On an Analytical Representation of an Integral Related to the Fock Integral That Appears in 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 for the theoretical analysis of 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 [A.V. 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 aims of this work are to derive an inhomogeneous linear first-order differential equation for this integral with the corresponding boundary conditions and to obtain the analytical representation of the Fock integral 1 by solving this equation. The result of this work will allow one to simplify the simulation of fields in a 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.