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9 Airy and Related FunctionsAiry Functions

§9.2 Differential Equation

Contents
  1. §9.2(i) Airy’s Equation
  2. §9.2(ii) Initial Values
  3. §9.2(iii) Numerically Satisfactory Pairs of Solutions
  4. §9.2(iv) Wronskians
  5. §9.2(v) Connection Formulas
  6. §9.2(vi) Riccati Form of Differential Equation

§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(ze2πi/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(ze2πi/3),Ai(ze2πi/3)}=12πi.

§9.2(v) Connection Formulas

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