SOME IMPLICATIONS OF THE LAPLACE TRANSFORM INVERSION ON SEM COUPLING COEFFICIENTS IN THE TIME DOMAIN - Wilson Pearson, L.; Wilton, Donald R.; Mittra, Raj (Electromagnetics, 1982)
The issues associated with the choice of.coupling coefficient forms in the singularity expansion and the closely-related subject of whether expansions may be written without an entire function present have persisted as major points of concern and confusion in the development of singularity expansion method theory. In this paper we show that the variety of choices available to one in constructing a singularity expansion relates directly to the large-s asymptotic behavior of the resolvent kernel for the integral equation from which the expansion is derived. The choices range from a cautious extreme in which causality is enforced explicitly to a bold extreme where one depends upon the expansion to sum to a causal result well ahead of the time of arrival of the excitation. By appealing to recently-reported estimates of this asymptotic behavior we define what the acceptable constructions are and discuss them on a comparative basis. A geometrical interpretation of the domain of integration for specific useful choices is provided. It is shown that one may choose the so-called turn-on time outside the acceptable range of choices, but at the expense of including an entire function contribution in the representation. The contribution of this entire function is generally significant enough that it cannot be neglected. It is shown, too, that the ordering of the pole series so that all poles associated with a given eigenvalue of the integral operator are grouped can admit to a choice of a bold coupling coefficient construction. The practicality of this eigenset summation, however, is questionable since the availability of eigenset information hinges on the separability of Maxwell's equations for a particular object shape and since for all but simply rotationally symmetric shapes the pole series may require augmentation by a branch integral for a given eigenvalue. An appendix is provided illustrating a number of the points made in the body of the paper for the simple example of a voltage wave excited on a uniform lossless transmission line.