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8 Incomplete Gamma and Related FunctionsComputation

§8.25 Methods of Computation

Contents

§8.25(i) Series Expansions

Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of z, they are cumbersome to use when |z| is large owing to slowness of convergence and cancellation. For large |z| the corresponding asymptotic expansions (generally divergent) are used instead. See also Luke (1975, pp. 101–102) and Temme (1994b).

§8.25(ii) Quadrature

See Allasia and Besenghi (1987b) for the numerical computation of Γ(a,z) from (8.6.4) by means of the trapezoidal rule.

§8.25(iii) Asymptotic Expansions

DiDonato and Morris (1986) describes an algorithm for computing P(a,x) and Q(a,x) for a0, x0, and a+x0 from the uniform expansions in §8.12. The algorithm supplies 14S accuracy. A numerical inversion procedure is also given for calculating the value of x (with 10S accuracy), when a and P(a,x) are specified, based on Newton’s rule (§3.8(ii)). See also Temme (1987, 1994b).

§8.25(iv) Continued Fractions

The computation of γ(a,z) and Γ(a,z) by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). See also Jacobsen et al. (1986) and Temme (1996b, p. 280).

§8.25(v) Recurrence Relations

Expansions involving incomplete gamma functions often require the generation of sequences P(a+n,x), Q(a+n,x), or γ*(a+n,x) for fixed a and n=0,1,2,. An efficient procedure, based partly on the recurrence relations (8.8.5) and (8.8.6), is described in Gautschi (1979b, 1999).

Stable recursive schemes for the computation of Ep(x) are described in Miller (1960) for x>0 and integer p. For x>0 and real p see Amos (1980b) and Chiccoli et al. (1987, 1988). See also Chiccoli et al. (1990a) and Stegun and Zucker (1974).