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33 Coulomb FunctionsVariables ρ,η

§33.12 Asymptotic Expansions for Large η


§33.12(i) Transition Region

When =0 and η>0, the outer turning point is given by ρtp(η,0)=2η; compare (33.2.2). Define

33.12.1 x =(2η-ρ)/(2η)1/3,
μ =(2η)2/3.

Then as η,

33.12.2 F0(η,ρ)G0(η,ρ)π1/2(2η)1/6{Ai(x)Bi(x)(1+B1μ+B2μ2+)+Ai(x)Bi(x)(A1μ+A2μ2+)},
33.12.3 F0(η,ρ)G0(η,ρ)-π1/2(2η)-1/6{Ai(x)Bi(x)(B1+xA1μ+B2+xA2μ2+)+Ai(x)Bi(x)(B1+A1μ+B2+A2μ2+)},

uniformly for bounded values of |(ρ-2η)/η1/3|. Here Ai and Bi are the Airy functions (§9.2), and

33.12.4 A1 =15x2,
A2 =135(2x3+6),
A3 =115750(21x7+370x4+580x),
33.12.5 B1 =-15x,
B2 =1350(7x5-30x2),
B3 =115750(264x6-290x3-560).

In particular,

33.12.6 F0(η,2η)3-1/2G0(η,2η)Γ(13)ω1/22π(1235Γ(23)Γ(13)1ω4-820251ω6579246 06875Γ(23)Γ(13)1ω10-),
33.12.7 F0(η,2η)3-1/2G0(η,2η)Γ(23)2πω1/2(±1+115Γ(13)Γ(23)1ω2±2141751ω6+143623 38875Γ(13)Γ(23)1ω8±),

where ω=(23η)1/3.

For derivations and additional terms in the expansions in this subsection see Abramowitz and Rabinowitz (1954) and Fröberg (1955). For asymptotic expansions of F(η,ρ) and G(η,ρ) when η± see Temme (2015, Chapter 31).

§33.12(ii) Uniform Expansions

With the substitution ρ=2ηz, Equation (33.2.1) becomes

33.12.8 d2wdz2=(4η2(1-zz)+(+1)z2)w.

Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for F(η,ρ) and G(η,ρ) when η. See Temme (2015, §31.7).

The first set is in terms of Airy functions and the expansions are uniform for fixed and δz<, where δ is an arbitrary small positive constant. They would include the results of §33.12(i) as a special case.

The second set is in terms of Bessel functions of orders 2+1 and 2+2, and they are uniform for fixed and 0z1-δ, where δ again denotes an arbitrary small positive constant.

Compare also §33.20(iv).