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9 Airy and Related FunctionsAiry Functions

§9.2 Differential Equation

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§9.2(i) Airy’s Equation

9.2.1 d2wdz2=zw.

All solutions are entire functions of z.

Standard solutions are:

9.2.2 w=Ai(z), Bi(z), Ai(ze2πi/3).

§9.2(ii) Initial Values

9.2.3 Ai(0) =132/3Γ(23)
=0.35502 80538,
9.2.4 Ai(0) =-131/3Γ(13)
=-0.25881 94037,
9.2.5 Bi(0) =131/6Γ(23)
=0.61492 66274,
9.2.6 Bi(0) =31/6Γ(13)
=0.44828 83573.

§9.2(iii) Numerically Satisfactory Pairs of Solutions

Table 9.2.1 lists numerically satisfactory pairs of solutions of (9.2.1) for the stated intervals or regions; compare §2.7(iv).

Table 9.2.1: Numerically satisfactory pairs of solutions of Airy’s equation.
Pair Interval or Region
Ai(x),Bi(x) -<x<
Ai(z),Bi(z) {|phz|13π-<z0
Ai(z),Ai(ze-2πi/3) -13πphzπ
Ai(z),Ai(ze2πi/3) -πphz13π
Ai(ze2πi/3) |ph(-z)|23π

§9.2(iv) Wronskians

9.2.7 𝒲{Ai(z),Bi(z)}=1π,
9.2.8 𝒲{Ai(z),Ai(ze2πi/3)}=e±πi/62π,
9.2.9 𝒲{Ai(ze-2πi/3),Ai(ze2πi/3)}=12πi.

§9.2(v) Connection Formulas

9.2.10 Bi(z)=e-πi/6Ai(ze-2πi/3)+eπi/6Ai(ze2πi/3).
9.2.11 Ai(ze2πi/3)=12eπi/3(Ai(z)±iBi(z)).
9.2.12 Ai(z)+e-2πi/3Ai(ze-2πi/3)+e2πi/3Ai(ze2πi/3)=0,
9.2.13 Bi(z)+e-2πi/3Bi(ze-2πi/3)+e2πi/3Bi(ze2πi/3)=0.
9.2.14 Ai(-z) =eπi/3Ai(zeπi/3)+e-πi/3Ai(ze-πi/3),
9.2.15 Bi(-z) =e-πi/6Ai(zeπi/3)+eπi/6Ai(ze-πi/3).

§9.2(vi) Riccati Form of Differential Equation

9.2.16 dWdz+W2=z,

W=(1/w)dw/dz, where w is any nontrivial solution of (9.2.1). See also Smith (1990).