On a special function used in the description of electromagnetic surface waves

The tangential component of the electric field of a surface wave at any distance from the transmitting antenna lying in the interface plane of two homogeneous media can be represented in terms of a function of two complex variables $\widehat W(q,\xi)$ for arbitrary parameters of the interface. In this paper, representations of the function $\widehat W(q,\xi)$ in the form of series are given that allow one to quickly calculate the values of $\widehat W(q,\xi)$ and to investigate the analytic properties of this function. The dependence of the field of the surface wave on time is determined using the inverse Laplace transform, where the path of integration is chosen in such a way that the integrand rapidly decreases at infinity, which drastically improves the computation speed compared with the method based on the Fourier transform.