Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, the third step of the author's regularization approach will be started by establishing the expansions of the functionX_n^(r) (, ,u) in terms of the sectorial variables_j⁽ⁱ⁾ introduced in Paper IV (Sharaf, 1982) to regularize the highly-oscillating perturbation force of some orbital systems. The literal analytical expressions for the Fourier expansion of the function will be explored in terms of_j⁽ⁱ⁾ for anyn positive integer,r any real number whatever the types and the number of sectors forming the divisions situation of the elliptic orbits may be. The basic computational materials of the theory will also be given and for which the method of solution, the recurrence formulae, and the general computational sequence for the coefficients are considered.