On an Analytical Representation of an Integral Related to the Fock Integral Appearing in the Calculations of the Electromagnetic Fields of Dipole Sources at the Interface between Two Half-Spaces

Abstract

The Fock integral is called after Fock who introduced it to theoretically analyze the electromagnetic field of magnetic dipoles at the boundary of a uniform conducting (nonmagnetic) half-space and obtained its analytical expression in terms of cylindrical functions. Detailed analytical representations of integrals, where all components of the fields of the vertical and horizontal magnetic dipoles are expressed, are reported in (Veshev et al., 1971). To obtain analytical expressions for similar integrals representing the components of the fields of electric dipoles in a similar model, it is necessary to consider not only the Fock integral but also another related integral conditionally called the Fock integral 1 whose analytical expression is still unknown. The aim of this work is to propose an original method to obtain the analytical representation of the Fock integral 1 by solving a derived inhomogeneous linear differential equation of the first order for this integral with the corresponding boundary condition. The result obtained in this work will allow one to simplify the simulation of fields in the uniform half-space and to improve the interpretation of electromagnetic data due to more accurate and reliable estimates of the normal field in such models of a host medium.