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9 Airy and Related FunctionsAiry Functions

§9.11 Products

Contents

§9.11(i) Differential Equation

9.11.1 3wz3-4zwz-2w=0,
w=w1w2,

where w1 and w2 are any solutions of (9.2.1). For example, w=Ai2(z), Ai(z)Bi(z), Ai(z)Ai(z2π/3), M2(z). Numerically satisfactory triads of solutions can be constructed where needed on or by inspection of the asymptotic expansions supplied in §9.7.

§9.11(ii) Wronskian

9.11.2 𝒲{Ai2(z),Ai(z)Bi(z),Bi2(z)}=2π-3.

§9.11(iii) Integral Representations

9.11.3 Ai2(x)=14π30J0(112t3+xt)tt,
x0,

where J0 is the Bessel function (§10.2(ii)).

9.11.4 Ai2(z)+Bi2(z)=1π3/20exp(zt-112t3)t-1/2t.

For an integral representation of the Dirac delta involving a product of two Ai functions see §1.17(ii).

For further integral representations see Reid (1995, 1997a, 1997b).

§9.11(iv) Indefinite Integrals

Let w1,w2 be any solutions of (9.2.1), not necessarily distinct. Then

9.11.5 w1w2z=-w1w2+zw1w2,
9.11.6 w1w2z=12(w1w2+z𝒲{w1,w2}),
9.11.7 w1w2z=13(w1w2+w1w2+zw1w2-z2w1w2),
9.11.8 zw1w2z=16(w1w2+w1w2)-13(zw1w2-z2w1w2),
9.11.9 zw1w2z=12w1w2+14z2𝒲{w1,w2},
9.11.10 zw1w2z=310(-w1w2+zw1w2+zw1w2)+15(z2w1w2-z3w1w2).

For znw1w2z, znw1w2z, znw1w2z, where n is any positive integer, see Albright (1977). For related integrals see Gordon (1969, Appendix B).

For any continuously-differentiable function f

9.11.11 1w12f(w2w1)z=1𝒲{w1,w2}f(w2w1).

Examples

9.11.12 zAi2(z) =πBi(z)Ai(z),
9.11.13 zAi(z)Bi(z) =πln(Bi(z)Ai(z)),
9.11.14 Ai(z)Bi(z)(Ai2(z)+Bi2(z))2z =π2Bi2(z)Ai2(z)+Bi2(z).

§9.11(v) Definite Integrals

9.11.15 0tα-1Ai2(t)t=2Γ(α)π1/212(2α+5)/6Γ(13α+56),
α>0.
9.11.16 -Ai3(t)t =Γ2(13)4π2,
9.11.17 -Ai2(t)Bi(t)t =Γ2(13)43π2.
9.11.18 0Ai4(t)t =ln324π2.
9.11.19 0tAi2(t)+Bi2(t)=0ttAi2(t)+Bi2(t)=π26.

For further definite integrals see Prudnikov et al. (1990, §1.8.2), Laurenzi (1993), Reid (1995, 1997a, 1997b), and Vallée and Soares (2010, Chapters 3, 4).