Expressing the coordinates of a planet in terms of the eccentric anomaly of another planet

The convergence region of perturbation function expansion is investigated for the case when the perturbing body coordinates are expressed through the eccentric anomaly of the perturbed body. A method of studying singular points is applied to the solution of the Kepler equation. A transcendental equation for the distance of the singular points from the real axis is obtained. The solution of this equation defines the convergence radius. Compact formulas are derived for the coefficients of the series expansions of the eccentric anomaly, using the anomaly as the independent variable.