An exact series representation for the EM field from a vertical electric dipole on an imperfectly conducting half-space

A rigorous approach is introduced for the exact integration of the complete expressions for the time-harmonic fields of a vertical electric dipole situated on a homogeneous lossy half-space. Analytical integration is made possible by applying the residue theorem after replacing the square root terms of the integrands in the field integrals with their rational function representations according to the Babylonian square root algorithm. As a result, the EM field is explicitly expressed as a superposition of cylindrical waves. The obtained formulas allow to relax all the assumptions underlying King’s formulation for the same Sommerfeld half-space problem. Numerical results are presented to show the validity of the proposed formulation.