Higher-order Newton-cotes and gauss-quadrature integration rules to solve carson and pollaczek integrals

The problem of electromagnetic wave propagation, despite its importance, does not have a simple analytical solution. Therefore, approximations—such as assuming a semi-infinite homogeneous ground plane—are of practical interest. Using quasi-static approximations, Carson and Pollaczek derived integral equations to calculate the electromagnetic field produced by a horizontal current over a lossy ground plane. Carson proposed the first solution to these expressions using power series expansions, but this approach lacks uniform convergence. Since then, various efforts have been made to obtain more accurate solutions. In this sense, main approaches have been developed to solve these integral equations. The first involves modifying the integrand to obtain an analytic solution, while the second relies on using numerical integration techniques, which are essential for solving integrals without closed-form solutions. The accuracy of numerical methods is influenced by the choice of integration scheme, order, and number of samples. However, theoretical expectations of improved accuracy with higher-order methods and increased sample points are often limited by numerical representation constraints. One of this constrain is the finite bit representation in binary calculations. Despite these limitations, numerical methods remain the only viable approach for certain integrals. This work presents the implementation of Newton and Gauss integration methods, analyzing their performance concerning method type, order, and sample count. Since Carson and Pollaczek’s equations include a decreasing exponential term, the infinite upper limit is replaced by a finite bound without exceeding a predefined error threshold. By applying this limit substitution and numerical techniques, we obtain a new solution that ensures uniform convergence across all cases.