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33 Coulomb FunctionsVariables ρ,η

Β§33.12 Asymptotic Expansions for Large Ξ·

  1. Β§33.12(i) Transition Region
  2. Β§33.12(ii) Uniform Expansions

Β§33.12(i) Transition Region

When β„“=0 and Ξ·>0, the outer turning point is given by ρtp⁑(Ξ·,0)=2⁒η; compare (33.2.2). Define

33.12.1 x =(2β’Ξ·βˆ’Ο)/(2⁒η)1/3,
μ =(2⁒η)2/3.

Then as Ξ·β†’βˆž,

33.12.2 F0⁑(Ξ·,ρ)G0⁑(Ξ·,ρ)βˆΌΟ€1/2⁒(2⁒η)1/6⁒{Ai⁑(x)Bi⁑(x)⁒(1+B1ΞΌ+B2ΞΌ2+β‹―)+Ai′⁑(x)Bi′⁑(x)⁒(A1ΞΌ+A2ΞΌ2+β‹―)},
33.12.3 F0′⁑(Ξ·,ρ)G0′⁑(Ξ·,ρ)βˆΌβˆ’Ο€1/2⁒(2⁒η)βˆ’1/6⁒{Ai⁑(x)Bi⁑(x)⁒(B1β€²+x⁒A1ΞΌ+B2β€²+x⁒A2ΞΌ2+β‹―)+Ai′⁑(x)Bi′⁑(x)⁒(B1+A1β€²ΞΌ+B2+A2β€²ΞΌ2+β‹―)},

uniformly for bounded values of |(Οβˆ’2⁒η)/Ξ·1/3|. Here Ai and Bi are the Airy functions (Β§9.2), and

33.12.4 A1 =15⁒x2,
A2 =135⁒(2⁒x3+6),
A3 =115750⁒(21⁒x7+370⁒x4+580⁒x),
33.12.5 B1 =βˆ’15⁒x,
B2 =1350⁒(7⁒x5βˆ’30⁒x2),
B3 =115750⁒(264⁒x6βˆ’290⁒x3βˆ’560).

In particular,

33.12.6 F0⁑(Ξ·,2⁒η)3βˆ’1/2⁒G0⁑(Ξ·,2⁒η)βˆΌΞ“β‘(13)⁒ω1/22⁒π⁒(1βˆ“235⁒Γ⁑(23)Γ⁑(13)⁒1Ο‰4βˆ’82025⁒1Ο‰6βˆ“579246 06875⁒Γ⁑(23)Γ⁑(13)⁒1Ο‰10βˆ’β‹―),
33.12.7 F0′⁑(Ξ·,2⁒η)3βˆ’1/2⁒G0′⁑(Ξ·,2⁒η)βˆΌΞ“β‘(23)2⁒π⁒ω1/2⁒(Β±1+115⁒Γ⁑(13)Γ⁑(23)⁒1Ο‰2Β±214175⁒1Ο‰6+143623 38875⁒Γ⁑(13)Γ⁑(23)⁒1Ο‰8Β±β‹―),

where Ο‰=(23⁒η)1/3.

For derivations and additional terms in the expansions in this subsection see Abramowitz and Rabinowitz (1954) and FrΓΆberg (1955). For asymptotic expansions of Fℓ⁑(Ξ·,ρ) and Gℓ⁑(Ξ·,ρ) when Ξ·β†’Β±βˆž see Temme (2015, Chapter 31).

Β§33.12(ii) Uniform Expansions

With the substitution ρ=2⁒η⁒z, Equation (33.2.1) becomes

33.12.8 d2wdz2=(4⁒η2⁒(1βˆ’zz)+ℓ⁒(β„“+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 Fℓ⁑(Ξ·,ρ) and Gℓ⁑(Ξ·,ρ) when Ξ·β†’βˆž. See Temme (2015, Β§31.7).

The first set is in terms of Airy functions and the expansions are uniform for fixed β„“ and δ≀z<∞, where Ξ΄ is an arbitrary small positive constant. They would include the results of Β§33.12(i) as a special case.

The second set is in terms of Bessel functions of orders 2⁒ℓ+1 and 2⁒ℓ+2, and they are uniform for fixed β„“ and 0≀z≀1βˆ’Ξ΄, where Ξ΄ again denotes an arbitrary small positive constant.

Compare also Β§33.20(iv).