A practical method for the numerical evaluation of Sommerfeld integrals

Abstract
Sommerfeld integrals, which appear frequently in dipole radiation problems, often must be evaluated numerically. Gauss-Laguerre quadrature is an effective integration method, provided the horizontal distance\rhofrom the source to the receiver is less than their vertical separation\zeta. A complementary method is to use Romberg adaptive quadrature to integrate over the positive and negative half-cycles of the integrand, then from a sequence of approximations by summing the contributions from successive half-cycle subintegrals, accelerating the convergence of this sequence using the Shanks transformation implemented via Wynn's algorithm. A comparison of the efficiencies of these techniques favors Gauss-Laguerre quadrature for\rho/\zeta < 1and the Romberg-Shanks method otherwise.