approximations

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11: 9.19 Approximations

§9.19 Approximations

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§9.19(i) Approximations in Terms of Elementary Functions

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Martín et al. (1992) provides two simple formulas for approximating $\operatorname{Ai}\left(x\right)$ to graphical accuracy, one for $-\infty<x\leq 0$ , the other for $0\leq x<\infty$ .

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§9.19(ii) Expansions in Chebyshev Series

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§9.19(iii) Approximations in the Complex Plane

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12: 6.20 Approximations

§6.20 Approximations

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§6.20(i) Approximations in Terms of Elementary Functions

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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

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13: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

… ►Semiclassical (WKBJ) approximations in terms of trigonometric or exponential functions are given in Varshalovich et al. (1988, §§8.9, 9.9, 10.7). Uniform approximations in terms of Airy functions for the

\mathit{3j}

and

\mathit{6j}

symbols are given in Schulten and Gordon (1975b). For approximations for the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

14: 8.16 Generalizations

… ►For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). …

15: 12.20 Approximations

§12.20 Approximations

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16: 18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

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18.29.2

Q_{n}(z;a,b,c,d\mid q)\sim\frac{z^{n}\left(az^{-1},bz^{-1},cz^{-1},dz^{-1};q% \right)_{\infty}}{\left(z^{-2},bc,bd,cd;q\right)_{\infty}},

n\to\infty

;

z, a, b, c, d, q

fixed.

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $z$ : complex variable, $q$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.29.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.29 and Ch.18

►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the

q

-Laguerre and continuous

q^{-1}

-Hermite polynomials see Chen and Ismail (1998).

17: 2 Asymptotic Approximations

Chapter 2 Asymptotic Approximations

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18: 9.14 Incomplete Airy Functions

… ►For information, including asymptotic approximations, computation, and applications, see Levey and Felsen (1969), Constantinides and Marhefka (1993), and Michaeli (1996).

19: Edward Neuman

… ►Neuman has published several papers on approximations and expansions, special functions, and mathematical inequalities. …

20: 3.11 Approximation Techniques

§3.11 Approximation Techniques

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§3.11(i) Minimax Polynomial Approximations

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§3.11(iv) Padé Approximations

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