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 $x\to\infty$ .

…

18: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

… ►

§33.10(i) Large $\rho$

… ►

§33.10(ii) Large Positive $\eta$

… ►

§33.10(iii) Large Negative $\eta$

…

19: 2.11 Remainder Terms; Stokes Phenomenon

… ►In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable

x

that is intended to be used. …

20: 33.12 Asymptotic Expansions for Large $\eta$

§33.12 Asymptotic Expansions for Large $\eta$

…

large variable

11—20 of 109 matching pages

11: 8.12 Uniform Asymptotic Expansions for Large Parameter

Inverse Function

12: 28.31 Equations of Whittaker–Hill and Ince

Asymptotic Behavior

13: Bibliography W

14: 30.11 Radial Spheroidal Wave Functions

§30.11(iii) Asymptotic Behavior

15: 33.11 Asymptotic Expansions for Large ρ

§33.11 Asymptotic Expansions for Large ρ

16: 33.18 Limiting Forms for Large ℓ

§33.18 Limiting Forms for Large ℓ

17: 8.27 Approximations

18: 33.10 Limiting Forms for Large ρ or Large | η |

§33.10(i) Large ρ

§33.10(ii) Large Positive η

§33.10(iii) Large Negative η

19: 2.11 Remainder Terms; Stokes Phenomenon

20: 33.12 Asymptotic Expansions for Large η

§33.12 Asymptotic Expansions for Large η

15: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

16: 33.18 Limiting Forms for Large $\ell$

§33.18 Limiting Forms for Large $\ell$