# §9.18 Tables

## §9.18(i) Introduction

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

## §9.18(ii) Real Variables

• Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, $\mathrm{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\mathrm{Ai}\left(x\right)$, $\mathrm{Ai}'\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, $\mathrm{Bi}'\left(x\right)$ for $x=-10(.1)2.5$; $\operatorname{log}_{10}\mathrm{Bi}\left(x\right)$, $\mathrm{Bi}'\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta\left(x\right)$, $\phi\left(x\right)$ (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.

• Fox (1960, Table 3) tabulates $2\pi^{1/2}x^{1/4}\*\exp(\tfrac{2}{3}x^{3/2})\*\mathrm{Ai}\left(x\right)$, $2\pi^{1/2}x^{-1/4}\*\exp(\tfrac{2}{3}x^{3/2})\*\mathrm{Ai}'\left(x\right)$, $\pi^{1/2}x^{1/4}\*\exp(-\tfrac{2}{3}x^{3/2})\*\mathrm{Bi}\left(x\right)$, and $\pi^{1/2}x^{-1/4}\*\exp(-\tfrac{2}{3}x^{3/2})\*\mathrm{Bi}'\left(x\right)$ for $\tfrac{3}{2}x^{-3/2}=0(.001)0.05$, together with similar auxiliary functions for negative values of $x$. Precision is 10D.

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

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

## §9.18(iii) Complex Variables

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

• 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 $-x_{0}\leq\Re z\leq x_{0}$, $0\leq\Im z\leq y_{0}$, $|x_{0}+iy_{0}|<6.1$, with interval 0.1 in $\Re z$ and $\Im z$. Precision is 8D. Here $h_{1}(z)=-2^{4/3}3^{1/6}i\mathrm{Ai}\left(e^{-\pi i/3}z\right)$, $h_{2}(z)=2^{4/3}3^{1/6}i\mathrm{Ai}\left(e^{\pi i/3}z\right)$.

## §9.18(iv) Zeros

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

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

• Zhang and Jin (1996, p. 339) tabulates $a_{k}$, $\mathrm{Ai}'\left(a_{k}\right)$, $a^{\prime}_{k}$, $\mathrm{Ai}\left(a^{\prime}_{k}\right)$, $b_{k}$, $\mathrm{Bi}'\left(b_{k}\right)$, $b^{\prime}_{k}$, $\mathrm{Bi}\left(b^{\prime}_{k}\right)$, $k=1(1)20$; 8D.

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

## §9.18(v) Integrals

• Rothman (1954b) tabulates $\int_{0}^{x}\mathrm{Ai}\left(t\right)\mathrm{d}t$ and $\int_{0}^{x}\mathrm{Bi}\left(t\right)\mathrm{d}t$ for $x=-10(.1)\infty$ and $-10(.1)2$, respectively; 7D. The entries in the columns headed $\int_{0}^{x}\mathrm{Ai}\left(-x\right)\mathrm{d}x$ and $\int_{0}^{x}\mathrm{Bi}\left(-x\right)\mathrm{d}x$ 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.

• National Bureau of Standards (1958) tabulates $\int_{0}^{x}\mathrm{Ai}\left(-t\right)\mathrm{d}t$ and $\int_{0}^{x}\int_{0}^{v}\mathrm{Ai}\left(-t\right)\mathrm{d}t\mathrm{d}v$ (see (9.10.20)) for $x=-2(.01)5$ to 8D and 7D, respectively.

• Zhang and Jin (1996, p. 338) tabulates $\int_{0}^{x}\mathrm{Ai}\left(t\right)\mathrm{d}t$ and $\int_{0}^{x}\mathrm{Bi}\left(t\right)\mathrm{d}t$ for $x=-10(.2)10$ to 8D or 8S.

## §9.18(vi) Scorer Functions

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

• Rothman (1954a) tabulates $\int_{0}^{x}\mathrm{Gi}\left(t\right)\mathrm{d}t$, $\mathrm{Gi}'\left(x\right)$, $\int_{0}^{x}\mathrm{Hi}\left(-t\right)\mathrm{d}t$, $-\mathrm{Hi}'\left(-x\right)$ for $x=0(.1)10$; 7D.

• National Bureau of Standards (1958) tabulates $A_{0}(x)\equiv\pi\mathrm{Hi}\left(-x\right)$ and $-A_{0}^{\prime}(x)\equiv\pi\mathrm{Hi}'\left(-x\right)$ 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.

• Nosova and Tumarkin (1965) tabulates $e_{0}(x)\equiv\pi\mathrm{Hi}\left(-x\right)$, $e^{\prime}_{0}(x)\equiv-\pi\mathrm{Hi}'\left(-x\right)$, $\widetilde{e}_{0}(-x)\equiv-\pi\mathrm{Gi}\left(x\right)$, $\widetilde{e}^{\mspace{2.0mu }\prime}_{0}(-x)\equiv\pi\mathrm{Gi}'\left(x\right)$ for $x=-1(.01)10$; 7D. Also included are the real and imaginary parts of $e_{0}(z)$ and $ie^{\prime}_{0}(z)$, where $z=iy$ and $y=0(.01)9$; 6-7D.

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

## §9.18(vii) Generalized Airy Functions

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