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11—20 of 239 matching pages

11: William P. Reinhardt

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… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

12: 7.24 Approximations

§7.24 Approximations

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§7.24(i) Approximations in Terms of Elementary Functions

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Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$ . The maximum relative precision is about 20S.

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Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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13: 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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Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$ , $x=-19(1)20$ , $y=0(1)20$ , 6D; $e^{z}E_{1}\left(z\right)$ , $x=-4(.5)-2$ , $y=0(.2)1$ , 6D; $E_{1}\left(z\right)+\ln z$ , $x=-2(.5)2.5$ , $y=0(.2)1$ , 6D.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

15: Peter L. Walker

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20 Theta Functions

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16: Staff

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William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

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William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

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17: 25.20 Approximations

§25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$ , $5\leq s\leq 11$ , $11\leq s\leq 25$ , $25\leq s\leq 55$ . Precision is varied, with a maximum of 20S.

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta\left(s+1\right)$ and $\zeta\left(s+k\right)$ , $k=2,3,4,5,8$ , for $0\leq s\leq 1$ (23D).

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Morris (1979) gives rational approximations for $\operatorname{Li}_{2}\left(x\right)$ (§25.12(i)) for $0.5\leq x\leq 1$ . Precision is varied with a maximum of 24S.

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Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty<x\leq 2$ and $2\leq x<\infty$ , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ . For each $s$ there are three sets of approximations, with relative maximum errors $10^{-4},10^{-8},10^{-12}$ .

18: 10.75 Tables

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Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

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Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

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Zhang and Jin (1996, p. 322) tabulates $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , $x=0(1)20$ , 7S.

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Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , 8D.

19: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …

20: 25.12 Polylogarithms

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See accompanying text — Figure 25.12.1: Dilogarithm function $\operatorname{Li}_{2}\left(x\right)$ , $-20\leq x<1.$ Magnify

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G_{s}(x)=\operatorname{Li}_{s+1}\left(e^{x}\right).

►For a uniform asymptotic approximation for

F_{s}(x)

see Temme and Olde Daalhuis (1990).