Bessel function expansions of Coulomb wave functions

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Abstract
From the convergence properties of the expansion of the function Φl∝r−l−1Fl in powers of the energy, we successively obtain the expansions of Fl and Gl as single series of modified Bessel functions I2l+1+n and K2l+1+n, respectively, as well as corresponding asymptotic approximations of Gl for ‖η‖→∞. Both repulsive and attractive fields are considered for real and complex energies as well. The expansion of Fl is not new, but its convergence is given a simpler and corrected proof. The simplest form of the asymptotic approximations obtained for Gl, in the case of a repulsive field and for low positive energies, is compared to an expansion obtained by Abramowitz.