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9 Airy and Related FunctionsAiry Functions

§9.6 Relations to Other Functions

Contents

§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions

For the notation see §§10.2(ii) and 10.25(ii). With

9.6.1 ζ=23z3/2,
9.6.2 Ai(z) =π-1z/3K±1/3(ζ)
=13z(I-1/3(ζ)-I1/3(ζ))
=12z/3e2πi/3H1/3(1)(ζeπi/2)
=12z/3eπi/3H-1/3(1)(ζeπi/2)
=12z/3e-2πi/3H1/3(2)(ζe-πi/2)
=12z/3e-πi/3H-1/3(2)(ζe-πi/2),
9.6.3 Ai(z) =-π-1(z/3)K±2/3(ζ)
=(z/3)(I2/3(ζ)-I-2/3(ζ))
=12(z/3)e-πi/6H2/3(1)(ζeπi/2)
=12(z/3)e-5πi/6H-2/3(1)(ζeπi/2)
=12(z/3)eπi/6H2/3(2)(ζe-πi/2)
=12(z/3)e5πi/6H-2/3(2)(ζe-πi/2),
9.6.4 Bi(z) =z/3(I1/3(ζ)+I-1/3(ζ))
=12z/3(eπi/6H1/3(1)(ζe-πi/2)+e-πi/6H1/3(2)(ζeπi/2))
=12z/3(e-πi/6H-1/3(1)(ζe-πi/2)+eπi/6H-1/3(2)(ζeπi/2)),
9.6.5 Bi(z) =(z/3)(I2/3(ζ)+I-2/3(ζ))
=12(z/3)(eπi/3H2/3(1)(ζe-πi/2)+e-πi/3H2/3(2)(ζeπi/2))
=12(z/3)(e-πi/3H-2/3(1)(ζe-πi/2)+eπi/3H-2/3(2)(ζeπi/2)),
9.6.6 Ai(-z)=(z/3)(J1/3(ζ)+J-1/3(ζ))=12z/3(eπi/6H1/3(1)(ζ)+e-πi/6H1/3(2)(ζ))=12z/3(e-πi/6H-1/3(1)(ζ)+eπi/6H-1/3(2)(ζ)),
9.6.7 Ai(-z)=(z/3)(J2/3(ζ)-J-2/3(ζ))=12(z/3)(e-πi/6H2/3(1)(ζ)+eπi/6H2/3(2)(ζ))=12(z/3)(e-5πi/6H-2/3(1)(ζ)+e5πi/6H-2/3(2)(ζ)),
9.6.8 Bi(-z)=z/3(J-1/3(ζ)-J1/3(ζ))=12z/3(e2πi/3H1/3(1)(ζ)+e-2πi/3H1/3(2)(ζ))=12z/3(eπi/3H-1/3(1)(ζ)+e-πi/3H-1/3(2)(ζ)),
9.6.9 Bi(-z)=(z/3)(J-2/3(ζ)+J2/3(ζ))=12(z/3)(eπi/3H2/3(1)(ζ)+e-πi/3H2/3(2)(ζ))=12(z/3)(e-πi/3H-2/3(1)(ζ)+eπi/3H-2/3(2)(ζ)).

§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions

Again, for the notation see §§10.2(ii) and 10.25(ii). With

9.6.10 z=(32ζ)2/3,
9.6.11 J±1/3(ζ) =123/z(3Ai(-z)Bi(-z)),
9.6.12 J±2/3(ζ) =12(3/z)(±3Ai(-z)+Bi(-z)),
9.6.13 I±1/3(ζ) =123/z(3Ai(z)+Bi(z)),
9.6.14 I±2/3(ζ) =12(3/z)(±3Ai(z)+Bi(z)),
9.6.15 K±1/3(ζ) =π3/zAi(z),
9.6.16 K±2/3(ζ) =-π(3/z)Ai(z),
9.6.17 H1/3(1)(ζ) =e-πi/3H-1/3(1)(ζ)
=e-πi/63/z(Ai(-z)-iBi(-z)),
9.6.18 H2/3(1)(ζ) =e-2πi/3H-2/3(1)(ζ)
=eπi/6(3/z)(Ai(-z)-iBi(-z)),
9.6.19 H1/3(2)(ζ) =eπi/3H-1/3(2)(ζ)
=eπi/63/z(Ai(-z)+iBi(-z)),
9.6.20 H2/3(2)(ζ) =e2πi/3H-2/3(2)(ζ)
=e-πi/6(3/z)(Ai(-z)+iBi(-z)).

§9.6(iii) Airy Functions as Confluent Hypergeometric Functions

For the notation see §§13.1, 13.2, and 13.14(i). With ζ as in (9.6.1),

9.6.21 Ai(z) =12π-1/2z-1/4W0,1/3(2ζ)
=3-1/6π-1/2ζ2/3e-ζU(56,53,2ζ),
9.6.22 Ai(z) =-12π-1/2z1/4W0,2/3(2ζ)
=-31/6π-1/2ζ4/3e-ζU(76,73,2ζ),
9.6.23 Bi(z) =121/3Γ(23)z-1/4M0,-1/3(2ζ)+325/3Γ(13)z-1/4M0,1/3(2ζ),
9.6.24 Bi(z) =21/3Γ(13)z1/4M0,-2/3(2ζ)+3210/3Γ(23)z1/4M0,2/3(2ζ),
9.6.25 Bi(z) =131/6Γ(23)e-ζF11(16;13;2ζ)+35/622/3Γ(13)ζ2/3e-ζF11(56;53;2ζ),
9.6.26 Bi(z) =31/6Γ(13)e-ζF11(-16;-13;2ζ)+37/627/3Γ(23)ζ4/3e-ζF11(76;73;2ζ).

To express Airy functions in terms of hypergeometric functions combine §9.6(i) with (10.39.9).