Note on ``Electromagnetic Response of a Large Circular Loop Source on a Layered Earth: A New Computation Method'' by N. P. Singh and T. Mogi

Abstract

Infinite range integrals of products of Bessel functions occur in a wide variety of problems from physics and engineering and are notoriously hard to evaluate. In SINGH and MOGI (2005) integrals of this type arise in the expressions of electromagnetic field components at a point on or above the surface of n-layered earth due to a circular loop. In the appendix to that article, the authors derive analytical expressions for three of these integrals, which allow a direct computation and are therefore of substantial interest. However, integrals such as these should be handled with care because of their highly discontinuous behavior and we believe it should be pointed out that the mathematical reasoning of the authors which led to these expressions is not entirely correct. For two of the three integrals this does not matter much since the final results are still valid, but one of the expressions obtained by the authors is incorrect. Moreover, if readers endeavor to use this (flawed) reasoning to compute similar integrals, the correctness of the results is unpredictable: They may be accidentally right (as in the case under consideration) or entirely wrong. The purpose of this note is twofold. Firstly, we wish to correct the errors and simplify the correct results given in SINGH and MOGI (2005). This simplification should make the computations more efficient. Secondly, we present an entirely different approach to compute integrals of this and a more general type, using an algorithm described in VAN DEUN and COOLS (2005, 2006a,b), which we believe may be of interest to people in this field.