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9 Airy and Related FunctionsAiry Functions

§9.7 Asymptotic Expansions

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§9.7(i) Notation

Here δ denotes an arbitrary small positive constant and

9.7.1 ζ=23z3/2.

Also u0=v0=1 and for k=1,2,,

9.7.2 uk =(2k+1)(2k+3)(2k+5)(6k-1)(216)k(k)!,
vk =6k+11-6kuk.

Lastly,

9.7.3 χ(n)=π1/2Γ(12n+1)/Γ(12n+12).

Numerical values of this function are given in Table 9.7.1 for n=1(1)20 to 2D. For large n,

9.7.4 χ(n)(12πn)1/2.
Table 9.7.1: χ(n).
n χ(n) n χ(n) n χ(n) n χ(n)
1 1.57 6 3.20 11 4.25 16 5.09
2 2.00 7 3.44 12 4.43 17 5.24
3 2.36 8 3.66 13 4.61 18 5.39
4 2.67 9 3.87 14 4.77 19 5.54
5 2.95 10 4.06 15 4.94 20 5.68

§9.7(ii) Poincaré-Type Expansions

As z the following asymptotic expansions are valid uniformly in the stated sectors.

9.7.5 Ai(z) -ζ2πz1/4k=0(-1)kukζk,
|phz|π-δ,
9.7.6 Ai(z) -z1/4-ζ2πk=0(-1)kvkζk,
|phz|π-δ,
9.7.7 Bi(z) ζπz1/4k=0ukζk,
|phz|13π-δ,
9.7.8 Bi(z) z1/4ζπk=0vkζk,
|phz|13π-δ.
9.7.9 Ai(-z) 1πz1/4(cos(ζ-14π)k=0(-1)ku2kζ2k+sin(ζ-14π)k=0(-1)ku2k+1ζ2k+1),
|phz|23π-δ,
9.7.10 Ai(-z) z1/4π(sin(ζ-14π)k=0(-1)kv2kζ2k-cos(ζ-14π)k=0(-1)kv2k+1ζ2k+1),
|phz|23π-δ,
9.7.11 Bi(-z) 1πz1/4(-sin(ζ-14π)k=0(-1)ku2kζ2k+cos(ζ-14π)k=0(-1)ku2k+1ζ2k+1),
|phz|23π-δ,
9.7.12 Bi(-z) z1/4π(cos(ζ-14π)k=0(-1)kv2kζ2k+sin(ζ-14π)k=0(-1)kv2k+1ζ2k+1),
|phz|23π-δ.
9.7.13 Bi(z±π/3)2π±π/6z1/4(cos(ζ-14π12ln2)k=0(-1)ku2kζ2k+sin(ζ-14π12ln2)k=0(-1)ku2k+1ζ2k+1,)
|phz|23π-δ,
9.7.14 Bi(z±π/3)2ππ/6z1/4(-sin(ζ-14π12ln2)k=0(-1)kv2kζ2k+cos(ζ-14π12ln2)k=0(-1)kv2k+1ζ2k+1,)
|phz|23π-δ.

§9.7(iii) Error Bounds for Real Variables

In (9.7.5) and (9.7.6) the nth error term, that is, the error on truncating the expansion at n terms, is bounded in magnitude by the first neglected term and has the same sign, provided that the following term is of opposite sign, that is, if n0 for (9.7.5) and n1 for (9.7.6).

In (9.7.7) and (9.7.8) the nth error term is bounded in magnitude by the first neglected term multiplied by 2χ(n)exp(σπ/(72ζ)) where σ=5 for (9.7.7) and σ=7 for (9.7.8), provided that n1 in both cases.

In (9.7.9)–(9.7.12) the nth error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign.

As special cases, when 0<x<

9.7.15 Ai(x) -ξ2πx1/4,
|Ai(x)| x1/4-ξ2π(1+772ξ),
9.7.16 Bi(x) ξπx1/4(1+5π72ξexp(5π72ξ)),
Bi(x) x1/4ξπ(1+7π72ξexp(7π72ξ)),

where ξ=23x3/2.

§9.7(iv) Error Bounds for Complex Variables

When n1 the nth error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17 2exp(σ36|ζ|),
2χ(n)exp(σπ72|ζ|)or
4χ(n)|cos(phζ)|nexp(σπ36|ζ|),

according as |phz|13π, 13π|phz|23π, or 23π|phz|π. Here σ=5 for (9.7.5) and σ=7 for (9.7.6).

Corresponding bounds for the errors in (9.7.7) to (9.7.14) may be obtained by use of these results and those of §9.2(v) and their differentiated forms.

For other error bounds see Boyd (1993).

§9.7(v) Exponentially-Improved Expansions

In (9.7.5) and (9.7.6) let

9.7.18 Ai(z) =-ζ2πz1/4(k=0n-1(-1)kukζk+Rn(z)),
9.7.19 Ai(z) =-z1/4-ζ2π(k=0n-1(-1)kvkζk+Sn(z)),

with n=2|ζ|. Then

9.7.20 Rn(z) =(-1)nk=0m-1(-1)kukGn-k(2ζ)ζk+Rm,n(z),
9.7.21 Sn(z) =(-1)n-1k=0m-1(-1)kvkGn-k(2ζ)ζk+Sm,n(z),

where

9.7.22 Gp(z)=z2πΓ(p)Γ(1-p,z).

(For the notation see §8.2(i).) And as z with m fixed

9.7.23 Rm,n(z),Sm,n(z)=O(-2|ζ|ζ-m),
|phz|23π.

For re-expansions of the remainder terms in (9.7.7)–(9.7.14) combine the results of this section with those of §9.2(v) and their differentiated forms, as in §9.7(iv).

For higher re-expansions of the remainder terms see Olde Daalhuis (1995, 1996), and Olde Daalhuis and Olver (1995a).