}+ \frac{c_1}{z^2} +\cdots+ \frac{c_n}{z^{n+1}}\) with \(c_0\neq 0\) real and \(c_k\) polynomials in \(a_0, a_1,\dots, a_n, \overline{a}_1,\dots, \overline{a}_n\). Thus, one sets up a map \(\eta\) by defining \(\eta(q)= p\). In this note, by calculating the Jacobian of \(\eta\), the author makes some interesting remarks about the nature of \(\eta\), useful in the inverse problem of recovering \(D\) from \(p(z)\).
reviewed by: Victor Anandam
Mathematics Subject Classification ID: 31A25
zbMATH DE Number: 1452682
zbMATH Keywords: potential of a domain
zbMATH Keywords: domain reconstruction
author: Stepan Orevkov
MaRDI profile type: Publication
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1452682