Asymptotic expansions of an integral of the free-space Green's function

Abstract

In this paper, we derive the expressions of the three first terms in the asymptotic expansions of the integral I_m(, z; ', z') = ∫x o R(') cos(m') d' as k→ ∞. R(') is the distance between point P = (p, z, = 0) and a point P' = (p', z', ') situated on a circular loop of radius ' and axis z. The numerical values of I_m derived from these asymptotic expansions are compared with the "exact" values computed with a Gauss-Legendre quadrature rule. Accurate results are obtained for sufficiently large values of k even when P is close to the loop, and a reliable estimate of the error is defined.