Integration of the general bivariate Gaussian distribution over an offset circle

Introduction. An efficient method is described in this paper for the numerical evaluation by a high-speed digital computer of the integral of any uncorrelated elliptical Gaussian distribution over the area of any arbitrarily centered circle in the plane. Actually, by introducing at most four linear transformations of coordinates, the present method can be used to cover the general problem of integrating any correlated or uncorrelated bivariate normal distribution over an arbitrary ellipse in the plane. The basic integral to be evaluated is expressible in the form ( 1 ) x p (r) exp [-2 2+ Y 2) dxdy, where the exponential represents an uncorrelated bivariate normal distribution centered at the origin with standard deviations o-x and o-. A point target T is assumed at some arbitrary point (h, k) in the Oxy plane. The distance r is ordinary distance [(x - h)2 + (y - k1)2]112, where (x, y) is an arbitrary burst point in the plane. The function p(r) by definition has the value one if r does not exceed a given constant R, and p(r) is zero otherwise. This has the effect of limiting the field of integration to a circle with center at (h, k) and radius R. Hence the integral can be written as