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A numerical methodology is proposed for the evaluation of the electromagnetic fields of an electric or magnetic dipole of arbitrary orientation in spherical stratified media. The proposed methodology is based on the numerical solution of the differential equation in the radial coordinate that is satisfied by each coefficient in the spherical harmonics expansion of the governing Helmholtz equation. More specifically, the finite-difference solution of this equation may be cast in a pole-residue form that allows the analytic evaluation of the spherical harmonics series by means of the Watson transformation. Thus, a closed-form expression is obtained for the electromagnetic fields in terms of a short series of associated Legendre functions Pmν(cos(θ)) of integer order m and complex degree ν. The number of terms in the series is strongly dependent on the angle θ, and decreases very fast when the point of observation moves away from the source. The method allows for arbitrary variation in the permittivity and permeability profiles in the radial directions.