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In this paper a new technique is presented to evaluate efficiently the Green's functions of filament sources radiating over a half-space. The technique is based on the annihilation of the asymptotic and branch-point singular behaviours of the spectral Green's function. The asymptotic and singularity terms are subtracted, then calculated analytically and finally re-added to the remaining integral. The annihilation leads to a very smooth remaining integral that is calculated adaptively by using Gaussian quadratures. In order to validate the accuracy and efficiency of the proposed technique, lossless and lossy half-space media are analyzed and the results are found to agree very well with literature.