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Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. (English) Zbl 1196.33020
Summary: The fast computation of the Gauss hypergeometric function ${}_{2}{F}_{1}$ with all its parameters complex is a difficult task. Although the ${}_{2}{F}_{1}$ 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 ${}_{2}{F}_{1}$, are inaccessible using ${}_{2}{F}_{1}$ 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 ${}_{2}{F}_{1}$ function with real parameters. As in real case transformation theory, the large canceling terms occurring in ${}_{2}{F}_{1}$ 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 Pöschl-Teller-Ginocchio potential involving ${}_{2}{F}_{1}$ evaluations is considered.
##### MSC:
 33F05 Numerical approximation and evaluation of special functions 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 81-04 Machine computation, programs (quantum theory) 81Q05 Closed and approximate solutions to quantum-mechanical equations
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