Expansion of spherical wave functions for electromagnetic fields of a vertical magnetic dipole over an anisotropic half-space

We derive an accurate and analytical expression for electromagnetic fields of a vertical magnetic dipole over an anisotropic uniaxial medium half-space. By using the expansion of spherical wave functions for a cylindrical wave function and with the help of the theory of hypergeometric functions, the Sommerfeld type integral contained in the electromagnetic fields is expressed as a rapidly and absolutely convergent series of spherical wave functions; and the coefficients of the series are the Legender Polynomials with the complex argument for the constitutive parameter. The results have explicit mathematical and physical implications and no restriction on the locations of dipole sources and observational points, medium parameters, and frequency. The expression can be used conveniently to calculate and analyze electromagnetic fields at any points.