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33 Coulomb FunctionsVariables r,Ο΅

Β§33.16 Connection Formulas

Contents
  1. Β§33.16(i) Fβ„“ and Gβ„“ in Terms of f and h
  2. Β§33.16(ii) f and h in Terms of Fβ„“ and Gβ„“ when Ο΅>0
  3. §33.16(iii) f and h in Terms of Wκ,μ⁑(z) when ϡ<0
  4. Β§33.16(iv) s and c in Terms of Fβ„“ and Gβ„“ when Ο΅>0
  5. §33.16(v) s and c in Terms of Wκ,μ⁑(z) when ϡ<0

Β§33.16(i) Fβ„“ and Gβ„“ in Terms of f and h

33.16.1 Fℓ⁑(Ξ·,ρ)=(2⁒ℓ+1)!⁒Cℓ⁑(Ξ·)(βˆ’2⁒η)β„“+1⁒f⁑(1/Ξ·2,β„“;βˆ’Ξ·β’Ο),
33.16.2 Gℓ⁑(Ξ·,ρ)=π⁒(βˆ’2⁒η)β„“(2⁒ℓ+1)!⁒Cℓ⁑(Ξ·)⁒h⁑(1/Ξ·2,β„“;βˆ’Ξ·β’Ο),

where Cℓ⁑(Ξ·) is given by (33.2.5) or (33.2.6).

Β§33.16(ii) f and h in Terms of Fβ„“ and Gβ„“ when Ο΅>0

When Ο΅>0 denote

33.16.3 Ο„=Ο΅1/2(>0),

and again define A⁑(Ο΅,β„“) by (33.14.11) or (33.14.12). Then for r>0

33.16.4 f⁑(Ο΅,β„“;r)=(2π⁒τ⁒1βˆ’eβˆ’2⁒π/Ο„A⁑(Ο΅,β„“))1/2⁒Fℓ⁑(βˆ’1/Ο„,τ⁒r),
33.16.5 h⁑(Ο΅,β„“;r)=(2π⁒τ⁒A⁑(Ο΅,β„“)1βˆ’eβˆ’2⁒π/Ο„)1/2⁒Gℓ⁑(βˆ’1/Ο„,τ⁒r).

Alternatively, for r<0

33.16.6 f⁑(Ο΅,β„“;r) =(βˆ’1)β„“+1⁒(2π⁒τ⁒e2⁒π/Ο„βˆ’1A⁑(Ο΅,β„“))1/2⁒Fℓ⁑(1/Ο„,βˆ’Ο„β’r),
33.16.7 h⁑(Ο΅,β„“;r) =(βˆ’1)ℓ⁒(2π⁒τ⁒A⁑(Ο΅,β„“)e2⁒π/Ο„βˆ’1)1/2⁒Gℓ⁑(1/Ο„,βˆ’Ο„β’r).

§33.16(iii) f and h in Terms of Wκ,μ⁑(z) when ϡ<0

When Ο΅<0 denote

33.16.8 Ξ½=1/(βˆ’Ο΅)1/2(>0),
33.16.9 ΢ℓ⁑(Ξ½,r) =WΞ½,β„“+12⁑(2⁒r/Ξ½),
ξℓ⁑(Ξ½,r) =β„œβ‘(ei⁒π⁒ν⁒Wβˆ’Ξ½,β„“+12⁑(ei⁒π⁒2⁒r/Ξ½)),

and again define A⁑(Ο΅,β„“) by (33.14.11) or (33.14.12). Then for r>0

33.16.10 f⁑(Ο΅,β„“;r) =(βˆ’1)ℓ⁒νℓ+1⁒(βˆ’cos⁑(π⁒ν)⁒΢ℓ⁑(Ξ½,r)Γ⁑(β„“+1+Ξ½)+sin⁑(π⁒ν)⁒Γ⁑(Ξ½βˆ’β„“)⁒ξℓ⁑(Ξ½,r)Ο€),
33.16.11 h⁑(Ο΅,β„“;r) =(βˆ’1)ℓ⁒νℓ+1⁒A⁑(Ο΅,β„“)⁒(sin⁑(π⁒ν)⁒΢ℓ⁑(Ξ½,r)Γ⁑(β„“+1+Ξ½)+cos⁑(π⁒ν)⁒Γ⁑(Ξ½βˆ’β„“)⁒ξℓ⁑(Ξ½,r)Ο€).

