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§10.41 Asymptotic Expansions for Large Order

  1. §10.41(i) Asymptotic Forms
  2. §10.41(ii) Uniform Expansions for Real Variable
  3. §10.41(iii) Uniform Expansions for Complex Variable
  4. §10.41(iv) Double Asymptotic Properties
  5. §10.41(v) Double Asymptotic Properties (Continued)

§10.41(i) Asymptotic Forms

§10.41(ii) Uniform Expansions for Real Variable

As ν through positive real values,

10.41.3 Iν(νz) eνη(2πν)12(1+z2)14k=0Uk(p)νk,
10.41.4 Kν(νz) (π2ν)12eνη(1+z2)14k=0(1)kUk(p)νk,
10.41.5 Iν(νz) (1+z2)14eνη(2πν)12zk=0Vk(p)νk,
10.41.6 Kν(νz)(π2ν)12(1+z2)14eνηzk=0(1)kVk(p)νk,

uniformly for 0<z<. Here

10.41.7 η=(1+z2)12+lnz1+(1+z2)12,
10.41.8 p=(1+z2)12,

where the branches assume their principal values. Also, Uk(p) and Vk(p) are polynomials in p of degree 3k, given by U0(p)=V0(p)=1, and

10.41.9 Uk+1(p) =12p2(1p2)Uk(p)+180p(15t2)Uk(t)dt,
Vk+1(p) =Uk+1(p)12p(1p2)Uk(p)p2(1p2)Uk(p),

For k=1,2,3,

10.41.10 U1(p) =124(3p5p3),
U2(p) =11152(81p2462p4+385p6),
U3(p) =14 14720(30375p33 69603p5+7 65765p74 25425p9),
10.41.11 V1(p) =124(9p+7p3),
V2(p) =11152(135p2+594p4455p6),
V3(p) =14 14720(42525p3+4 51737p58 83575p7+4 75475p9).

For U4(p), U5(p), U6(p), see Bickley et al. (1952, p. xxxv).

For numerical tables of η=η(z) and the coefficients Uk(p), Vk(p), see Olver (1962, pp. 43–51).

§10.41(iii) Uniform Expansions for Complex Variable

The expansions (10.41.3)–(10.41.6) also hold uniformly in the sector |phz|12πδ (<12π), with the branches of the fractional powers in (10.41.3)–(10.41.8) extended by continuity from the positive real z-axis.

Figures 10.41.1 and 10.41.2 show corresponding points of the mapping of the z-plane and the η-plane. The curve E1BE2 in the z-plane is the upper boundary of the domain 𝐊 depicted in Figure 10.20.3 and rotated through an angle 12π. Thus B is the point z=c, where c is given by (10.20.18).

For derivations of the results in this subsection, and also error bounds, see Olver (1997b, pp. 374–378). For extensions of the regions of validity in the z-plane and extensions to complex values of ν see Olver (1997b, pp. 378–382).

See accompanying text
Figure 10.41.1: z-plane. Magnify
See accompanying text
Figure 10.41.2: η-plane. Magnify

For expansions in inverse factorial series see Dunster et al. (1993).

§10.41(iv) Double Asymptotic Properties

The series (10.41.3)–(10.41.6) can also be regarded as generalized asymptotic expansions for large |z|. Thus as z with (1) and ν (>0) both fixed,

10.41.12 Iν(νz)=eνη(2πν)12(1+z2)14(k=01Uk(p)νk+O(1z)),
10.41.13 Kν(νz)=(π2ν)12eνη(1+z2)14(k=01(1)kUk(p)νk+O(1z)),

Similarly for (10.41.5) and (10.41.6).

In the case of (10.41.13) with positive real values of z the result is a consequence of the error bounds given in Olver (1997b, pp. 377–378). Then by expanding the quantities η, (1+z2)14, and Uk(p), k=0,1,,1, and rearranging, we arrive at an expansion of the right-hand side of (10.41.13) in powers of z1. Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with z replaced by νz, up to and including the term in z(1). It also enjoys the same sector of validity.

To establish (10.41.12) we substitute into (10.34.3), with m=0 and z replaced by νz, by means of (10.41.13) observing that when |z| is large the effect of replacing z by ze±πi is to replace η, (1+z2)14, and p by η, ±i(1+z2)14, and p, respectively.

§10.41(v) Double Asymptotic Properties (Continued)

Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. We first prove that for the expansions (10.20.6) for the Hankel functions Hν(1)(νz) and Hν(2)(νz) the z-asymptotic property applies when z±i, respectively. This is a consequence of the error bounds associated with these expansions. We then extend the validity of this property from z±i to z in the sector π+δphz2πδ in the case of Hν(1)(νz), and to z in the sector 2π+δphzπδ in the case of Hν(2)(νz). This is done by re-expansion with the aid of (10.20.10), (10.20.11), and §10.41(ii), followed by comparison with (10.17.5) and (10.17.6), with z replaced by νz. Lastly, we substitute into (10.4.4), again with z replaced by νz. The final results are:

10.41.14 Jν(νz) =(4ζ1z2)14(Ai(ν23ζ)ν13(k=0Ak(ζ)ν2k+O(1ζ3+3))+Ai(ν23ζ)ν53(k=01Bk(ζ)ν2k+O(1ζ3+1))),
10.41.15 Yν(νz) =(4ζ1z2)14(Bi(ν23ζ)ν13(k=0Ak(ζ)ν2k+O(1ζ3+3))+Bi(ν23ζ)ν53(k=01Bk(ζ)ν2k+O(1ζ3+1))),

as z in |phz|πδ, or equivalently as ζ in |ph(ζ)|23πδ, for fixed (0) and fixed ν (>0).

It needs to be noted that the results (10.41.14) and (10.41.15) do not apply when z0+ or equivalently ζ+. This is because Ak(ζ) and ζ12Bk(ζ),k=0,1,, do not form an asymptotic scale (§2.1(v)) as ζ+; see Olver (1997b, pp. 422–425).