§9.18 Tables

Permalink:: http://dlmf.nist.gov/9.18

§9.18(i) Introduction
§9.18(ii) Real Variables
§9.18(iii) Complex Variables
§9.18(iv) Zeros
§9.18(v) Integrals
§9.18(vi) Scorer Functions
§9.18(vii) Generalized Airy Functions

§9.18(i) Introduction

Permalink:: http://dlmf.nist.gov/9.18.i

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

Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.18.ii

Miller (1946) tabulates $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)$ for $x=-20(.01)2;$ $\mathop{\mathrm{log}_{{10}}\/}\nolimits\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)/\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ for $x=0(.1)25(1)75$ ; $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)$ for $x=-10(.1)2.5$ ; $\mathop{\mathrm{log}_{{10}}\/}\nolimits\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)/\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ for $x=0(.1)10$ ; $\mathop{M\/}\nolimits\!\left(x\right)$ , $\mathop{N\/}\nolimits\!\left(x\right)$ , $\mathop{\theta\/}\nolimits\!\left(x\right)$ , $\mathop{\phi\/}\nolimits\!\left(x\right)$ (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 $2\pi^{{1/2}}x^{{1/4}}\*\mathop{\exp\/}\nolimits(\tfrac{2}{3}x^{{3/2}})\*\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ , $2\pi^{{1/2}}x^{{-1/4}}\*\mathop{\exp\/}\nolimits(\tfrac{2}{3}x^{{3/2}})\*{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\pi^{{1/2}}x^{{1/4}}\*\mathop{\exp\/}\nolimits(-\tfrac{2}{3}x^{{3/2}})\*\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ , and $\pi^{{1/2}}x^{{-1/4}}\*\mathop{\exp\/}\nolimits(-\tfrac{2}{3}x^{{3/2}})\*{\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)$ 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 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.
Yakovleva (1969) tabulates Fock’s functions $U(x)\equiv\sqrt{\pi}\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ , $U^{{\prime}}(x)\equiv\sqrt{\pi}{\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $V(x)\equiv\sqrt{\pi}\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ , $V^{{\prime}}(x)\equiv\sqrt{\pi}{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)$ for $x=-9(.001)9$ . Precision is 7S.

§9.18(iii) Complex Variables

Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.18.iii

Woodward and Woodward (1946) tabulates the real and imaginary parts of $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ , ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(z\right)$ for $\realpart{z}=-2.4(.2)2.4$ , $\imagpart{z}=-2.4(.2)0$ . 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 $-x_{0}\leq\realpart{z}\leq x_{0}$ , $0\leq\imagpart{z}\leq y_{0}$ , $|x_{0}+iy_{0}|<6.1$ , with interval 0.1 in $\realpart{z}$ and $\imagpart{z}$ . Precision is 8D. Here $h_{1}(z)=-2^{{4/3}}3^{{1/6}}i\mathop{\mathrm{Ai}\/}\nolimits\!\left(e^{{-\pi i/3}}z\right)$ , $h_{2}(z)=2^{{4/3}}3^{{1/6}}i\mathop{\mathrm{Ai}\/}\nolimits\!\left(e^{{\pi i/3}}z\right)$ .

§9.18(iv) Zeros

Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.18.iv

Miller (1946) tabulates $\mathop{a_{{k}}\/}\nolimits$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(\mathop{a_{{k}}\/}\nolimits\right)$ , $\mathop{a^{{\prime}}_{{k}}\/}\nolimits$ , $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\mathop{a^{{\prime}}_{{k}}\/}\nolimits\right)$ , $k=1(1)50$ ; $\mathop{b_{{k}}\/}\nolimits$ , ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(\mathop{b_{{k}}\/}\nolimits\right)$ , $\mathop{b^{{\prime}}_{{k}}\/}\nolimits$ , $\mathop{\mathrm{Bi}\/}\nolimits\!\left(\mathop{b^{{\prime}}_{{k}}\/}\nolimits\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).
Sherry (1959) tabulates $\mathop{a_{{k}}\/}\nolimits$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(\mathop{a_{{k}}\/}\nolimits\right)$ , $\mathop{a^{{\prime}}_{{k}}\/}\nolimits$ , $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\mathop{a^{{\prime}}_{{k}}\/}\nolimits\right)$ , $k=1(1)50$ ; 20S.
Zhang and Jin (1996, p. 339) tabulates $\mathop{a_{{k}}\/}\nolimits$ , ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(\mathop{a_{{k}}\/}\nolimits\right)$ , $\mathop{a^{{\prime}}_{{k}}\/}\nolimits$ , $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\mathop{a^{{\prime}}_{{k}}\/}\nolimits\right)$ , $\mathop{b_{{k}}\/}\nolimits$ , ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(\mathop{b_{{k}}\/}\nolimits\right)$ , $\mathop{b^{{\prime}}_{{k}}\/}\nolimits$ , $\mathop{\mathrm{Bi}\/}\nolimits\!\left(\mathop{b^{{\prime}}_{{k}}\/}\nolimits\right)$ , $k=1(1)20$ ; 8D.
Corless et al. (1992) gives the real and imaginary parts of $\mathop{\beta _{{k}}\/}\nolimits$ for $k=1(1)13$ ; 14S.
See also §9.9(v).

