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9 Airy and Related FunctionsRelated Functions

§9.12 Scorer Functions

  1. §9.12(i) Differential Equation
  2. §9.12(ii) Graphs
  3. §9.12(iii) Initial Values
  4. §9.12(iv) Numerically Satisfactory Solutions
  5. §9.12(v) Connection Formulas
  6. §9.12(vi) Maclaurin Series
  7. §9.12(vii) Integral Representations
  8. §9.12(viii) Asymptotic Expansions
  9. §9.12(ix) Zeros

§9.12(i) Differential Equation

9.12.1 d2wdz2zw=1π.

Solutions of this equation are the Scorer functions and can be found by the method of variation of parameters (§1.13(iii)). The general solution is given by

9.12.2 w(z)=Aw1(z)+Bw2(z)+p(z),

where A and B are arbitrary constants, w1(z) and w2(z) are any two linearly independent solutions of Airy’s equation (9.2.1), and p(z) is any particular solution of (9.12.1). Standard particular solutions are

9.12.3 Gi(z),


9.12.4 Gi(z)=Bi(z)zAi(t)dt+Ai(z)0zBi(t)dt,
9.12.5 Hi(z)=Bi(z)zAi(t)dtAi(z)zBi(t)dt.

Gi(z) and Hi(z) are entire functions of z.

§9.12(ii) Graphs

See Figures 9.12.1 and 9.12.2.

See accompanying text
Figure 9.12.1: Gi(x), Gi(x). Magnify
See accompanying text
Figure 9.12.2: Hi(x), Hi(x). Magnify

§9.12(iii) Initial Values

9.12.6 Gi(0) =12Hi(0)=13Bi(0)=1/(37/6Γ(23))=0.20497 55424,
9.12.7 Gi(0) =12Hi(0)=13Bi(0)=1/(35/6Γ(13))=0.14942 94524.

§9.12(iv) Numerically Satisfactory Solutions

Gi(x) is a numerically satisfactory companion to the complementary functions Ai(x) and Bi(x) on the interval 0x<. Hi(x) is a numerically satisfactory companion to Ai(x) and Bi(x) on the interval <x0.

In , numerically satisfactory sets of solutions are given by

9.12.8 Gi(z),Ai(z),Bi(z),
9.12.9 Hi(z),Ai(ze2πi/3),Ai(ze2πi/3),


9.12.10 e2πi/3Hi(ze2πi/3),Ai(z),Ai(ze±2πi/3),

§9.12(v) Connection Formulas

9.12.11 Gi(z)+Hi(z)=Bi(z),
9.12.12 Gi(z)=12eπi/3Hi(ze2πi/3)+12eπi/3Hi(ze2πi/3),
9.12.13 Gi(z)=eπi/3Hi(ze±2πi/3)±iAi(z),
9.12.14 Hi(z)=e±2πi/3Hi(ze±2πi/3)+2eπi/6Ai(ze2πi/3).

§9.12(vi) Maclaurin Series

9.12.15 Gi(z)=32/3πk=0cos(2k13π)Γ(k+13)(31/3z)kk!,
9.12.16 Gi(z)=31/3πk=0cos(2k+13π)Γ(k+23)(31/3z)kk!.
9.12.17 Hi(z)=32/3πk=0Γ(k+13)(31/3z)kk!,
9.12.18 Hi(z)=31/3πk=0Γ(k+23)(31/3z)kk!.

§9.12(vii) Integral Representations

9.12.19 Gi(x)=1π0sin(13t3+xt)dt,
9.12.20 Hi(z)=1π0exp(13t3+zt)dt,
9.12.21 Gi(z)=1π0exp(13t312zt)cos(123zt+23π)dt.

If ζ=23z3/2 or 23x3/2, and K1/3 is the modified Bessel function (§10.25(ii)), then

9.12.22 Hi(z) =4z233/2π20K1/3(t)ζ2+t2dt,
9.12.23 Gi(x) =4x233/2π20K1/3(t)ζ2t2dt,

where the last integral is a Cauchy principal value (§1.4(v)).

Mellin–Barnes Type Integral

9.12.24 Hi(z)=32/32π2iiiΓ(13+13t)Γ(t)(31/3eπiz)tdt,

where the integration contour separates the poles of Γ(13+13t) from those of Γ(t).

§9.12(viii) Asymptotic Expansions

Functions and Derivatives

As z, and with δ denoting an arbitrary small positive constant,

9.12.25 Gi(z)1πzk=0(3k)!k!(3z3)k,
9.12.26 Gi(z)1πz2k=0(3k+1)!k!(3z3)k,
9.12.27 Hi(z)1πzk=0(3k)!k!(3z3)k,
9.12.28 Hi(z)1πz2k=0(3k+1)!k!(3z3)k,

For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. For example, with the notation of §9.7(i),

9.12.29 Hi(z)1πzk=0(3k)!k!(3z3)k+eζπz1/4k=0ukζk,


9.12.30 0zGi(t)dt1πlnz+2γ+ln33π1πk=1(3k1)!k!(3z3)k,
9.12.31 0zHi(t)dt1πlnz+2γ+ln33π+1πk=1(1)k1(3k1)!k!(3z3)k,

where γ is Euler’s constant (§5.2(ii)).

§9.12(ix) Zeros

All zeros, real or complex, of Gi(z) and Hi(z) are simple.

Neither Hi(z) nor Hi(z) has real zeros.

Gi(z) has no nonnegative real zeros and Gi(z) has exactly one nonnegative real zero, given by z=0.60907 54170 7. Both Gi(z) and Gi(z) have an infinity of negative real zeros, and they are interlaced.

For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c).

For graphical illustration of the real zeros see Figures 9.12.1 and 9.12.2.