On a modified form of Pocklington equation for thin, bent wires

Classical integral or integro-differential equations of the Pocklington and Hallen type, describing radiation and scattering of electromagnetic fields by thin, ideally conducting wires, are of significant practical interest and have been extensively studied. These equations follow from the boundary condition that requires a vanishing of the tangential component of the total electric field at the wire surface. The total electric field consists of both a known incident field and a scattered field that is due to a generally unknown current induced in the wire. The scattered electric field for a given point on the wire surface consists both of a “far” field at distant points significantly exceeding the wire's radius $a$, and by a “near” field due to arbitrarily nearby points. Expressions for the “near” field include a logarithmic singularity in the kernel of the associated Pocklington equation. This singularity is an important feature that makes the Pocklington equation solvable and well-posed. Thus, the Pocklington equation in its standard form can be considered as a Fredholm integral equation of the first kind with a singular kernel.