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9 Airy and Related FunctionsAiry Functions

§9.10 Integrals

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§9.10(i) Indefinite Integrals

9.10.1 zAi(t)dt=π(Ai(z)Gi(z)-Ai(z)Gi(z)),
9.10.2 -zAi(t)dt=π(Ai(z)Hi(z)-Ai(z)Hi(z)),
9.10.3 -zBi(t)dt=0zBi(t)dt=π(Bi(z)Gi(z)-Bi(z)Gi(z))=π(Bi(z)Hi(z)-Bi(z)Hi(z)).

For the functions Gi and Hi see §9.12.

§9.10(ii) Asymptotic Approximations

9.10.4 xAi(t)dt 12π-1/2x-3/4exp(-23x3/2),
x,
9.10.5 0xBi(t)dt π-1/2x-3/4exp(23x3/2),
x.
9.10.6 -xAi(t)dt=π-1/2(-x)-3/4cos(23(-x)3/2+14π)+O(|x|-9/4),
x-,
9.10.7 -xBi(t)dt=π-1/2(-x)-3/4sin(23(-x)3/2+14π)+O(|x|-9/4),
x-.

For higher terms in (9.10.4)–(9.10.7) see Vallée and Soares (2010, §3.1.3). For error bounds see Boyd (1993).

See also Muldoon (1970).

§9.10(iii) Other Indefinite Integrals

Let w(z) be any solution of Airy’s equation (9.2.1). Then

9.10.8 zw(z)dz =w(z),
9.10.9 z2w(z)dz =zw(z)-w(z),
9.10.10 zn+3w(z)dz=zn+2w(z)-(n+2)zn+1w(z)+(n+1)(n+2)znw(z)dz,
n=0,1,2,.

See also §9.11(iv).

§9.10(iv) Definite Integrals

9.10.11 0Ai(t)dt =13,
-0Ai(t)dt =23,
9.10.12 -0Bi(t)dt=0.

§9.10(v) Laplace Transforms

9.10.13 -eptAi(t)dt=ep3/3,
p>0.
9.10.14 0e-ptAi(t)dt=e-p3/3(13-pF11(13;43;13p3)34/3Γ(43)+p2F11(23;53;13p3)35/3Γ(53)),
p.
9.10.15 0e-ptAi(-t)dt=13ep3/3(Γ(13,13p3)Γ(13)+Γ(23,13p3)Γ(23)),
p>0,
9.10.16 0e-ptBi(-t)dt=13ep3/3(Γ(23,13p3)Γ(23)-Γ(13,13p3)Γ(13)),
p>0.

For the confluent hypergeometric function F11 and the incomplete gamma function Γ see §§13.1, 13.2, and 8.2(i).

For Laplace transforms of products of Airy functions see Shawagfeh (1992).

§9.10(vi) Mellin Transform

9.10.17 0tα-1Ai(t)dt=Γ(α)3(α+2)/3Γ(13α+23),
α>0.

§9.10(vii) Stieltjes Transforms

9.10.18 Ai(z)=z5/4e-(2/3)z3/227/2π0t-1/2e-(2/3)t3/2Ai(t)z3/2+t3/2dt,
|phz|<23π.
9.10.19 Bi(x)=x5/4e(2/3)x3/225/2π0t-1/2e-(2/3)t3/2Ai(t)x3/2-t3/2dt,
x>0,

where the last integral is a Cauchy principal value (§1.4(v)).

§9.10(viii) Repeated Integrals

9.10.20 0x0vAi(t)dtdv=x0xAi(t)dt-Ai(x)+Ai(0),
9.10.21 0x0vBi(t)dtdv=x0xBi(t)dt-Bi(x)+Bi(0),
9.10.22 0ttAi(-t)(dt)n=2cos(13(n-1)π)3(n+2)/3Γ(13n+23),
n=1,2,.

§9.10(ix) Compendia

For further integrals, including the Airy transform, see §9.11(iv), Widder (1979), Prudnikov et al. (1990, §1.8.1), Prudnikov et al. (1992a, pp. 405–413), Prudnikov et al. (1992b, §4.3.25), Vallée and Soares (2010, Chapters 3, 4).