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

TL;DR

This paper presents a novel, stable algorithm for rapidly computing the Gauss hypergeometric function with complex parameters, overcoming analytical and numerical challenges, with applications to quantum wave functions.

Contribution

It generalizes transformation theory to complex parameters, enabling stable, fast computation of 2F1 functions in complex domains, useful for physical applications.

Findings

Developed a new complex-plane transformation method for 2F1

Achieved faster and more stable computations of hypergeometric functions

Applied the method to calculate wave functions in quantum potentials

Abstract

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 are inaccessible using only 2F1 power series formulas, thus rendering 2F1 evaluations impossible on a purely analytical basis. 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 Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.