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The fast computation of the Gauss hypergeometric function 2F1 with all itsparameters complex is a difficult task. Although the 2F1 function verifiesnumerous analytical properties involving power series expansions whoseimplementation is apparently immediate, their use is thwarted by instabilitiesinduced by cancellations between very large terms. Furthermore, small areas ofthe complex plane are inaccessible using only 2F1 power series formulas, thusrendering 2F1 evaluations impossible on a purely analytical basis. In order tosolve these problems, a generalization of R.C. Forrey's transformation theoryhas been developed. The latter has been successful in treating the 2F1 functionwith real parameters. As in real case transformation theory, the largecanceling terms occurring in 2F1 analytical formulas are rigorously dealt with,but by way of a new method, directly applicable to the complex plane. Taylorseries expansions are employed to enter complex areas outside the domain ofvalidity of power series analytical formulas. The proposed algorithm, however,becomes unstable in general when |a|,|b|,|c| are moderate or large. As aphysical application, the calculation of the wave functions of the analyticalPoschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.

computer physics communications

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