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

Lastly, for x>0 we define

9.7.3 χ(x)=π1/2Γ(12x+1)/Γ(12x+12).

For large x,

9.7.4 χ(x)(12πx)1/2.

Numerical values of χ(n) are given in Table 9.7.1 for n=1(1)20 to 2D.

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) e-ζ2πz1/4k=0(-1)kukζk,
|phz|π-δ,
9.7.6 Ai(z) -z1/4e-ζ2πk=0(-1)kvkζk,
|phz|π-δ,
9.7.7 Bi(z) eζπz1/4k=0ukζk,
|phz|13π-δ,
9.7.8 Bi(z) z1/4eζπk=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(ze±πi/3)2πe±πi/6z1/4(cos(ζ-14π12iln2)k=0(-1)ku2kζ2k+sin(ζ-14π12iln2)k=0(-1)ku2k+1ζ2k+1),
|phz|23π-δ,
9.7.14 Bi(ze±πi/3)2πeπi/6z1/4(-sin(ζ-14π12iln2)k=0(-1)kv2kζ2k+cos(ζ-14π12iln2)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 χ(n+σ)+1 where σ=16 for (9.7.7) and σ=0 for (9.7.8), provided that n0 in the first case and n1 in the second case.

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) e-ξ2πx1/4,
|Ai(x)| x1/4e-ξ2π(1+772ξ),
9.7.16 Bi(x) eξπx1/4(1+(χ(76)+1)572ξ),
Bi(x) x1/4eξπ(1+(π2+1)772ξ),

where ξ=23x3/2.

§9.7(iv) Error Bounds for Complex Variables

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 {1,|phz|13π,min(|csc(phζ)|,χ(n+σ)+1),13π|phz|23π,2π(n+σ)|cos(phζ)|n+σ+χ(n+σ)+1,23π|phz|<π,

provided that n0, σ=16 for (9.7.5) and n1, σ=0 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.

§9.7(v) Exponentially-Improved Expansions

In (9.7.5) and (9.7.6) let

9.7.18 Ai(z) =e-ζ2πz1/4(k=0n-1(-1)kukζk+Rn(z)),
9.7.19 Ai(z) =-z1/4e-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)=ez2πΓ(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(e-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).