Pole contributions to electromagnetic fields in the light of a modified saddle point technique

The content of this paper is of interest to those dealing with complex integrals which contain pole singularities, for instance in problems of electromagnetic theory where the pales correspond to modes of a structure, and where one is interested in the effect of these modes on the total field of the excited structure. Examples of such modes could be Zenneck waves, leaky waves, or trapped slow surface waves. Although mathematical techniques for dealing with complex integrals are well developed, the interpretation of the results is not. The subject under discussion in this paper is the interpretation of the mathematical results obtained when the presence of a pole is accounted for in the saddle point evaluation of a complex integral. Consider the following integral in the complex
\phi
-plane.
I = \int\liminf{c} F(\phi)e^{-jkR \cos(\phi-\theta)+j\omega t} d\phi
where
kR
is a large parameter, where the exponential term has a saddle point at
\phi = \theta
, and where
F (\phi)
has a pole at
\phi = \phi_{p}
. If the parameter kR is large enough, the integral may be approximated by the ordinary saddle point expansion plus a residue term that is added if in deforming the path of integration the pole is crossed. Interpretation of this approximation will be discussed.