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33 Coulomb FunctionsVariables ρ,η

§33.8 Continued Fractions

With arguments η,ρ suppressed,

33.8.1 FF=S+1-R+12T+1-R+22T+2-.

For R, S, and T see (33.4.1).

33.8.2 H±H±=c±iρab2(ρ-η±i)+(a+1)(b+1)2(ρ-η±2i)+,

where

33.8.3 a =1+±iη,
b =-±iη,
c =±i(1-(η/ρ)).

The continued fraction (33.8.1) converges for all finite values of ρ, and (33.8.2) converges for all ρ0.

If we denote u=F/F and p+iq=H+/H+, then

33.8.4 F =±(q-1(u-p)2+q)-1/2,
F =uF,
33.8.5 G =q-1(u-p)F,
G =q-1(up-p2-q2)F.

The ambiguous sign in (33.8.4) has to agree with that of the final denominator in (33.8.1) when the continued fraction has converged to the required precision. For proofs and further information see Barnett et al. (1974) and Barnett (1996).