Generalization of Levi-Civita regularization in the restricted three-body problem

A family of polynomial coupled function of $n$ degree is proposed, in order to generalize the Levi-Civita regularization method, in the restricted three-body problem. Analytical relationship between polar radii in the physical plane and in the regularized plane are established; similar for polar angles. As a numerical application, trajectories of the test particle using polynomial functions of $2,3,\mathrm{.}\mathrm{.}\mathrm{.},8$ degree are obtained. For the polynomial of second degree, the Levi-Civita regularization method is found.