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

Abstract

The fast computation of the Gauss hypergeometric function F 1 2 with all its parameters complex is a difficult task. Although the F 1 2 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 ± i π 3 , are inaccessible using F 1 2 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 F 1 2 function with real parameters. As in real case transformation theory, the large canceling terms occurring in F 1 2 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 F 1 2 evaluations is considered. Program summary Program title: hyp_2F1, PTG_wf Catalogue identifier: AEAE_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAE_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 6839 No. of bytes in distributed program, including test data, etc.: 63 334 Distribution format: tar.gz Programming language: C++, Fortran 90 Computer: Intel i686 Operating system: Linux, Windows Word size: 64 bits Classification: 4.7 Nature of problem: The Gauss hypergeometric function F 1 2 , with all its parameters complex, is uniquely calculated in the frame of transformation theory with power series summations, thus providing a very fast algorithm. The evaluation of the wave functions of the analytical Pöschl–Teller–Ginocchio potential is treated as a physical application. Solution method: The Gauss hypergeometric function F 1 2 verifies linear transformation formulas allowing consideration of arguments of a small modulus which then can be handled by a power series. They, however, give rise to indeterminate or numerically unstable cases, when b − a and c − a − b are equal or close to integers. They are properly dealt with through analytical manipulations of the Lanczos expression providing the Gamma function. The remaining zones of the complex plane uncovered by transformation formulas are dealt with Taylor expansions of the F 1 2 function around complex points where linear transformations can be employed. The Pöschl–Teller–Ginocchio potential wave functions are calculated directly with F 1 2 evaluations. Restrictions: The algorithm provides full numerical precision in almost all cases for | a | , | b | , and | c | of the order of one or smaller, but starts to be less precise or unstable when they increase, especially through a, b, and c imaginary parts. While it is possible to run the code for moderate or large | a | , | b | , and | c | and obtain satisfactory results for some specified values, the code is very likely to be unstable in this regime. Unusual features: Two different codes, one for the hypergeometric function and one for the Pöschl–Teller–Ginocchio potential wave functions, are provided in C++ and Fortran 90 versions. Running time: 20,000 F 1 2 function evaluations take an average of one second.