Numerical evaluation of the three-dimensional Coulomb wave function using contour integral representations

The main purpose of this work is to show that direct numerical integration of the contour integral representation of special functions, in this case the 3-D Coulomb wave function, can yield accurate numerical results. This approach to these problems of the numerical evaluation of special functions has not been considered before to our knowledge. We compare our numerical evaluation of the 3-D Coulomb wave function to the built-in Mathematica confluent hypergeometric function. The parameters of the our 3-D Coulomb wave function are complex and purely imaginary respectively, making them difficult to evaluate by simple techniques. We have found a contour integral representation of the 3-D Coulomb wave function which we can evaluate very accurately and quite efficiently. We have investigated several contour integral representations of this function from the point of view of finding the one that consistently yields accurate results over a wide range of parameters.