  We investigate the Coulomb phase shift, and derive and analyze new and more
precise analytical formulae. We consider next to leading order terms to the
Stirling approximation, and show that they are important at small values of the
angular momentum $l$ and other regimes. We employ the uniform approximation.
The use of our expressions in low energy scattering of charged particles is
discussed and some comparisons are made with other approximation methods.
