Digital Library of Mathematical Functions
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33 Coulomb FunctionsVariables ρ,η

§33.13 Complex Variable and Parameters

The functions F(η,ρ), G(η,ρ), and H±(η,ρ) may be extended to noninteger values of by generalizing (2+1)!=Γ(2+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, η, and . The quantities C(η), σ(η), and R, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that

33.13.1 C(η)=2σ(η)-(πη/2)Γ(+1-η)/Γ(2+2),

and

33.13.2 R=(2+1)C(η)/C-1(η).

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