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8 Incomplete Gamma and Related FunctionsComputation

§8.27 Approximations

  1. §8.27(i) Incomplete Gamma Functions
  2. §8.27(ii) Generalized Exponential Integral

§8.27(i) Incomplete Gamma Functions

  • DiDonato (1978) gives a simple approximation for the function F(p,x)=xpex2/2xet2/2tpdt (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(x2p)+(x2p)2+8(x2+p) and is shown to produce an absolute error O(x7) as x.

  • 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 Γ(a,ωz) (by specifying parameters) with 1ω<, and γ(a,ωz) with 0ω1; 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 |phz|<π.

§8.27(ii) Generalized Exponential Integral

  • Luke (1975, p. 103) gives Chebyshev-series expansions for E1(x) and related functions for x5.

  • Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for E1(z) and z10zt1(1et)dt for complex z with |phz|π.

  • Verbeeck (1970) gives polynomial and rational approximations for Ep(x)=(ex/x)P(z), approximately, where P(z) denotes a quotient of polynomials of equal degree in z=x1.