A vertical antenna radiation in the presence of a sphere such as the earth has been a common problem in many science and engineering applications.The resulted electric and magnetic fields in such a geometry are usually expressed in terms of the spherical vector wave functions, regardless of whether the sphere is perfectly electric conducting or lossy dielectric.While the electromagnetic fields can be obtained analytically, numerical discontinuity exists in the closed form solutions of the direct wave radiated, which forms the well known Gibbs phenomenon.In the past many years, this problem was left over there and it resulted a lot of concerns and questions when the numerical results of electromagnetic fields were obtained.Although several some attempts were tried to use numerical approaches to overcome this problem.In this work, an analytic removal of the Gibbs phenomenon is proposed and derived.With this simple approach, all the field components can be obtained analytically and rigorously.The results can be widely applied to accurate numerical solutions to various problems in physics,engineering, chemistry, biology etc.