On the Surface Limit in the Boundary Integral Equations of Electromagnetics

Boundary integral equations of electromagnetics are enforced at a current-carrying surface, where the kernels are strongly singular and give rise to discontinuous integrals, which must be evaluated in the principal-value sense, after the contributions from the neighborhood of the singularity are analytically extracted. There exist two basic techniques of taking the surface limit: one is to make the field point approach a small disk on the current-carrying surface and to subsequently let the disk radius vanish; another is to deform the surface by creating a hemispherical indentation about the field point and to shrink the radius of the dent to zero. In this paper, both methodologies are examined and it is demonstrated that the often used hemispherical indentation approach is flawed and can lead to incorrect results.