This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, reference or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. The authors present an example in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI).

In this paper, we introduce a novel method for calculating the electromagnetic (EM) field below a flat interface between two lossless media when a radiating vertical Hertzian dipole (VHD) is located far from the interface. The method uses a Geometrical Optics (GOs) approach based on the concept of ‘equal optical lengths’, in the framework of which the location of the ‘virtual image’ of the original source is calculated. Using the well-known formulae for the far EM field of a vertical Hertzian dipole, the EM field at a point below the flat interface is calculated in a closed mathematical form.

In this paper, the boundary value problem of wave diffraction on a semi-infinite circular pin is solved using the Wiener–Hopf method with compensation of eigenmodes. The solution to the problem is presented as an infinite series defined by a recurrence formula. The reliability and accuracy of the solution are verified numerically in terms of fulfillment of the law of energy conservation. Sufficiently reliable results are obtained at the first iteration. The method used for solving this problem can be applied to solving diffraction problems on axisymmetric volumetric structures.

A Hertz radiator’s Sommerfeld boundary value problem is considered for the case when its electric moment is directed horizontally relative to the plane interface between two media with different values of magnetic permeability. An integral representation of the exact expression for the Hertz potential, which generalizes the classical solution for non-magnetic media, both in cylindrical and spherical coordinate systems, is obtained. The corresponding expressions for the scattered wave fields are given in the form of Sommerfeld integrals. It is shown that the potential components can be represented as the sum of an infinite series in powers of the Green function.

The boundary-value problem of TM-wave diffraction on a semi-infinite circular pin is studied using the mode-damping method derived from theWiener–Hopf analytical source. The solution is expressed as an infinite series, with subsequent terms obtained through a recurrence relation, ensuring systematic refinement of the approximation. Numerical verification based on the conservation of energy confirms the validity of the solution. As expected, the diffraction process generates a TEM wave in the coaxial region of the waveguide, which highlights the physical consistency of the approach. The analysis demonstrates that the proposed methodology achieves sufficient accuracy even at the first approximation, making it both efficient and reliable. Unlike conventional techniques, the presented method provides a mathematically rigorous framework that combines analytical tractability with numerical validation. Furthermore, the approach can be extended to diffraction problems involving volumetric axisymmetric structures, thereby broadening its applicability in electromagnetic wave analysis. The results contribute to the advancement of diffraction theory and provide a foundation for applications in antenna design, scattering studies, and modern wireless communication systems.