§9.18(v) Integrals

Keywords:: Airy functions, integrals
Permalink:: http://dlmf.nist.gov/9.18.v

Rothman (1954b) tabulates $\int _{0}^{x}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt$ and $\int _{0}^{x}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)dt$ for $x=-10(.1)\infty$ and $-10(.1)2$ , respectively; 7D. The entries in the columns headed $\int _{0}^{x}\mathop{\mathrm{Ai}\/}\nolimits\!\left(-x\right)dx$ and $\int _{0}^{x}\mathop{\mathrm{Bi}\/}\nolimits\!\left(-x\right)dx$ 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.
National Bureau of Standards (1958) tabulates $\int _{0}^{x}\mathop{\mathrm{Ai}\/}\nolimits\!\left(-t\right)dt$ and $\int _{0}^{x}\int _{0}^{v}\mathop{\mathrm{Ai}\/}\nolimits\!\left(-t\right)dtdv$ (see (9.10.20)) for $x=-2(.01)5$ to 8D and 7D, respectively.
Zhang and Jin (1996, p. 338) tabulates $\int _{0}^{x}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt$ and $\int _{0}^{x}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)dt$ for $x=-10(.2)10$ to 8D or 8S.

§9.18(vi) Scorer Functions

Keywords:: Scorer functions, integrals
Permalink:: http://dlmf.nist.gov/9.18.vi

Scorer (1950) tabulates $\mathop{\mathrm{Gi}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{Hi}\/}\nolimits\!\left(-x\right)$ for $x=0(.1)10$ ; 7D.
Rothman (1954a) tabulates $\int _{0}^{x}\mathop{\mathrm{Gi}\/}\nolimits\!\left(t\right)dt$ , ${\mathop{\mathrm{Gi}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\int _{0}^{x}\mathop{\mathrm{Hi}\/}\nolimits\!\left(-t\right)dt$ , $-{\mathop{\mathrm{Hi}\/}\nolimits^{{\prime}}}\!\left(-x\right)$ for $x=0(.1)10$ ; 7D.
National Bureau of Standards (1958) tabulates $A_{0}(x)\equiv\pi\mathop{\mathrm{Hi}\/}\nolimits\!\left(-x\right)$ and $-A_{0}^{{\prime}}(x)\equiv\pi{\mathop{\mathrm{Hi}\/}\nolimits^{{\prime}}}\!\left(-x\right)$ 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.
Nosova and Tumarkin (1965) tabulates $e_{0}(x)\equiv\pi\mathop{\mathrm{Hi}\/}\nolimits\!\left(-x\right)$ , $e^{{\prime}}_{0}(x)\equiv-\pi{\mathop{\mathrm{Hi}\/}\nolimits^{{\prime}}}\!\left(-x\right)$ , $\widetilde{e}_{0}(-x)\equiv-\pi\mathop{\mathrm{Gi}\/}\nolimits\!\left(x\right)$ , $\widetilde{e}^{{\mspace{0.733333mu}\prime}}_{0}(-x)\equiv\pi{\mathop{\mathrm{Gi}\/}\nolimits^{{\prime}}}\!\left(x\right)$ 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 ${\mathop{\mathrm{Gi}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , the first 10 negative real zeros of $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$ and ${\mathop{\mathrm{Gi}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , and the first 10 complex zeros of $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$ , ${\mathop{\mathrm{Gi}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$ , and ${\mathop{\mathrm{Hi}\/}\nolimits^{{\prime}}}\!\left(z\right)$ . Precision is 11 or 12S.

§9.18(vii) Generalized Airy Functions

Keywords:: generalized Airy functions
Permalink:: http://dlmf.nist.gov/9.18.vii

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.

§9.18 Tables

Contents

§9.18(i) Introduction

§9.18(ii) Real Variables

§9.18(iii) Complex Variables

§9.18(iv) Zeros

§9.18(v) Integrals

§9.18(vi) Scorer Functions

§9.18(vii) Generalized Airy Functions