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33 Coulomb FunctionsVariables r,ϵ

§33.19 Power-Series Expansions in r

33.19.1 f(ϵ,;r)=r+1k=0αkrk,

where

33.19.2 α0 =2+1/(2+1)!,
α1 =-α0/(+1),
k(k+2+1)αk+2αk-1+ϵαk-2 =0,
k=2,3,.
33.19.3 2πh(ϵ,;r)=k=02(2-k)!γkk!(2r)k--k=0δkrk++1-A(ϵ,)(2ln|2r/κ|+ψ(+1+κ)+ψ(-+κ))f(ϵ,;r),
r0.

Here κ is defined by (33.14.6), A(ϵ,) is defined by (33.14.11) or (33.14.12), γ0=1, γ1=1, and

33.19.4 γk-γk-1+14(k-1)(k-2-2)ϵγk-2=0,
k=2,3,.

Also,

33.19.5 δ0 =(β2+1-2(ψ(2+2)+ψ(1))A(ϵ,))α0,
δ1 =(β2+2-2(ψ(2+3)+ψ(2))A(ϵ,))α1,
33.19.6 k(k+2+1)δk+2δk-1+ϵδk-2+2(2k+2+1)A(ϵ,)αk=0,
k=2,3,,

with β0=β1=0, and

33.19.7 βk-βk-1+14(k-1)(k-2-2)ϵβk-2+12(k-1)ϵγk-2=0,
k=2,3,.

The expansions (33.19.1) and (33.19.3) converge for all finite values of r, except r=0 in the case of (33.19.3).