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A mathematically exact formulation for the vector potential and corresponding electromagnetic fields of a triangular current cylindrical dipole are presented for the first time in this paper. These exact expressions converge rapidly in the near-field region of the antenna allowing them to be used for the efficient and accurate computational modeling of electrically short cylindrical antennas. The exact series expansion of the triangular current vector potential is shown to contain two fundamental exponential integrals and their higher-order associated integrals. Numerically stable forward recurrence relations have been derived which may be used for the efficient evaluation of these higher-order integrals in addition to the cylindrical wire kernel. These recursions may also be employed in the computation of the electric and magnetic fields. It is demonstrated that the classical thin wire forms of the vector potential and electromagnetic fields are actually special cases of the more general exact expansions. Finally, the exact formulation was used to investigate the near-field behavior of traditional thin wire as well as moderately thick wire dipoles. Several near-field plots are presented including a vector plot of the total electric field in the vicinity of a moderately thick quarter-wavelength dipole.