Techniques for Obtaining Non-Divergent Series as Alternatives to Divergent Asymptotic Series for Special Functions and Integrals

This work discusses approximate evaluations of some types of integrals arising in electromagnetic problems. An integral can be transformed into a divergent asymptotic series by replacing the function within the integral with a power series and then evaluating it term by term. Despite their divergence, asymptotic series are inevitably employed in computations, particularly in high-frequency electromagnetics. We develop series in various forms using two new techniques, demonstrating that their accuracy significantly surpasses that of conventional asymptotic series. We focus on the type of approximation applied to a function within an integral. Approximations in the form of power series often diverge outside a limited area known as the region of convergence (ROC), which negatively affects the accuracy of the resulting asymptotic series. In our earlier studies, an engaging method was successfully employed to approximate functions using non-divergent exponential functions. Our new techniques extend this method for evaluating integrals. In the examples provided, we derive new series by applying the techniques to the integral definitions of various special functions. Unlike conventional asymptotic series, these new series do not diverge and exhibit significantly greater accuracy. This suggests their potential to replace many commonly used asymptotic series in high-frequency electromagnetics.