The coulomb wavefunctions, originally constructed for real > 0, real η and integer λ 0, are defined for , η, and λ all complex. We examine the complex continuation of a variety of series and continued-fraction expansions for the Coulomb functions and their logarithmic derivatives, and then see how these expansions may be selectively combined to calculate both the regular and irregular functions and their derivatives. The resulting algorithm [46] is a complex generalisation of Steed's method [6, 7] as it appears in the real procedure COULFG [10]. Complex Whittaker, confluent hypergeometric and Bessel functions can also be calculated.