Coulomb wave functions with complex values of the variable and the parameters

The motivation for the present paper lies in the fact that the literature concerning the Coulomb wave functions F-L(eta,rho) and G(L)(eta,rho) is a jungle in which it may be hard to find a safe way when one needs general formulas for the Coulomb wave functions with complex values of the variable rho and the parameters L and eta. For the Coulomb wave functions and certain linear combinations of these functions we discuss the connection with the Whittaker function, the Coulomb phase shift, Wronskians, reflection formulas (L -->-L-1), integral representations, series expansions, circuital relations (rho -->rho e(+/- i pi)) and asymptotic formulas on a Riemann surface for the variable rho. The parameters L and eta are allowed to assume complex values. (C) 1999 American Institute of Physics. [S0022-2488(99)01710-7].