  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.
