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Sommerfeld integrals appear in the solution of radiation and scattering problems involving antennas in planar multilayered media. In the conventional approach it is quite difficult to numerically integrate the tails related to Sommerfeld integrals as they are not only oscillatory but also slowly decaying. Numerous research efforts have been developed to accelerate the accurate computation of such integrals, for example, by changing the integration path in the complex plane, or by using extrapolation methods. In this paper, the physical origin of the problem of the Sommerfeld integral tails is studied. Based on the physical description of the problem, a new Green's function for the radiation of a vertical electric dipole over an imperfect ground plane is derived. The new Green's function involves what is called in this paper Schelkunoff integrals. The new formulation is compared to the conventional Sommerfeld formulation, mainly with respect to the speed of convergence when the fields are calculated near the ground plane. The characteristics of the new formulation show that if Schelkunoff integrals are used in the appropriate region, the problem of Sommerfeld integral tails, which plagued the electromagnetic community for decades, can be totally abolished.