# §8.25 Methods of Computation

## §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).

See Allasia and Besenghi (1987b) for the numerical computation of $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ from (8.6.4) by means of the trapezoidal rule.

## §8.25(iii) Asymptotic Expansions

DiDonato and Morris (1986) describes an algorithm for computing $\mathop{P\/}\nolimits\!\left(a,x\right)$ and $\mathop{Q\/}\nolimits\!\left(a,x\right)$ for $a\geq 0$, $x\geq 0$, and $a+x\neq 0$ 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 $\mathop{P\/}\nolimits\!\left(a,x\right)$ are specified, based on Newton’s rule (§3.8(ii)). See also Temme (1987, 1994b).

## §8.25(iv) Continued Fractions

The computation of $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ and $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ 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 $\mathop{P\/}\nolimits\!\left(a+n,x\right)$, $\mathop{Q\/}\nolimits\!\left(a+n,x\right)$, or $\mathop{\gamma^{*}\/}\nolimits\!\left(a+n,x\right)$ for fixed $a$ and $n=0,1,2,\dots$. 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 $\mathop{E_{p}\/}\nolimits\!\left(x\right)$ 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).