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33 Coulomb FunctionsVariables ρ,η

§33.6 Power-Series Expansions in ρ

33.6.1 F(η,ρ)=C(η)k=+1Ak(η)ρk,
33.6.2 F(η,ρ)=C(η)k=+1kAk(η)ρk-1,

where A+1=1, A+2=η/(+1), and

33.6.3 (k+)(k--1)Ak=2ηAk-1-Ak-2,

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

33.6.4 Ak(η)=(-i)k--1(k--1)!F12(+1-k,+1-iη;2+2;2).
33.6.5 H±(η,ρ)=e±iθ(η,ρ)(2+1)!Γ(-+iη)×(k=0(a)k(2+2)kk!(2iρ)a+k×(ln(2iρ)+ψ(a+k)-ψ(1+k)-ψ(2+2+k))-k=12+1(2+1)!(k-1)!(2+1-k)!(1-a)k(2iρ)a-k),

where a=1+±iη and ψ(x)=Γ(x)/Γ(x)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 H±(η,ρ) can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).