# §8.27 Approximations

## §8.27(i) Incomplete Gamma Functions

• DiDonato (1978) gives a simple approximation for the function (which is related to the incomplete gamma function by a change of variables) for real and large positive . This takes the form , approximately, where and is shown to produce an absolute error as .

• 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 -axis. See also Temme (1994b, §3).

• Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for (by specifying parameters) with , and with ; see also Temme (1994b, §3).

• Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the -plane that exclude and are valid for .

## §8.27(ii) Generalized Exponential Integral

• Luke (1975, p. 103) gives Chebyshev-series expansions for and related functions for .

• Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for and for complex with .

• Verbeeck (1970) gives polynomial and rational approximations for , approximately, where denotes a quotient of polynomials of equal degree in .