The motivation for the present paper lies in the fact that the literature concerning the Coulomb wave functions FL(η,ρ) and GL(η,ρ) 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 ρ and the parameters L and η. 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 (ρ→ρe±iπ) and asymptotic formulas on a Riemann surface for the variable ρ. The parameters L and η are allowed to assume complex values.

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