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A new frequency-domain algorithm, the planar Taylor expansion through the fast Fourier transform (FFT) method, has been developed to speed the computation of the Green's function related formulas in the half-space scenario for both the near-field (NF) and the far-field (FF). Two types of Taylor-FFT algorithms are presented in this paper: the spatial Taylor-FFT and the spectral Taylor-FFT. The former is for the computation of the NF and the latter is for the computation of the FF or the Fourier spectrum. The planar Taylor-FFT algorithm has a computational complexity of O(N2 log2 N2) for an N times N computational grid, comparable to the multilevel fast multipole method (MLFMM). What's more important is that, the narrowband property of many electromagnetic fields allows the Taylor-FFT algorithm to use larger sampling spacing, which is limited by the transverse wave number. In addition, the algorithm is free of singularities. An accuracy of -50 for the planar Taylor-FFT algorithm is easily obtained and an accuracy of -80 dB is possible when the algorithm is optimized. The algorithm works particularly well for narrowband fields and quasi-planar geometries.