Analytical images of the coordinates time dependence of Keplerian motion

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In order to calculate the time dependence of polar coordinates of relative motion in the problem of two bodies in the analytical form it was proposed variants of iterative algorithms with fast convergence, which are based on the usage of approximating functions. It was shown that an independent determining the time dependence of radial coordinate in the elliptical motion, as well as at large distance from the pericenter in the case of the hyperbolic motion yields good convergence using the method of ordinary successive iterations.