A method for computing the circular coverage function

A Method for Computing the Circular Coverage Function

“…In this case, in which A and k are arbitrary, the center of the circle of integration can always be taken as offset a distance of <tD from the origin along the positive x axis by simply introducing a rotation of axes through the angle arc tan ( -I. Moreover, by introducing the integral expression for Io(x) as given by equation (4), the circular coverage function, P(R, D), [1], [4], [6], [7], [9] where R = R/ax , D2 m (A2 + k ) /a/'-The function dP(R, D)/dR is required for computing the inverse function, R(P, D), by the Newton-Raphson procedure (Appendix C, [4]) and is also of use in computing P(R, D) itself (see equation (9)). This function is obtained straightforwardly from equation (5) as (6) g = ßexp(-^±^)/0(ßö).…”

“…FUß -D (ID The detailed derivations of equations (10), (11) are given in [4]. Briefly, to obtain equation ( (See page 76, [8]).…”

“…Fuller details are given in [4]. From equations (13) and (19), S2n is the general term in the series obtained by multiplying every term of the Taylor series for I0(RD) by exp [-(R2 + D2)/2], and equations (23) and (25) are obtained immediately.…”

“…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.…”

“…If h = k = 0, a special case identified as the V(K, c) or elliptical normal probability function (sometimes known by other titles, for example, the generalized circular error function) [4] …”