Miller (1946) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)/\operatorname{Ai}\left(x\right)$ for $x=0(.1)25(1)75$ ; $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=-10(.1)2.5$ ; $\operatorname{log}_{10}\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)/\operatorname{Bi}\left(x\right)$ for $x=0(.1)10$ ; $M\left(x\right)$ , $N\left(x\right)$ , $\theta\left(x\right)$ , $\phi\left(x\right)$ (respectively $F(x)$ , $G(x)$ , $\chi(x)$ , $\psi(x)$ ) for $x=-80(1)-30(.1)0$ . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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•

Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Yakovleva (1969) tabulates Fock’s functions $U(x)\equiv\sqrt{\pi}\operatorname{Bi}\left(x\right)$ , $U^{\prime}(x)=\sqrt{\pi}\operatorname{Bi}'\left(x\right)$ , $V(x)\equiv\sqrt{\pi}\operatorname{Ai}\left(x\right)$ , $V^{\prime}(x)=\sqrt{\pi}\operatorname{Ai}'\left(x\right)$ for $x=-9(.001)9$ . Precision is 7S.

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•

Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

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•

National Bureau of Standards (1958) tabulates $A_{0}(x)\equiv\pi\operatorname{Hi}\left(-x\right)$ and $-A_{0}^{\prime}(x)\equiv\pi\operatorname{Hi}'\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$ ; $\int_{0}^{x}A_{0}(t)\,\mathrm{d}t$ for $x=0.5,1(1)11$ . Precision is 8D.

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14: 8.23 Statistical Applications

… ►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of

Q\left(a,x\right)

; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

15: Errata

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Equation (25.15.6)

25.15.6

G(\chi)\equiv\sum_{r=1}^{k-1}\chi(r){\mathrm{e}}^{2\pi ir/k}.

The upper-index of the finite sum which originally was $k$ , was replaced with $k-1$ since $\chi(k)=0$ .

Reported by Gergő Nemes on 2021-08-23

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Subsection 5.2(iii)

Three new identities for Pochhammer’s symbol (5.2.6)–(5.2.8) have been added at the end of this subsection.

Suggested by Tom Koornwinder.

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Equation (9.6.26)

9.6.26

\operatorname{Bi}'\left(z\right)=\frac{3^{1/6}}{\Gamma\left(\tfrac{1}{3}\right% )}{\mathrm{e}}^{-\zeta}{{}_{1}F_{1}}\left(-\tfrac{1}{6};-\tfrac{1}{3};2\zeta% \right)+\frac{3^{7/6}}{2^{7/3}\Gamma\left(\tfrac{2}{3}\right)}\zeta^{4/3}{% \mathrm{e}}^{-\zeta}{{}_{1}F_{1}}\left(\tfrac{7}{6};\tfrac{7}{3};2\zeta\right)

Originally the second occurrence of the function ${{}_{1}F_{1}}$ was given incorrectly as ${{}_{1}F_{1}}\left(\tfrac{7}{6};\tfrac{7}{3};\zeta\right)$ .

Reported 2014-05-21 by Hanyou Chu.

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Chapters 8, 20, 36

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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References

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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