Comments on the analogy between sommerfeld and Schelkunoff integrals for the analysis of dipoles over imperfect half-planes

Summary form only given. Recently, Schelkunoff integrals were exploited to formulate a physics-based Green´s function for analysis of vertical electric dipole radiation over an imperfect ground plane (W. Dyab, T. Sarkar, and M. Salazar-Palma, IEEE transactions on Ant. & Prop. vol. 61, no. 8, August 2013). Schelkunoff integrals were proved to be much more suitable for numerical computation than Sommerfeld integrals which are used conventionally to solve problems in multi-layered media. Schelkunoff integrals have no convergence problem on the tail of the contour of integration, especially when the fields are calculated near the boundary separating the media and for large source-receiver separations. On the other hand, however, Schelkunoff integrals suffer from convergence problems when the fields are to be calculated on the axis of the source. Since it is more practical for the fields to be calculated near the interface and not on the axis of the source, Schelkunoff integrals gain some research interest due to its numerical behavior in those regions. In this paper, Schelkunoff integrals are utilized to derive a Green´s function for the case of a horizontal electric dipole radiating over an imperfect ground plane.Originally, Schelkunoff derived the modified Sommerfeld integrals to solve problems with cylindrical rather than planar boundaries. The reason for which Schelkunoff did not use his modified integrals to solve for planar boundaries was not clear in his book (S. A. Schelkunoff, Electromagnetic waves, D. Van Nostrand Company, Inc. 1943), and was never investigated after him. Also due to the wide acceptance of Sommerfeld integrals for the analysis of problems with planar boundaries, the use of the modified Sommerfeld integrals (Schelkunoff integrals) for such kind of problems needs to be justified. However, a direct comparison between the results of the two integrals is not a trivial task, especially in the regions where one of the integrals is divergent. - n this paper, a rigorous mathematical analysis is presented to attempt to answer the previously mentioned question.