Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions Original Research Article

The fast computation of the Gauss hypergeometric function image with all its parameters complex is a difficult task. Although the image function verifies numerous analytical properties involving power series expansions whose implementation is apparently immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane, in the vicinity of image, are inaccessible using image power series linear transformations. In order to solve these problems, a generalization of R.C. Forreyʹs transformation theory has been developed. The latter has been successful in treating the image function with real parameters. As in real case transformation theory, the large canceling terms occurring in image analytical formulas are rigorously dealt with, but by way of a new method, directly applicable to the complex plane. Taylor series expansions are employed to enter complex areas outside the domain of validity of power series analytical formulas. The proposed algorithm, however, becomes unstable in general when image, image, image are moderate or large. As a physical application, the calculation of the wave functions of the analytical Pöschl–Teller–Ginocchio potential involving image evaluations is considered.