A method for computing the circular coverage function

In this paper an efficient method is described for the numerical evaluation, with a high-speed digital computer, of a special case of the integral of an uncorrelated bivariate Gaussian distribution centered at the origin over the area of an arbitrarily placed circle in the plane.This function, popularly known as the circular coverage function or as the non-central chi-square distribution for two degrees of freedom*, can be written as where S is the circle: (x -h)2 + (y -k)2 = (o-R)2, where ax = arv = -o-, and oD is the radial distance from the origin to the center (h, k) of the circle of integration, S. Because of the equivalence mentioned above, a great deal of published literature applies.The papers [13], [15], suggested by the referee, list a large number of such references.The average computing time for the calculation of the integral in equation ( 1) to six decimal digits, by the method of this paper, is six milliseconds on the IBM 7090 and ten milliseconds on NORC.An extensive inverse table, which is described in the last section of this paper and which is given in [4], has been computed with R as a function of P and D. A condensed version, Table 1, is presented herein.In the general case [3], [11] suppose the uncorrelated bivariate Gaussian distribution centered at the origin of an Oxy Cartesian coordinate system has standard deviations ax ,