§9.18 Tables

Additional listings of early tables of the functions treated in this chapter are given in Fletcher et al. (1962) and Lebedev and Fedorova (1960).

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  Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-20(.01)2;$ ${\log }_{10}\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=-10(.1)2.5$; ${\log }_{10}\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ (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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  Fox (1960, Table 3) tabulates $2{\pi }^{1/2}{x}^{1/4}\exp (\frac{2}{3}{x}^{3/2})\mathrm{Ai}\left(x\right)$, $2{\pi }^{1/2}{x}^{-1/4}\exp (\frac{2}{3}{x}^{3/2}){\mathrm{Ai}}^{\prime }\left(x\right)$, ${\pi }^{1/2}{x}^{1/4}\exp (-\frac{2}{3}{x}^{3/2})\mathrm{Bi}\left(x\right)$, and ${\pi }^{1/2}{x}^{-1/4}\exp (-\frac{2}{3}{x}^{3/2}){\mathrm{Bi}}^{\prime }\left(x\right)$ for $\frac{3}{2}{x}^{-3/2}=0(.001)0.05$, together with similar auxiliary functions for negative values of $x$. Precision is 10D.

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  Zhang and Jin (1996, p. 337) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ 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 $U(x)\equiv \sqrt{\pi }\mathrm{Bi}\left(x\right)$, $U^{\prime }(x)=\sqrt{\pi }{\mathrm{Bi}}^{\prime }\left(x\right)$, $V(x)\equiv \sqrt{\pi }\mathrm{Ai}\left(x\right)$, $V^{\prime }(x)=\sqrt{\pi }{\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-9(.001)9$. Precision is 7S.

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  Woodward and Woodward (1946) tabulates the real and imaginary parts of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$, $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$ for $\mathrm{\Re }z=-2.4(.2)2.4$, $\mathrm{\Im }z=-2.4(.2)0$. Precision is 4D.

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  Harvard University (1945) tabulates the real and imaginary parts of $h_1(z)$, $h_1^{\prime }(z)$, $h_2(z)$, $h_2^{\prime }(z)$ for $-x_0\le \mathrm{\Re }z\le x_0$, $0\le \mathrm{\Im }z\le y_0$, , with interval 0.1 in $\mathrm{\Re }z$ and $\mathrm{\Im }z$. Precision is 8D. Here $h_1(z)=-2^{4/3}{3}^{1/6}\mathrm{i}\mathrm{Ai}\left({\mathrm{e}}^{-\pi \mathrm{i}/3}z\right)$, $h_2(z)=2^{4/3}{3}^{1/6}\mathrm{i}\mathrm{Ai}\left({\mathrm{e}}^{\pi \mathrm{i}/3}z\right)$.

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  Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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  Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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  Zhang and Jin (1996, p. 339) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$; 8D.

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  Corless et al. (1992) gives the real and imaginary parts of ${\beta }_k$ for $k=1(1)13$; 14S.

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  See also §9.9(v).

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  Rothman (1954b) tabulates ${\int }_0^x\mathrm{Ai}\left(t\right)dt$ and ${\int }_0^x\mathrm{Bi}\left(t\right)dt$ for $x=-10(.1)\mathrm{\infty }$ and $-10(.1)2$, respectively; 7D. The entries in the columns headed ${\int }_0^x\mathrm{Ai}\left(-x\right)dx$ and ${\int }_0^x\mathrm{Bi}\left(-x\right)dx$ all have the wrong sign. The tables are reproduced in Abramowitz and Stegun (1964, Chapter 10), and the sign errors are corrected in later reprintings.

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  National Bureau of Standards (1958) tabulates ${\int }_0^x\mathrm{Ai}\left(-t\right)dt$ and ${\int }_0^x{\int }_0^v\mathrm{Ai}\left(-t\right)dtdv$ (see (9.10.20)) for $x=-2(.01)5$ to 8D and 7D, respectively.

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  Zhang and Jin (1996, p. 338) tabulates ${\int }_0^x\mathrm{Ai}\left(t\right)dt$ and ${\int }_0^x\mathrm{Bi}\left(t\right)dt$ for $x=-10(.2)10$ to 8D or 8S.

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  Scorer (1950) tabulates $\mathrm{Gi}\left(x\right)$ and $\mathrm{Hi}\left(-x\right)$ for $x=0(.1)10$; 7D.

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  Rothman (1954a) tabulates ${\int }_0^x\mathrm{Gi}\left(t\right)dt$, ${\mathrm{Gi}}^{\prime }\left(x\right)$, ${\int }_0^x\mathrm{Hi}\left(-t\right)dt$, $-{\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.1)10$; 7D.

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  National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

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  Nosova and Tumarkin (1965) tabulates $e_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$, $e_0^{\prime }(x)=-\pi {\mathrm{Hi}}^{\prime }\left(-x\right)$, ${\tilde{e}}_0(-x)\equiv -\pi \mathrm{Gi}\left(x\right)$, ${\tilde{e}}_0^{\prime }(-x)=\pi {\mathrm{Gi}}^{\prime }\left(x\right)$ for $x=-1(.01)10$; 7D. Also included are the real and imaginary parts of $e_0(z)$ and $\mathrm{i}{e}_0^{\prime }(z)$, where $z=\mathrm{i}y$ and $y=0(.01)9$; 6-7D.

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  Gil et al. (2003c) tabulates the only positive zero of ${\mathrm{Gi}}^{\prime }\left(z\right)$, the first 10 negative real zeros of $\mathrm{Gi}\left(z\right)$ and ${\mathrm{Gi}}^{\prime }\left(z\right)$, and the first 10 complex zeros of $\mathrm{Gi}\left(z\right)$, ${\mathrm{Gi}}^{\prime }\left(z\right)$, $\mathrm{Hi}\left(z\right)$, and ${\mathrm{Hi}}^{\prime }\left(z\right)$. Precision is 11 or 12S.

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  Smirnov (1960) tabulates $U_1(x,\alpha )$, $U_2(x,\alpha )$, defined by (9.13.20), (9.13.21), and also $\partial U_1(x,\alpha )/\partial x$, $\partial U_2(x,\alpha )/\partial x$, for $\alpha =1$, $x=-6(.01)10$ to 5D or 5S, and also for $\alpha =\pm \frac{1}{4}$, $\pm \frac{1}{3}$, $\pm \frac{1}{2}$, $\pm \frac{2}{3}$, $\pm \frac{3}{4}$, $\frac{5}{4}$, $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\frac{7}{4}$, 2, $x=0(.01)6$; 4D.