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To solve problems of bodies of revolution (BoR) using the method of moments, the modal Green's function is used, which is defined as the radiation field of a circular loop antenna with sinusoidal current distribution. It is a group of oscillating integrals that forms the bottleneck for further accelerating the computation of BoR problems. In this paper, we propose closed-form expressions for near-axis far-distance modal Green's functions, which can be used to accelerate the computation of BoR problems especially when the object is slim. A criterion is presented based on a rigorous error analysis to guarantee its range of applications. Both the scattering and radiation examples are presented to demonstrate the validity and efficiency of the proposed expressions.