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9 Airy and Related FunctionsAiry Functions

§9.8 Modulus and Phase

  1. §9.8(i) Definitions
  2. §9.8(ii) Identities
  3. §9.8(iii) Monotonicity
  4. §9.8(iv) Asymptotic Expansions

§9.8(i) Definitions

Throughout this section x is real and nonpositive.

9.8.1 Ai(x) =M(x)sinθ(x),
9.8.2 Bi(x) =M(x)cosθ(x),
9.8.3 M(x) =Ai2(x)+Bi2(x),
9.8.4 θ(x) =arctan(Ai(x)/Bi(x)).
9.8.5 Ai(x) =N(x)sinϕ(x),
9.8.6 Bi(x) =N(x)cosϕ(x),
9.8.7 N(x) =Ai2(x)+Bi2(x),
9.8.8 ϕ(x) =arctan(Ai(x)/Bi(x)).

Graphs of M(x) and N(x) are included in §9.3(i). The branches of θ(x) and ϕ(x) are continuous and fixed by θ(0)=ϕ(0)=16π. (These definitions of θ(x) and ϕ(x) differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).)

In terms of Bessel functions, and with ξ=23|x|3/2,

9.8.9 |x|1/2M2(x) =12ξ(J1/32(ξ)+Y1/32(ξ)),
9.8.10 |x|1/2N2(x) =12ξ(J2/32(ξ)+Y2/32(ξ)),
9.8.11 θ(x) =23π+arctan(Y1/3(ξ)/J1/3(ξ)),
9.8.12 ϕ(x) =13π+arctan(Y2/3(ξ)/J2/3(ξ)).

§9.8(ii) Identities

Primes denote differentiations with respect to x, which is continued to be assumed real and nonpositive.

9.8.13 M(x)N(x)sin(θ(x)ϕ(x))=π1,
9.8.14 M2(x)θ(x) =π1,
N2(x)ϕ(x) =π1x,
N(x)N(x) =xM(x)M(x),
9.8.15 N2(x) =M2(x)+M2(x)θ2(x)=M2(x)+π2M2(x),
9.8.16 x2M2(x) =N2(x)+N2(x)ϕ2(x)=N2(x)+π2x2N2(x),
9.8.17 tan(θ(x)ϕ(x))=1/(πM(x)M(x))=M(x)θ(x)/M(x),
9.8.18 M′′(x)=xM(x)+π2M3(x),
9.8.19 θ2(x)+12(θ′′′(x)/θ(x))34(θ′′(x)/θ(x))2=x.

§9.8(iii) Monotonicity

As x increases from to 0 each of the functions M(x), M(x), |x|1/4N(x), M(x)N(x), θ(x), ϕ(x) is increasing, and each of the functions |x|1/4M(x), θ(x), ϕ(x) is decreasing.

§9.8(iv) Asymptotic Expansions

As x

9.8.20 M2(x) 1π(x)1/2k=0135(6k1)k!(96)k1x3k,
9.8.21 N2(x) (x)1/2πk=0135(6k1)k!(96)k1+6k16k1x3k,
9.8.22 θ(x) π4+23(x)3/2(1+5321x3+110561441x6+82825655361x9+12820 31525587 202561x12+),
9.8.23 ϕ(x) π4+23(x)3/2(17321x3146361441x64 952713 276801x92065 3042983 886081x12).

The remainder after n terms does not exceed the (n+1)th term in absolute value and is of the same sign, provided that n0 for (9.8.20), (9.8.22) and (9.8.23), and n1 for (9.8.21).

For higher terms in (9.8.22) and (9.8.23) see Fabijonas et al. (2004). Also, approximate values (25S) of the coefficients of the powers x15, x18, , x56 are available in Sherry (1959).