Coulomb Functions for Large Charges and Small Velocities

Expansions in powers of ${\eta }^{-\frac{1}{3}}$, where $\eta$ is defined in the introduction below, for the Coulomb wave functions $F_L\left(\rho \right)$ and $G_L\left(\rho \right)$ and their derivatives are given for special values of $\rho =2\mathrm{}\eta$ and $\rho ={\rho }_L=\eta +{[{\eta }^2+L(L+1\mathrm{}\left)\right]}^{\frac{1}{2}}$, the classical turning points for $L=0\mathrm{}$ and any $L$, respectively. Expansions applicable in the vicinity of the turning point are given as a series involving Bessel functions of order $\pm \frac{n}{3}$ with the expansion parameter ${{\rho }_L}^{-\frac{1}{3}}$. Approximations valid for large values of $\eta$ are given and discussed.