Complete Closed-Form Expression of Dyadic Green´s Function and Its Far- and Near-Field Approximations for an Impedance Half-Plane

A closed-form expression of the dyadic Green´s function is formulated for an impedance half-plane, which is written in terms of the incomplete cylindrical function of Poisson form. Due to the branch-cut of the logarithm function that is required to calculate the input argument of the incomplete cylindrical function, the closed-form representation consists of two formulations. Since the closed-form expression contains a singularity at ρ = 0, the small argument expansion of the expression is also derived to rigorously characterize the behavior of the function at ρ ≈ 0. The previously-reported complete asymptotic expansion for the Sommerfeld integral for an impedance half-plane is not accurate for practically important cases such as near-earth propagation and/or when the surface is highly conductive. Hence, in this paper, a new complete asymptotic series of the Sommerfeld integral are derived for the case that the existing asymptotic series is not accurate. The two asymptotic series not only allow efficient numerical computation but also provide more accurate results for virtually all propagation scenarios. Based on the two asymptotic series, the complete asymptotic series of the dyadic Green´s function is derived. All derived formulations are numerically verified, and their accuracies are investigated.