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In the branch of mathematics called potential theory, a quadrature domain in two dimensional real Euclidean space is a domain D together witha 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,
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.
It is known that quadrature domains exist for all values of k. There is an analogous definition of quadrature domains in Euclidean space of dimension d larger than 2. There is also an alternative, electrostatic interpretation of quadrature domains: a domain D is a quadrature domain if a uniform distribution of electric charge on D creates the same electrostatic field outside D as does a k-tuple of point charges at the points z1, …, zk.