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1: 10.75 Tables

… ►Also, for additional listings of tables pertaining to complex arguments see Babushkina et al. (1997). … ►

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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British Association for the Advancement of Science (1937) tabulates $I_0\left(x\right)$, $I_1\left(x\right)$, $x=0(.001)5$, 7–8D; $K_0\left(x\right)$, $K_1\left(x\right)$, $x=0.01(.01)5$, 7–10D; ${\mathrm{e}}^{-x}{I}_0\left(x\right)$, ${\mathrm{e}}^{-x}{I}_1\left(x\right)$, ${\mathrm{e}}^x{K}_0\left(x\right)$, ${\mathrm{e}}^x{K}_1\left(x\right)$, $x=5(.01)10(.1)20$, 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $K_0\left(x\right)$, $K_1\left(x\right)$ for small values of $x$.

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Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.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 $K_n^{\prime }\left(z\right)$, for $n=2(1)20$, 9S.

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2: 9.7 Asymptotic Expansions

… ►Here $\delta$ denotes an arbitrary small positive constant and … ►

9.7.3 $$\chi (x)\equiv {\pi }^{1/2}\mathrm{\Gamma }\left(\frac{1}{2}x+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}x+\frac{1}{2}\right).$$

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Defines:

$\chi (x)$: function (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $\equiv$: equals by definition

Source:

Olver (1997b, (13.02), p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

Originally the function $\chi$ was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

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9.7.4 $$\chi (x)\sim {(\frac{1}{2}\pi x)}^{1/2}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $\chi (x)$: function

Source:

Olver (1997b, p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E4

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

Originally the formula was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. … ►

9.7.5 $$\mathrm{Ai}\left(z\right)\sim \frac{{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

(9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

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3: Errata

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Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${}_2{}^{}F_1^{}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

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The arguments of some of the functions in (33.11.2)–(33.11.7) were included to improve clarity of the presentation.

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Several small revisions have been made. For details see §§5.11(ii), 10.12, 10.19(ii), 18.9(i), 18.16(iv), 19.7(ii), 22.2, 32.11(v), 32.13(ii).

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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