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

The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 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 z = e π 3 , are inaccessible using 2F1 power series linear transformations. In order to solve these problems, a generalization of R.C. Forrey’s transformation theory has been developed. The latter has been successful in treating the 2F1 function with real parameters. As in real case transformation theory, the large canceling terms occurring in 2F1 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 |a|, |b|, |c| are moderate or large. As a physical application, the calculation of the wave functions of the analytical PöschlTeller-Ginocchio potential involving 2F1 evaluations is considered. Program Summary Title of the programs: hyp 2F1, PTG wf Catalogue number: Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland 1 E-mail address: [email protected] Phone : 81-75-753-3857 Fax : 81-75-753-3886