High-precision evaluation of the regular and irregular Coulomb wavefunctions

Steed's method for the calculation of both the regular and irregular Coulomb functions for positive energy, F_λ(η,x),G_λ(η,x), and their derivatives, is extended into the region of very high precision (～ 30S). Other methods in general use result in less than one half of this precision. The test-case results of Strecok and Gregory for G₀(η,x) and G′₀(η,x) to 22S over a restricted range of parameter values close to the transition line are almost completely verified. Limiting forms of the functions are given for x < x_TP (the turning point) and an heuristic estimate of the errors in the functions is obtained. The method is valid for real η, including η = 0 (i.e. the spherical Bessel functions), and for real λ > ?1; thus cylindrical Bessel functions, Airy functions, and even the real Gamma function can also be obtained to high accuracy. For each of these, in the oscillating region (where appropriate) results are available to within about 2S of the machine accuracy; in the monotonic region the loss of accuracy is quantitatively predictable.