Abstract
In this paper, an analytical approach is described for the evaluation of the Sommerfeld type integrals. The basis of technique is that the solutions for Helmholtz equation in the spherical coordinate system become discrete spectrum in semi-infinite space region, rather than a continuum of eigenfunctions for the cylindrical coordinate system as arising in the Sommerfeld integrals. By using the present method, the Sommerfeld integral contained in the Hertz potential of the vertical electric dipole over the constant-impedance plane is expanded as a rapidly and absolutely convergent series of spherical waves functions. The general term of the series can be interpreted as an elementary spherical wave arising from an image equipotent source located at the mirror-image point with respect to the constant-impedance plane; and the expansion coefficients are cast into the Legnder function of the second kind with argument for the complex surface impedance of the ground. The present result of the Sommerfeld half-space problem is exact solution relative to the impedance boundary condition (valid for general source and observation points) and can conveniently be used to analyze and calculate the electromagnetic field.
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