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§10.20 Uniform Asymptotic Expansions for Large Order

  1. §10.20(i) Real Variables
  2. §10.20(ii) Complex Variables
  3. §10.20(iii) Double Asymptotic Properties

§10.20(i) Real Variables

Define ζ=ζ(z) to be the solution of the differential equation

10.20.1 (dζdz)2=1z2ζz2

that is infinitely differentiable on the interval 0<z<, including z=1. Then

10.20.2 23ζ32=z11t2tdt=ln(1+1z2z)1z2,
10.20.3 23(ζ)32=1zt21tdt=z21arcsecz,

all functions taking their principal values, with ζ=,0,, corresponding to z=0,1,, respectively.

As ν through positive real values

10.20.4 Jν(νz) (4ζ1z2)14(Ai(ν23ζ)ν13k=0Ak(ζ)ν2k+Ai(ν23ζ)ν53k=0Bk(ζ)ν2k),
10.20.5 Yν(νz) (4ζ1z2)14(Bi(ν23ζ)ν13k=0Ak(ζ)ν2k+Bi(ν23ζ)ν53k=0Bk(ζ)ν2k),
10.20.6 Hν(1)(νz)Hν(2)(νz)}2eπi/3(4ζ1z2)14(Ai(e±2πi/3ν23ζ)ν13k=0Ak(ζ)ν2k+e±2πi/3Ai(e±2πi/3ν23ζ)ν53k=0Bk(ζ)ν2k),
10.20.7 Jν(νz) 2z(1z24ζ)14(Ai(ν23ζ)ν43k=0Ck(ζ)ν2k+Ai(ν23ζ)ν23k=0Dk(ζ)ν2k),
10.20.8 Yν(νz) 2z(1z24ζ)14(Bi(ν23ζ)ν43k=0Ck(ζ)ν2k+Bi(ν23ζ)ν23k=0Dk(ζ)ν2k),
10.20.9 Hν(1)(νz)Hν(2)(νz)}4e2πi/3z(1z24ζ)14(e2πi/3Ai(e±2πi/3ν23ζ)ν43k=0Ck(ζ)ν2k+Ai(e±2πi/3ν23ζ)ν23k=0Dk(ζ)ν2k),

uniformly for z (0,) in all cases, where Ai and Bi are the Airy functions (§9.2).

In the following formulas for the coefficients Ak(ζ), Bk(ζ), Ck(ζ), and Dk(ζ), uk, vk are the constants defined in §9.7(i), and Uk(p), Vk(p) are the polynomials in p of degree 3k defined in §10.41(ii).

Interval 0<z<1

10.20.10 Ak(ζ)=j=02k(32)jvjζ3j/2U2kj((1z2)12),
10.20.11 Bk(ζ)=ζ12j=02k+1(32)jujζ3j/2U2kj+1((1z2)12),
10.20.12 Ck(ζ)=ζ12j=02k+1(32)jvjζ3j/2V2kj+1((1z2)12),
10.20.13 Dk(ζ)=j=02k(32)jujζ3j/2V2kj((1z2)12).

Interval 1<z<

In formulas (10.20.10)–(10.20.13) replace ζ12, ζ12, ζ3j/2, and (1z2)12 by i(ζ)12, i(ζ)12, i3j(ζ)3j/2, and i(z21)12, respectively.

Note: Another way of arranging the above formulas for the coefficients Ak(ζ),Bk(ζ),Ck(ζ), and Dk(ζ) would be by analogy with (12.10.42) and (12.10.46). In this way there is less usage of many-valued functions.

Values at ζ=0

10.20.14 A0(0) =1,
A1(0) =1225,
A2(0) =1 514392182 95000,
A3(0) =8872 78009250 49351 25000,
B0(0) =170213,
B1(0) =121310 23750213,
B2(0) =1 65425 378333774 32055 00000213,
B3(0) =959 71711 8460325 47666 37125 00000213.

Each of the coefficients Ak(ζ), Bk(ζ), Ck(ζ), and Dk(ζ), k=0,1,2,, is real and infinitely differentiable on the interval <ζ<. For (10.20.14) and further information on the coefficients see Temme (1997).

For numerical tables of ζ=ζ(z), (4ζ/(1z2))14 and Ak(ζ), Bk(ζ), Ck(ζ), and Dk(ζ) see Olver (1962, pp. 28–42).

§10.20(ii) Complex Variables

The function ζ=ζ(z) given by (10.20.2) and (10.20.3) can be continued analytically to the z-plane cut along the negative real axis. Corresponding points of the mapping are shown in Figures 10.20.1 and 10.20.2.

The equations of the curved boundaries D1E1 and D2E2 in the ζ-plane are given parametrically by

10.20.15 ζ=(32)23(τiπ)23,


The curves BP1E1 and BP2E2 in the z-plane are the inverse maps of the line segments

10.20.16 ζ=eiπ/3τ,

respectively. They are given parametrically by

10.20.17 z=±(τcothττ2)12±i(τ2τtanhτ)12,

where τ0=1.19968 is the positive root of the equation τ=cothτ. The points P1,P2 where these curves intersect the imaginary axis are ±ic, where

10.20.18 c=(τ021)12=0.66274.

The eye-shaped closed domain in the uncut z-plane that is bounded by BP1E1 and BP2E2 is denoted by 𝐊; see Figure 10.20.3.

See accompanying text
Figure 10.20.1: z-plane. P1 and P2 are the points ±ic. c=0.66274. Magnify
See accompanying text
Figure 10.20.2: ζ-plane. E1 and E2 are the points eπi/3(3π/2)2/3. Magnify
See accompanying text
Figure 10.20.3: z-plane. Domain 𝐊 (unshaded). c=0.66274. Magnify

As ν through positive real values the expansions (10.20.4)–(10.20.9) apply uniformly for |phz|πδ, the coefficients Ak(ζ), Bk(ζ), Ck(ζ), and Dk(ζ), being the analytic continuations of the functions defined in §10.20(i) when ζ is real.

For proofs of the above results and for error bounds and extensions of the regions of validity see Olver (1997b, pp. 419–425). For extensions to complex ν see Olver (1954). For resurgence properties of the coefficients (§2.7(ii)) see Howls and Olde Daalhuis (1999). For further results see Dunster (2001a), Wang and Wong (2002), and Paris (2004).

§10.20(iii) Double Asymptotic Properties

For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of z see §10.41(v).