Multiply connected quadrature domains and the Bergman kernel function
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.
scientific article; zbMATH DE number 5604852