Expressions for electromagnetic fields generated by vertical and horizontal electric dipoles located in the air or in a lossy half-space near its boundary with air are obtained from Hertz vectors by the method of operators under the condition |n|31, where 1/n is the refractive constant of the lossy space. These can be applied up to the near fields under the additional conditions, |n|21 and cos2θ1, where θ is the zenith angle of the point of observation. As for recent works inclusive of expressions of lateral waves their weak points are pointed out.

Extending the domain of the vector potential in the so-called Hallen's equation, four unknown constants are determined to satisfy the boundary conditions in the same way as the circuit theory, where the vector potential plays the leading role, from which the current density and the current itself are derived. Vanishing of the current density just outside the ends of the antenna is required. For a tube-shaped antenna with walls of infinitesimal thickness, further the current just inside the ends of the antenna should vanish, as a result, the current distribution becomes sinusoidal. Adoption of either the surface current distribution or axial current distribution incurs a crucial effect on the value of the currents calculated from the vector potential. The numerical results of the radiation impedance of a hslf-wave antenna show a tendency of consistency with that relatively newly obtained by employing the exact kernel. The problem on the nonsolvability of Hallen's equation is cleared up. Comments are given on the moment method in relation to the boundary value problems to recommend to add two more undecided constants to Hallen's equation.

Ideal style of arguments of the error function complement contained in the expression for the Norton's surface wave of a vertical dipole over the plane earth is discussed, and then it is pointed out that new formulas have not necessarily desired form as compared with old ones.