Alternatively, for r<0

33.16.12 f⁑(Ο΅,β„“;r)=(βˆ’1)ℓ⁒νℓ+1π⁒(βˆ’Ο€β’ΞΎβ„“β‘(βˆ’Ξ½,r)Γ⁑(β„“+1+Ξ½)+sin⁑(π⁒ν)⁒cos⁑(π⁒ν)⁒Γ⁑(Ξ½βˆ’β„“)⁒΢ℓ⁑(βˆ’Ξ½,r)),
33.16.13 h⁑(Ο΅,β„“;r)=(βˆ’1)ℓ⁒νℓ+1⁒A⁑(Ο΅,β„“)⁒Γ⁑(Ξ½βˆ’β„“)⁒΢ℓ⁑(βˆ’Ξ½,r)/Ο€.

Β§33.16(iv) s and c in Terms of Fβ„“ and Gβ„“ when Ο΅>0

When Ο΅>0, again denote Ο„ by (33.16.3). Then for r>0

33.16.14 s⁑(Ο΅,β„“;r) =(π⁒τ)βˆ’1/2⁒Fℓ⁑(βˆ’1/Ο„,τ⁒r),
c⁑(Ο΅,β„“;r) =(π⁒τ)βˆ’1/2⁒Gℓ⁑(βˆ’1/Ο„,τ⁒r).

Alternatively, for r<0

33.16.15 s⁑(Ο΅,β„“;r) =(π⁒τ)βˆ’1/2⁒Fℓ⁑(1/Ο„,βˆ’Ο„β’r),
c⁑(Ο΅,β„“;r) =(π⁒τ)βˆ’1/2⁒Gℓ⁑(1/Ο„,βˆ’Ο„β’r).

§33.16(v) s and c in Terms of Wκ,μ⁑(z) when ϡ<0

When Ο΅<0 denote Ξ½, ΢ℓ⁑(Ξ½,r), and ξℓ⁑(Ξ½,r) by (33.16.8) and (33.16.9). Also denote

33.16.16 K⁑(Ξ½,β„“)=(Ξ½2⁒Γ⁑(Ξ½+β„“+1)⁒Γ⁑(Ξ½βˆ’β„“))βˆ’1/2.

Then for r>0

33.16.17 s⁑(Ο΅,β„“;r) =(βˆ’1)β„“2⁒ν1/2⁒(sin⁑(π⁒ν)π⁒K⁑(Ξ½,β„“)⁒ξℓ⁑(Ξ½,r)βˆ’cos⁑(π⁒ν)⁒ν2⁒K⁑(Ξ½,β„“)⁒΢ℓ⁑(Ξ½,r)),
c⁑(Ο΅,β„“;r) =(βˆ’1)β„“2⁒ν1/2⁒(cos⁑(π⁒ν)π⁒K⁑(Ξ½,β„“)⁒ξℓ⁑(Ξ½,r)+sin⁑(π⁒ν)⁒ν2⁒K⁑(Ξ½,β„“)⁒΢ℓ⁑(Ξ½,r)).

Alternatively, for r<0

33.16.18 s⁑(Ο΅,β„“;r) =(βˆ’1)β„“+121/2⁒(Ξ½3/2K⁑(Ξ½,β„“)⁒ξℓ⁑(βˆ’Ξ½,r)βˆ’sin⁑(π⁒ν)⁒cos⁑(π⁒ν)π⁒ν1/2⁒K⁑(Ξ½,β„“)⁒΢ℓ⁑(βˆ’Ξ½,r)),
c⁑(Ο΅,β„“;r) =(βˆ’1)ℓπ⁒(2⁒ν)1/2⁒K⁑(Ξ½,β„“)⁒΢ℓ⁑(βˆ’Ξ½,r).