Research Article TM Electromagnetic Scattering from PEC Polygonal Cross-Section Cylinders: A New Analytical Approach for the Efficient Evaluation of Improper Integrals Involving Oscillating and Slowly Decaying Functions Mario Lucido , Chiara Santomassimo, Fulvio Schettino, Marco Donald Migliore , Daniele Pinchera , and Gaetano Panariello D.I.E.I. and ELEDIA Research Center (ELEDIA@UniCAS), University of Cassino and Southern Lazio, 03043, Cassino, Italy Correspondence should be addressed to Mario Lucido; lucido@unicas.it Received 30 July 2018; Accepted 15 November 2018; Published 10 January 2019 Academic Editor: Ping Li Copyright © 2019 Mario Lucido et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te analysis of the TM electromagnetic scattering from perfectly electrically conducting polygonal cross-section cylinders is successfully carried out by means of an electric feld integral equation formulation in the spectral domain and the method of analytical preconditioning which leads to a matrix equation at which Fredholm’s theory can be applied. Hence, the convergence of the discretization scheme is guaranteed. Unfortunately, the matrix coefcients are improper integrals involving oscillating and, in the worst cases, slowly decaying functions. Moreover, the classical analytical asymptotic acceleration technique leads to faster decaying integrands without overcoming the most important problem of their oscillating nature. Tus, the computation time rapidly increases as higher is the accuracy required for the solution. Te aim of this paper is to show a new analytical technique for the efcient evaluation of such kind of integrals even when high accuracy is required for the solution. 1. Introduction Spectral domain formulations are particularly suitable for the analysis of a wide class of electromagnetic problems ranging from the propagation in planar guides and waveguides or the radiation by planar antennas to the scattering from cylin- drical structures or planar surfaces involving homogeneous or stratifed media, just to give some examples. In general, the obtained integral equation in the spectral domain does not admit a closed form solution; hence, numerical schemes have to be adopted. Te fast convergence of such methods is a key point. When dealing with polygonal cross-section cylindrical structures or canonical shape planar surfaces, just for examples, a well-posed matrix operator equation can be obtained by means of the method of analytical precondi- tioning [1]. It consists of the discretization of the integral equation by means of Galerkin’s method with a suitable set of expansion functions leading to a matrix equation at which Fredholm’s or Steinberg’s theorems can be applied [2, 3]. In the literature, it has been widely shown that this goal can be fully reached by selecting expansion functions reconstructing the physical behaviour of the felds on the involved objects with a closed-form spectral domain counterpart [4–15]. With such a choice, few expansion functions are needed to achieve highly accurate results and the convolution integrals are reduced to algebraic products. However, the obtained matrix coefcients are improper integrals of oscillating and, in the worst cases, slowly decaying functions to be numerically evaluated. Te classical analytical asymptotic acceleration technique (CAAAT), consisting of the extraction from the kernels of such kind of integrals of their asymptotic behaviour while the slowly converging integrals of the extracted parts are expressed in closed form, allows us to obtain faster decaying integrands without overcoming the most important problem of their oscillating nature. Consequently, the conver- gence of the accelerated integrals becomes slower and slower as higher is the accuracy required for the solution. In order to overcome this problem, a novel technique has been proposed for the analysis of the propagation in Hindawi Advances in Mathematical Physics Volume 2019, Article ID 7902836, 9 pages https://doi.org/10.1155/2019/7902836