The classical solution of a definite integral occurring in the satellite theory in the extended phase space

A method of integrating the functions defined by Scheifele and Graf (1974), arising in the elimination of time from the arguments of satellite theory using Delaunay elements in the extended phase space, is proposed. By repeated applications of an identity of Stiefel and Scheifele (1971), neglecting certain terms, and assuming a solution in the form of a product of Bessel functions, a solution expanded in the eccentricity is obtained.