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§11.11 Asymptotic Expansions of Anger–Weber Functions


§11.11(i) Large |z|, Fixed ν

Let F0(ν)=G0(ν)=1, and for k=1,2,3,,

11.11.1 Fk(ν) =(ν2-12)(ν2-32)(ν2-(2k-1)2)=(-4)k(12-ν2)k(12+ν2)k,
Gk(ν) =(ν2-22)(ν2-42)(ν2-(2k)2)=(-4)k(1-ν2)k(1+ν2)k.

Then as z in |phz|π-δ

11.11.2 Jν(z)Jν(z)+sin(πν)πz(k=0Fk(ν)z2k-νzk=0Gk(ν)z2k),
11.11.3 Eν(z)-Yν(z)-1+cos(πν)πzk=0Fk(ν)z2k-ν(1-cos(πν))πz2k=0Gk(ν)z2k,
11.11.4 Aν(z)1πzk=0Fk(ν)z2k-νπz2k=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(1-νzν2-1+O(1ν2)),
11.11.6 Eν(z)=2πν(sin2(12πν)+νzν2-1cos2(12πν)+O(1ν2)).

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

11.11.7 E2n(z) 2z(4n2-1)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|π-δ,


11.11.9 a0(λ) =11+λ,
a1(λ) =-λ2(1+λ)4,
a2(λ) =9λ2-λ24(1+λ)7,
a3(λ) =-225λ3-54λ2+λ720(1+λ)10.

In general,

11.11.9_5 ak+1(λ)=λ1-λ2λak′′(λ)+ak(λ)(2k+1)(2k+2),

For fixed λ(>1),

11.11.10 A-ν(λν)-1πk=0(2k)!ak(-λ)ν2k+1,
ν, |phν|π-δ.

For fixed λ, 0<λ<1,

11.11.11 A-ν(λν)(2πν)1/2e-νμk=0(12)kbk(λ)νk,
ν, |phν|π2-δ,


11.11.12 μ=1-λ2-ln(1+1-λ2λ),


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

In general,

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

with the Uk defined in §10.41(ii).

In particular, as ν,

11.11.14 A-ν(λν)1πν(λ-1),
λ>1, |phν|π-δ,
11.11.15 A-ν(λν)(2πν)1/2(1+1-λ2λ)νe-ν1-λ2(1-λ2)1/4,
0<λ<1, |phν|π2-δ.

Also, as ν in |phν|2π-δ,

11.11.16 A-ν(ν)24/337/6Γ(23)ν1/3,


11.11.17 A-ν(ν+aν1/3)=21/3ν-1/3Hi(-21/3a)+O(ν-1),

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 A-ν(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|π-δ,
11.11.19 Eν(ν)21/337/6Γ(23)ν1/3,
ν, |phν|π-δ.