§8.27 Approximations

§8.27(i) Incomplete Gamma Functions

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{{-p}}e^{{x^{2}/2}}\int _{x}^{\infty}e^{{-t^{2}/2}}t^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$ . This takes the form $F(p,x)=4x/h(p,x)$ , approximately, where $h(p,x)=3(x^{2}-p)+\sqrt{(x^{2}-p)^{2}+8(x^{2}+p)}$ and is shown to produce an absolute error $\mathop{O\/}\nolimits\!\left(x^{{-7}}\right)$ as $x\to\infty$ .
Luke (1975, §4.3) gives Padé approximation methods, combined with a detailed analysis of the error terms, valid for real and complex variables except on the negative real $z$ -axis. See also Temme (1994b, §3).
Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\mathop{\Gamma\/}\nolimits\!\left(a,\omega z\right)$ (by specifying parameters) with $1\leq\omega<\infty$ , and $\mathop{\gamma\/}\nolimits\!\left(a,\omega z\right)$ with $0\leq\omega\leq 1$ ; see also Temme (1994b, §3).
Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the $z$ -plane that exclude $z=0$ and are valid for $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|<\pi$ .

Luke (1975, p. 103) gives Chebyshev-series expansions for $\mathop{E_{1}\/}\nolimits\!\left(x\right)$ and related functions for $x\geq 5$ .
Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ and $z^{{-1}}\int _{0}^{z}t^{{-1}}(1-e^{{-t}})dt$ for complex $z$ with $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|\leq\pi$ .
Verbeeck (1970) gives polynomial and rational approximations for $\mathop{E_{{p}}\/}\nolimits\!\left(x\right)=(e^{{-x}}/x)P(z)$ , approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{{-1}}$ .