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The electromagnetic fields generated by a vertical electric dipole, expressed in the form of the Sommerfeld integral, are important in the study of lightning. In this paper, an algorithm, which can analytically approximate the Sommerfeld tail integral, is proposed. Due to the fact that the tail integral is expressed by a sum of several error functions, the tail integral of a vertical electric dipole can be evaluated without the truncation of the infinite upper limit. The comparison indicates that the results calculated by the proposed algorithm are accurate, and the computational time, compared to the one using direct numerical quadrature method, is much shorter.