Quadrature domains
In the branch of mathematics called potential theory, a quadrature domain in two dimensional real Euclidean space is a domain D (an open connected set) together with a finite subset {z1, …, zk} of D such that, for every function u harmonic and integrable over D with respect to area measure, the integral of u with respect to this measure is given by a "quadrature formula"; that is, ∬ D u d x d y = ∑ j = 1 k c j u ( z j ) , {\displaystyle \iint _{D}u\,dxdy=\sum _{j=1}^{k}c_{j}u(z_{j}),} where the cj are nonzero complex constants independent of u. The most obvious example is when D is a circular disk: here k = 1, z1 is the center of the circle, and c1 equals the area of D. That quadrature formula expresses the mean value property of harmonic functions with respect to disks.