About the Project
33 Coulomb FunctionsVariables ρ,η

§33.6 Power-Series Expansions in ρ

33.6.1 Fℓ⁑(Ξ·,ρ)=Cℓ⁑(Ξ·)β’βˆ‘k=β„“+1∞Akℓ⁒(Ξ·)⁒ρk,
33.6.2 Fℓ′⁑(Ξ·,ρ)=Cℓ⁑(Ξ·)β’βˆ‘k=β„“+1∞k⁒Akℓ⁒(Ξ·)⁒ρkβˆ’1,

where Aβ„“+1β„“=1, Aβ„“+2β„“=Ξ·/(β„“+1), and

33.6.3 (k+β„“)⁒(kβˆ’β„“βˆ’1)⁒Akβ„“=2⁒η⁒Akβˆ’1β„“βˆ’Akβˆ’2β„“,
k=β„“+3,β„“+4,…,

or in terms of the hypergeometric function (§§15.1, 15.2(i)),

33.6.4 Akℓ⁒(Ξ·)=(βˆ’i)kβˆ’β„“βˆ’1(kβˆ’β„“βˆ’1)!⁒F12⁑(β„“+1βˆ’k,β„“+1βˆ’i⁒η;2⁒ℓ+2;2).
33.6.5 Hℓ±⁑(Ξ·,ρ)=eΒ±i⁒θℓ⁑(Ξ·,ρ)(2⁒ℓ+1)!⁒Γ⁑(βˆ’β„“Β±i⁒η)Γ—(βˆ‘k=0∞(a)k(2⁒ℓ+2)k⁒k!⁒(βˆ“2⁒i⁒ρ)a+kΓ—(ln⁑(βˆ“2⁒i⁒ρ)+ψ⁑(a+k)βˆ’Οˆβ‘(1+k)βˆ’Οˆβ‘(2⁒ℓ+2+k))βˆ’βˆ‘k=12⁒ℓ+1(2⁒ℓ+1)!⁒(kβˆ’1)!(2⁒ℓ+1βˆ’k)!⁒(1βˆ’a)k⁒(βˆ“2⁒i⁒ρ)aβˆ’k),

where a=1+β„“Β±i⁒η and ψ⁑(x)=Γ′⁑(x)/Γ⁑(x) (Β§5.2(i)).

The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of ρ. Corresponding expansions for Hℓ±′⁑(Ξ·,ρ) can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).