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9 Airy and Related FunctionsAiry Functions

§9.10 Integrals

Contents
  1. §9.10(i) Indefinite Integrals
  2. §9.10(ii) Asymptotic Approximations
  3. §9.10(iii) Other Indefinite Integrals
  4. §9.10(iv) Definite Integrals
  5. §9.10(v) Laplace Transforms
  6. §9.10(vi) Mellin Transform
  7. §9.10(vii) Stieltjes Transforms
  8. §9.10(viii) Repeated Integrals
  9. §9.10(ix) Compendia

§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 12π1/2x3/4exp(23x3/2),
x,
9.10.5 0xBi(t)dt π1/2x3/4exp(23x3/2),
x.
9.10.6 xAi(t)dt=π1/2(x)3/4cos(23(x)3/2+14π)+O(|x|9/4),
x,
9.10.7 xBi(t)dt=π1/2(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 0eptAi(t)dt=ep3/3(13pF11(13;43;13p3)34/3Γ(43)+p2F11(23;53;13p3)35/3Γ(53)),
p.
9.10.15 0eptAi(t)dt=13ep3/3(Γ(13,13p3)Γ(13)+Γ(23,13p3)Γ(23)),
p>0,
9.10.16 0eptBi(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 0tα1Ai(t)dt=Γ(α)3(α+2)/3Γ(13α+23),
α>0.

§9.10(vii) Stieltjes Transforms

9.10.18 Ai(z)=3z5/4e(2/3)z3/24π0t3/4e(2/3)t3/2Ai(t)z3/2+t3/2dt,
|phz|<23π.
9.10.19 Bi(x)=3x5/4e(2/3)x3/22π0t3/4e(2/3)t3/2Ai(t)x3/2t3/2dt,
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)dtAi(x)+Ai(0),
9.10.21 0x0vBi(t)dtdv=x0xBi(t)dtBi(x)+Bi(0),
9.10.22 0ttAi(t)(dt)n=2cos(13(n1)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).