Digital Library of Mathematical Functions
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9 Airy and Related FunctionsComputation

§9.18 Tables

Contents

§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 Ai(x), Ai(x) for x=-20(.01)2; log10Ai(x), Ai(x)/Ai(x) for x=0(.1)25(1)75; Bi(x), Bi(x) for x=-10(.1)2.5; log10Bi(x), Bi(x)/Bi(x) for x=0(.1)10; M(x), N(x), θ(x), ϕ(x) (respectively F(x), G(x), χ(x), ψ(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 2π1/2x1/4exp(23x3/2)Ai(x), 2π1/2x-1/4exp(23x3/2)Ai(x), π1/2x1/4exp(-23x3/2)Bi(x), and π1/2x-1/4exp(-23x3/2)Bi(x) for 32x-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 Ai(x), Ai(x), Bi(x), Bi(x) for x=0(1)20 to 8S and for x=-20(1)0 to 9D.

  • Yakovleva (1969) tabulates Fock’s functions U(x)πBi(x), U(x)πBi(x), V(x)πAi(x), V(x)πAi(x) for x=-9(.001)9. Precision is 7S.

§9.18(iii) Complex Variables

  • Woodward and Woodward (1946) tabulates the real and imaginary parts of Ai(z), Ai(z), Bi(z), Bi(z) for z=-2.4(.2)2.4, z=-2.4(.2)0. Precision is 4D.

  • Harvard University (1945) tabulates the real and imaginary parts of h1(z), h1(z), h2(z), h2(z) for -x0zx0, 0zy0, |x0+y0|<6.1, with interval 0.1 in z and z. Precision is 8D. Here h1(z)=-24/331/6Ai(-π/3z), h2(z)=24/331/6Ai(π/3z).

§9.18(iv) Zeros

  • Miller (1946) tabulates ak, Ai(ak), ak, Ai(ak), k=1(1)50; bk, Bi(bk), bk, Bi(bk), 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 ak, Ai(ak), ak, Ai(ak), k=1(1)50; 20S.

  • Zhang and Jin (1996, p. 339) tabulates ak, Ai(ak), ak, Ai(ak), bk, Bi(bk), bk, Bi(bk), k=1(1)20; 8D.

  • Corless et al. (1992) gives the real and imaginary parts of βk for k=1(1)13; 14S.

  • See also §9.9(v).

§9.18(v) Integrals

  • Rothman (1954b) tabulates 0xAi(t)t and 0xBi(t)t for x=-10(.1) and -10(.1)2, respectively; 7D. The entries in the columns headed 0xAi(-x)x and 0xBi(-x)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 0xAi(-t)t and 0x0vAi(-t)tv (see (9.10.20)) for x=-2(.01)5 to 8D and 7D, respectively.

  • Zhang and Jin (1996, p. 338) tabulates 0xAi(t)t and 0xBi(t)t for x=-10(.2)10 to 8D or 8S.

§9.18(vi) Scorer Functions

  • Scorer (1950) tabulates Gi(x) and Hi(-x) for x=0(.1)10; 7D.

  • Rothman (1954a) tabulates 0xGi(t)t, Gi(x), 0xHi(-t)t, -Hi(-x) for x=0(.1)10; 7D.

  • National Bureau of Standards (1958) tabulates A0(x)πHi(-x) and -A0(x)πHi(-x) for x=0(.01)1(.02)5(.05)11 and 1/x=0.01(.01)0.1; 0xA0(t)t for x=0.5,1(1)11. Precision is 8D.

  • Nosova and Tumarkin (1965) tabulates e0(x)πHi(-x), e0(x)-πHi(-x), e~0(-x)-πGi(x), e~0(-x)πGi(x) for x=-1(.01)10; 7D. Also included are the real and imaginary parts of e0(z) and e0(z), where z=y and y=0(.01)9; 6-7D.

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

§9.18(vii) Generalized Airy Functions

  • Smirnov (1960) tabulates U1(x,α), U2(x,α), defined by (9.13.20), (9.13.21), and also U1(x,α)/x, U2(x,α)/x, for α=1, x=-6(.01)10 to 5D or 5S, and also for α=±14, ±13, ±12, ±23, ±34, 54, 43, 32, 53, 74, 2, x=0(.01)6; 4D.