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On the computation of incomplete gamma functions in the complex domain. (English) Zbl 0577.65015
The authors discuss in detail the usefulness of continued fraction expansions for the numerical computation of the incomplete gamma function $\gamma$ (a,z) for complex values of a and z. Using functional relations for $\gamma$ (a,z), the complementary function ${\Gamma }$ (a,z) and the (related) special Kummer function M(l,b,z), they introduce what they call ”additional (or extended)” continued fraction expansions, which they show to be stable when evaluated by the backward recurrence algorithm. A thorough error analysis is presented and the special cases of $\gamma$ (a,z) and ${\Gamma }$ (a,z) which correspond to the error functions erf(z), erfc(z) and the exponential integral ${E}_{n}\left(z\right)$ are discussed in detail. In connections with the complex error function, the ”anomalous convergence” of general T-fractions (the two point Padé approximants) is discussed.
Reviewer: K.S.Kölbig
##### MSC:
 65D20 Computation of special functions, construction of tables 33B15 Gamma, beta and polygamma functions 33B20 Incomplete beta and gamma functions 33C60 Hypergeometric integrals and functions defined by them