The Mellin Transform Method: Electromagnetics, Complex Analysis, and Educational Potential [Education Corner]

Abstract

The Mellin Transform Method (MTM) is a method of analytical evaluation of definite integrals that utilizes key concepts of Complex Analysis (CA). Though sometimes laborious, the algorithmic, step-wise character of the method as well as its vast field of applications render it powerful and versatile. In particular, certain integrals that are presented in the literature without proof (or with proofs that are ingenious and non-intuitive) can be directly attacked in a standard manner. This is true for the three examples presented herein, all taken from standard and advanced electromagnetics textbooks. The particular goal of this paper is two-fold. For one, the effectiveness of the MTM is demonstrated for the three aforesaid examples. It becomes obvious that the treatment of the most elementary example is no way different than that of the other two. For another, the educational virtues of the MTM in demonstrating key concepts of CA are highlighted. CA is a difficult mathematical area, usually abstract, and often sidelined in worldwide engineering curricula. In this work, it is proposed to incorporate the MTM evaluations-as a mathematical digression-into typical microwaves or advanced electromagnetics courses by breaking down the method into 7 steps, with each one exhibiting rich-elementary or advanced-CA concepts such as analytic functions, the gamma function, generalized hypergeometric functions, contour integrals involving meromorphic function, as well as the important idea of analytic continuation. The MTM is more likely to benefit students who will end up working in research environments (rather than engineers who will work in industrial settings).