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§11.11 Asymptotic Expansions of Anger–Weber Functions

Contents

§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),
Gk(ν) =(ν2-22)(ν2-42)(ν2-(2k)2).

Then as z in |phz|π-δ (<π)

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.

§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,
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 (<π),

where

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

For fixed λ(>1),

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

For fixed λ, 0<λ<1,

11.11.11 A-ν(λν)2πν-νμk=0(12)kbk(λ)νk,
ν+,

where

11.11.12 μ=1-λ2-ln(1+1-λ2λ),

and

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 particular, as ν+,

11.11.14 A-ν(λν)1πν(λ-1),
λ>1,
11.11.15 A-ν(λν)(2πν)1/2(1+1-λ2λ)ν-ν1-λ2(1-λ2)1/4,
0<λ<1.

Also, as ν+,

11.11.16 A-ν(ν)24/337/6Γ(23)ν1/3,

and

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

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

All of (11.11.10)–(11.11.17) can be regarded as special cases of two asymptotic expansions given in Olver (1997b, pp. 352–357) for A-ν(λν) as ν+, one being uniform for δλ1, where δ again denotes an arbitrary small positive constant, and the other being uniform for 1λ<. (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) 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). In particular,

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