The differential equation

${f}^{\text{'}\text{'}}\left(\rho \right)+(1-2\eta /\rho -\lambda (\lambda +1)/{\rho}^{2})f\left(\rho \right)=0$ has two linearly independent solutions

${F}_{\lambda}(\eta ,\rho )$ and

${G}_{\lambda}(\eta ,\rho )$ called Coulomb wavefunctions. The real functions

${F}_{\lambda}$,

${G}_{\lambda}$,

$(d/d\rho ){G}_{\lambda}$ and

$(d/d\rho ){F}_{\lambda}$ have been studied extensively and methods of calculations have been developed by several authors. The most important technique has been developed by the second author et al. in 1974, where the authors calculate regular and irregular functions and their derivatives for any real

$\lambda $ or for a range of integer-spaced

$\lambda $-values. In the present paper the authors define Columb wavefunctions in the complex case and generalize the earlier method to calculate the functions

${F}_{\lambda}$,

${G}_{\lambda}$,

$(d/d\rho ){F}_{\lambda}$ and

$(d/d\rho ){G}_{\lambda}$ where all variables are complex.