Accurate evaluation of an integral involving the product of two bessel functions and a gaussian

Abstract

Abstract The integral to be evaluated is of the form G(L)=∫ R 0 K 1 K 2 r 2 j L (K 1 r)j L (K 2 r)e −r 2 d 2 dr , where R and a are real, and the wave numbers K 1 and K 2 can be complex. The js are spherical Bessel functions. Our method makes use of a recursion relation between G ( L ), G ( L + 1), and G ( L − 1), together with the values of G for L = 0 and L = −1. The latter are expressed in terms of error integrals of complex argument and are evaluated numerically with high accuracy by means of a continued fraction. Four methods are presented for calculating G at integer values of L . One consists of solving analytically the inhomogeneous finite difference recursion equation in terms of sums from 0 to L and from L + 1 to ∞ of quantities which involve the two linearly independent spherical Bessel functions of argument z 3 = (K 1 a)(K 2 a) 2i . The other three methods consist in numerically evaluating the recursion relation, either upwards in L , starting with the known values G (0) and G (−1), or downwards in L , either starting with two G values taken equal to zero, or following a method described by Olver. The method of Olver is found to be the one generally most useful, in that it gives a reliable estimate of the truncation error. Accuracies of twelve significant digits are usually achieved, on a computer (IBM 3081) using 16 bit words, as is demonstrated from the comparison of the four methods in numerical examples. The computing time is much less than the methods involving radial mesh sums, by factors of 5 or more, depending on the values of K 1 R and K 2 R .