Multiply connected quadrature domains and the Bergman kernel function
type: Item
instance of: scholarly article
title: Multiply connected quadrature domains and the Bergman kernel function
zbMATH Open document ID: 1186.30006
published in: Complex Analysis and Operator Theory
publication date: 2009-09-18T00:00:00Z
review text: It is well known that a quadrature domain is a domain in the complex plane satisfying an identity of the form \[ \int \int_{D}\;f(z)dxdy=\sum_{k=0}^{N}\sum_{j=0}^{n_{k}-1}a_{jk}f^{(j)}(z_{k}), \] where \(f(z)\) is an arbitrary analytic function in \(D\), \(f^{(j)}(z_{k})\) denotes the \(j-th\) derivative of \(f\) at \(z_{k}\), \(\{ a_{jk}\}\) is a set of complex constants, \(N\) is an integer, and \(\{n_{k}\}\) is a set of integers. The reviewer characterized doubly connected quadrature domains by showing that these are the conformal images of an annulus under a certain type of functions described in his dissertation in 1977. The authors generalize this result to domains of arbitrary connectivity. This nice and technical paper also contains useful information about the domain functions such as Green's function, Bergman's kernel function, the Schottky-Klein prime function, and the Schwarz function for multiply connected canonical domains bounded by circles.
reviewed by: Yusuf Avci
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30C40
zbMATH DE Number: 5604852
zbMATH Keywords: multiply connected domains
zbMATH Keywords: quadrature domains
zbMATH Keywords: Bergman kernel.
author: Darren G. Crowdy
author: Jonathan S. Marshall
MaRDI profile type: Publication
full work available at URL: s11785-008-0090-x
OpenAlex ID: W2035028971
DOI: 10.1007/S11785-008-0090-X
description: scientific article; zbMATH DE number 5604852