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9 Airy and Related FunctionsRelated Functions

§9.13 Generalized Airy Functions

  1. §9.13(i) Generalizations from the Differential Equation
  2. §9.13(ii) Generalizations from Integral Representations

§9.13(i) Generalizations from the Differential Equation

Equations of the form

9.13.1 d2wdz2=znw,

are used in approximating solutions to differential equations with multiple turning points; see §2.8(v). The general solution of (9.13.1) is given by

9.13.2 w=z1/2𝒵p(ζ),


9.13.3 p =1n+2,
ζ =2n+2z(n+2)/2=2pz1/(2p),

and 𝒵p is any linear combination of the modified Bessel functions Ip and epπiKp10.25(ii)).

Swanson and Headley (1967) define independent solutions An(z) and Bn(z) of (9.13.1) by

9.13.4 An(z) =(2p/π)sin(pπ)z1/2Kp(ζ),
Bn(z) =(pz)1/2(Ip(ζ)+Ip(ζ)),

when z is real and positive, and by analytic continuation elsewhere. (All solutions of (9.13.1) are entire functions of z.) When n=1, An(z) and Bn(z) become Ai(z) and Bi(z), respectively.

Properties of An(z) and Bn(z) follow from the corresponding properties of the modified Bessel functions. They include:

9.13.5 An(0) =p1/2Bn(0)=p1pΓ(1p),
An(0) =p1/2Bn(0)=ppΓ(p).
9.13.6 An(z) ={pz1/2(Jp(ζ)+Jp(ζ)),n odd,p1/2Bn(z),n even,
9.13.7 Bn(z) ={(pz)1/2(Jp(ζ)Jp(ζ)),n odd,p1/2An(z),n even.
9.13.8 𝒲{An(z),Bn(z)}=2πp1/2sin(pπ).

As z

9.13.9 An(z) =p/πsin(pπ)zn/4eζ(1+O(ζ1)),
9.13.10 An(z) ={2p/πcos(12pπ)zn/4(cos(ζ14π)+e|ζ|O(ζ1)),|phz|2pπδn odd,p/πzn/4eζ(1+O(ζ1)),|phz|pπδn even,
9.13.11 Bn(z) =π1/2zn/4eζ(1+O(ζ1)),
9.13.12 Bn(z) ={(2/π)sin(12pπ)zn/4(sin(ζ14π)+e|ζ|O(ζ1)),|phz|2pπδ,n odd,(1/π)sin(pπ)zn/4eζ(1+O(ζ1)),|phz|3pπδ,n even.

The distribution in and asymptotic properties of the zeros of An(z), An(z), Bn(z), and Bn(z) are investigated in Swanson and Headley (1967) and Headley and Barwell (1975).

In Olver (1977a, 1978) a different normalization is used. In place of (9.13.1) we have

9.13.13 d2wdt2=14m2tm2w,

where m=3,4,5,. For real variables the solutions of (9.13.13) are denoted by Um(t), Um(t) when m is even, and by Vm(t), V¯m(t) when m is odd. (The overbar has nothing to do with complex conjugates.) Their relations to the functions An(z) and Bn(z) are given by

9.13.14 m =n+2=1/p,
t =(12m)2/mz=ζ2/m,
9.13.15 2π(12m)(m1)/mcsc(π/m)An(z)={Um(t),m even,Vm(t),m odd,
9.13.16 π(12m)(m2)/(2m)csc(π/m)Bn(z)={Um(t),m even,V¯m(t),m odd.

Properties and graphs of Um(t), Vm(t), V¯m(t) are included in Olver (1977a) together with properties and graphs of real solutions of the equation

9.13.17 d2wdt2=14m2tm2w,
m even,

which are denoted by Wm(t), Wm(t).

In , the solutions of (9.13.13) used in Olver (1978) are

9.13.18 w=Um(te2jπi/m),

The function on the right-hand side is recessive in the sector (2j1)π/mphz(2j+1)π/m, and is therefore an essential member of any numerically satisfactory pair of solutions in this region.

Another normalization of (9.13.17) is used in Smirnov (1960), given by

9.13.19 d2wdx2+xαw=0,

where α>2 and x>0. Solutions are w=U1(x,α), U2(x,α), where

9.13.20 U1(x,α)=1(α+2)1/(α+2)Γ(α+1α+2)x1/2J1/(α+2)(2α+2x(α+2)/2),
9.13.21 U2(x,α)=(α+2)1/(α+2)Γ(α+3α+2)x1/2J1/(α+2)(2α+2x(α+2)/2),

and J denotes the Bessel function (§10.2(ii)).

When α is a positive integer the relation of these functions to Wm(t), Wm(t) is as follows:

9.13.22 α =m2,
x =(m/2)2/mt,
9.13.23 U1(x,α)=π1/22(m+2)/(2m)Γ(1/m)(Wm(t)+Wm(t)),
9.13.24 U2(x,α)=π1/2m2/m2(m+2)/(2m)Γ(1/m)(Wm(t)Wm(t)).

For properties of the zeros of the functions defined in this subsection see Laforgia and Muldoon (1988) and references given therein.

§9.13(ii) Generalizations from Integral Representations

Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow:

9.13.25 Ak(z,p)=12πiktpexp(zt13t3)dt,
k=1,2,3, p,
9.13.26 B0(z,p)=12πi0tpexp(zt13t3)dt,
9.13.27 Bk(z,p)=ktpexp(zt13t3)dt,
k=1,2,3, p=0,±1,±2,,

with z in all cases. The integration paths 0, 1, 2, 3 are depicted in Figure 9.13.1. 1, 2, 3 are depicted in Figure 9.13.2. When p is not an integer the branch of tp in (9.13.25) is usually chosen to be exp(p(ln|t|+ipht)) with 0pht<2π.

See accompanying text
Figure 9.13.1: t-plane. Paths 0, 1, 2, 3. Magnify
See accompanying text
Figure 9.13.2: t-plane. Paths 1, 2, 3. Magnify

When p=0

9.13.28 A1(z,0)=Ai(z),
9.13.29 A2(z,0) =e2πi/3Ai(ze2πi/3),
A3(z,0) =e2πi/3Ai(ze2πi/3),


9.13.30 B0(z,0) =0,
B1(z,0) =πHi(z).

Each of the functions Ak(z,p) and Bk(z,p) satisfies the differential equation

9.13.31 d3wdz3zdwdz+(p1)w=0,

and the difference equation

9.13.32 f(p3)zf(p1)+(p1)f(p)=0.

The Ak(z,p) are related by

9.13.33 A2(z,p) =e2(p1)πi/3A1(ze2πi/3,p),
A3(z,p) =e2(p1)πi/3A1(ze2πi/3,p).

Connection formulas for the solutions of (9.13.31) include

9.13.34 A1(z,p)+A2(z,p)+A3(z,p)+B0(z,p)=0,
9.13.35 B2(z,p)B3(z,p)=2πiA1(z,p),
9.13.36 B3(z,p)B1(z,p)=2πiA2(z,p),
9.13.37 B1(z,p)B2(z,p)=2πiA3(z,p).

Further properties of these functions, and also of similar contour integrals containing an additional factor (lnt)q, q=1,2,, in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). These properties include Wronskians, asymptotic expansions, and information on zeros.

For further generalizations via integral representations see Chin and Hedstrom (1978), Janson et al. (1993, §10), and Kamimoto (1998).