An Analytical Approach for the Efficient Evaluation of a Class of Improper Integrals Involving Oscillating and Slowly Decaying Functions

The analysis of a wide class of electromagnetic propagation, radiation and scattering problems can be readily carried out by means of suitable full-wave spectral domain formulations and the method of analytical preconditioning which allows us to transform the obtained integral equation in a matrix equation at which Fredholm/Steinberg theory can be applied. Hence, the convergence of the discretization scheme is guaranteed. Moreover, the choice of expansion functions reconstructing the physical behavior of the unknowns leads to a fast convergence. Unfortunately, the matrix coefficients are improper integrals involving oscillating and, in the worst cases, slowly decaying functions. Thus, the computation time rapidly increases as higher is the accuracy required for the solution. The aim of this paper is to show an analytical technique for the efficient evaluation of such kind of integrals even when high accuracy for the solution is required.