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§11.11 Asymptotic Expansions of Anger–Weber Functions

  1. §11.11(i) Large |z|, Fixed ν
  2. §11.11(ii) Large |ν|, Fixed z
  3. §11.11(iii) Large ν, Fixed z/ν

§11.11(i) Large |z|, Fixed ν

Let F0(ν)=G0(ν)=1, and for k=1,2,3,,

11.11.1 Fk(ν) =(ν212)(ν232)(ν2(2k1)2)=(4)k(12ν2)k(12+ν2)k,
Gk(ν) =(ν222)(ν242)(ν2(2k)2)=(4)k(1ν2)k(1+ν2)k.

Then as z in |phz|πδ

11.11.2 𝐉ν(z)Jν(z)+sin(πν)πz(k=0Fk(ν)z2kνzk=0Gk(ν)z2k),
11.11.3 𝐄ν(z)Yν(z)1+cos(πν)πzk=0Fk(ν)z2kν(1cos(πν))πz2k=0Gk(ν)z2k,
11.11.4 𝐀ν(z)1πzk=0Fk(ν)z2kνπz2k=0Gk(ν)z2k.

For sharp error bounds and exponentially-improved extensions, see Nemes (2018).

§11.11(ii) Large |ν|, Fixed z

If z is fixed, and ν in |phν|π in such a way that ν is bounded away from the set of all integers, then

11.11.5 𝐉ν(z)=sin(πν)πν(1νzν21+O(1ν2)),
11.11.6 𝐄ν(z)=2πν(sin2(12πν)+νzν21cos2(12πν)+O(1ν2)).

If ν=n(), then (11.10.29) applies for 𝐉n(z), and

11.11.7 𝐄2n(z) 2z(4n21)π,
𝐄2n+1(z) 2(2n+1)π,

as n±.

§11.11(iii) Large ν, Fixed z/ν

For fixed λ (>0),

11.11.8 𝐀ν(λν)1πk=0(2k)!ak(λ)ν2k+1,
ν, |phν|πδ,


11.11.9 a0(λ) =11+λ,
a1(λ) =λ2(1+λ)4,
a2(λ) =9λ2λ24(1+λ)7,
a3(λ) =225λ354λ2+λ720(1+λ)10.

In general,

11.11.9_5 ak+1(λ)=λ1λ2λak′′(λ)+ak(λ)(2k+1)(2k+2),

For fixed λ(>1),

11.11.10 𝐀ν(λν)1πk=0(2k)!ak(λ)ν2k+1,
ν, |phν|πδ.

For fixed λ, 0<λ<1,

11.11.11 𝐀ν(λν)(2πν)1/2eνμk=0(12)kbk(λ)νk,
ν, |phν|π2δ,


11.11.12 μ=1λ2ln(1+1λ2λ),


11.11.13 b0(λ) =1(1λ2)1/4,
b1(λ) =2+3λ212(1λ2)7/4,
b2(λ) =4+300λ2+81λ4864(1λ2)13/4.

In general,

11.11.13_5 (12)kbk(λ)=(1)k(1λ2)1/4Uk(11λ2),

with the Uk defined in §10.41(ii).

In particular, as ν,

11.11.14 𝐀ν(λν)1πν(λ1),
λ>1, |phν|πδ,
11.11.15 𝐀ν(λν)(2πν)1/2(1+1λ2λ)νeν1λ2(1λ2)1/4,
0<λ<1, |phν|π2δ.

Also, as ν in |phν|2πδ,

11.11.16 𝐀ν(ν)24/337/6Γ(23)ν1/3,


11.11.17 𝐀ν(ν+aν1/3)=21/3ν1/3Hi(21/3a)+O(ν1),

uniformly for bounded complex values of a. For the Scorer function Hi see §9.12(i).

Error bounds for (11.11.8) and (11.11.10) are given in Meijer (1932) and Nemes (2014b, c). The later references also contain exponentially-improved extensions of (11.11.8) and (11.11.10). For an extension of (11.11.17) (and (11.11.16)) into a complete asymptotic expansion, see Nemes (2020).

When ν is real and positive, all of (11.11.10)–(11.11.17) can be regarded as special cases of two asymptotic expansions given in Olver (1997b, pp. 352–360) for 𝐀ν(λν) as ν+, one being uniform for 0<λ1, and the other being uniform for λ1. (Note that Olver’s definition of 𝐀ν(z) omits the factor 1/π in (11.10.4).) See also Watson (1944, §10.15).

Lastly, corresponding asymptotic approximations and expansions for 𝐉ν(λν) and 𝐄ν(λν), with 0<λ<1 or λ>1, follow from (11.10.15) and (11.10.16) and the corresponding asymptotic expansions for the Bessel functions Jν(z) and Yν(z); see §10.19(ii). Furthermore,

11.11.18 𝐉ν(ν)21/332/3Γ(23)ν1/3,
ν, |phν|πδ,
11.11.19 𝐄ν(ν)21/337/6Γ(23)ν1/3,
ν, |phν|πδ.