Coulomb Functions for Large Charges and Small Velocities

Abstract

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