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33 Coulomb FunctionsVariables r,ϵ

§33.16 Connection Formulas

Contents

§33.16(i) F and G in Terms of f and h

§33.16(ii) f and h in Terms of F and G when ϵ>0

When ϵ>0 denote

33.16.3 τ=ϵ1/2(>0),

and again define A(ϵ,) by (33.14.11) or (33.14.12). Then for r>0

33.16.4 f(ϵ,;r)=(2πτ1--2π/τA(ϵ,))1/2F(-1/τ,τr),
33.16.5 h(ϵ,;r)=(2πτA(ϵ,)1--2π/τ)1/2G(-1/τ,τr).

Alternatively, for r<0

33.16.6 f(ϵ,;r) =(-1)+1(2πτ2π/τ-1A(ϵ,))1/2F(1/τ,-τr),
33.16.7 h(ϵ,;r) =(-1)(2πτA(ϵ,)2π/τ-1)1/2G(1/τ,-τr).

§33.16(iii) f and h in Terms of Wκ,μ(z) when ϵ<0

When ϵ<0 denote

33.16.8 ν=1/(-ϵ)1/2(>0),
33.16.9 ζ(ν,r) =Wν,+12(2r/ν),
ξ(ν,r) =(πνW-ν,+12(π2r/ν)),

and again define A(ϵ,) by (33.14.11) or (33.14.12). Then for r>0

33.16.10 f(ϵ,;r) =(-1)ν+1(-cos(πν)ζ(ν,r)Γ(+1+ν)+sin(πν)Γ(ν-)ξ(ν,r)π),
33.16.11 h(ϵ,;r) =(-1)ν+1A(ϵ,)(sin(πν)ζ(ν,r)Γ(+1+ν)+cos(πν)Γ(ν-)ξ(ν,r)π).

Alternatively, for r<0

33.16.12 f(ϵ,;r)=(-1)ν+1π(-πξ(-ν,r)Γ(+1+ν)+sin(πν)cos(πν)Γ(ν-)ζ(-ν,r)),
33.16.13 h(ϵ,;r)=(-1)ν+1A(ϵ,)Γ(ν-)ζ(-ν,r)/π.

§33.16(iv) s and c in Terms of F and G when ϵ>0

When ϵ>0, again denote τ by (33.16.3). Then for r>0

33.16.14 s(ϵ,;r) =(πτ)-1/2F(-1/τ,τr),
c(ϵ,;r) =(πτ)-1/2G(-1/τ,τr).

Alternatively, for r<0

33.16.15 s(ϵ,;r) =(πτ)-1/2F(1/τ,-τr),
c(ϵ,;r) =(πτ)-1/2G(1/τ,-τr).

§33.16(v) s and c in Terms of Wκ,μ(z) when ϵ<0

When ϵ<0 denote ν, ζ(ν,r), and ξ(ν,r) by (33.16.8) and (33.16.9). Also denote

33.16.16 K(ν,)=(ν2Γ(ν++1)Γ(ν-))-1/2.

Then for r>0

33.16.17 s(ϵ,;r) =(-1)2ν1/2(sin(πν)πK(ν,)ξ(ν,r)-cos(πν)ν2K(ν,)ζ(ν,r)),
c(ϵ,;r) =(-1)2ν1/2(cos(πν)πK(ν,)ξ(ν,r)+sin(πν)ν2K(ν,)ζ(ν,r)).

Alternatively, for r<0

33.16.18 s(ϵ,;r) =(-1)+121/2(ν3/2K(ν,)ξ(-ν,r)-sin(πν)cos(πν)πν1/2K(ν,)ζ(-ν,r)),
c(ϵ,;r) =(-1)π(2ν)1/2K(ν,)ζ(-ν,r).