Accurate evaluation of an integral involving the product of two Bessel functions and a Gaussian
An integral of the form \(G(L)=\int^{R}_{0}f_ 1(r)V(r)f_ 2(r)dr\) is numerically evaluated, where \(f_ i(r)=K_ irJ_ L(K_ ir)\), \(i=1,2\), and \(V(r)=\exp (-r^ 2/a^ 2)\) with real numbers R and r, complex numbers \(K_ i\), and spherical Bessel functions \(J_ L\) of \(K_ ir\). This algorithm 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 \(L=1\). This idea comes from the fact that the values of G(-1) and G(0) are analytically expressed in terms of error functions and easily calculatable by means of a continued fraction with few error.
scientific article; zbMATH DE number 3978261