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33 Coulomb FunctionsVariables ρ,η

Β§33.13 Complex Variable and Parameters

The functions Fℓ⁑(Ξ·,ρ), Gℓ⁑(Ξ·,ρ), and Hℓ±⁑(Ξ·,ρ) may be extended to noninteger values of β„“ by generalizing (2⁒ℓ+1)!=Γ⁑(2⁒ℓ+2), and supplementing (33.6.5) by a formula derived from (33.2.8) with U⁑(a,b,z) expanded via (13.2.42).

These functions may also be continued analytically to complex values of ρ, Ξ·, and β„“. The quantities Cℓ⁑(Ξ·), σℓ⁑(Ξ·), and Rβ„“, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that

33.13.1 Cℓ⁑(Ξ·)=2ℓ⁒ei⁒σℓ⁑(Ξ·)βˆ’(π⁒η/2)⁒Γ⁑(β„“+1βˆ’i⁒η)/Γ⁑(2⁒ℓ+2),

and

33.13.2 Rβ„“=(2⁒ℓ+1)⁒Cℓ⁑(Ξ·)/Cβ„“βˆ’1⁑(Ξ·).

For further information see Dzieciol et al. (1999), Thompson and Barnett (1986), and Humblet (1984).