# §8.27 Approximations

## §8.27(i) Incomplete Gamma Functions

• DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}t^{p}\mathrm{d}t$ (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 $O\left(x^{-7}\right)$ 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 $\Gamma\left(a,\omega z\right)$ (by specifying parameters) with $1\leq\omega<\infty$, and $\gamma\left(a,\omega z\right)$ 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 $\left|\operatorname{ph}z\right|<\pi$.

## §8.27(ii) Generalized Exponential Integral

• Luke (1975, p. 103) gives Chebyshev-series expansions for $E_{1}\left(x\right)$ and related functions for $x\geq 5$.

• Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for $E_{1}\left(z\right)$ and $z^{-1}\int_{0}^{z}t^{-1}(1-e^{-t})\mathrm{d}t$ for complex $z$ with $\left|\operatorname{ph}z\right|\leq\pi$.

• Verbeeck (1970) gives polynomial and rational approximations for $E_{p}\left(x\right)=(e^{-x}/x)P(z)$, approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$.