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33 Coulomb FunctionsVariables ρ,η

§33.5 Limiting Forms for Small ρ, Small |η|, or Large


§33.5(i) Small ρ

As ρ0 with η fixed,

33.5.1 F(η,ρ) C(η)ρ+1,
F(η,ρ) (+1)C(η)ρ.
33.5.2 G(η,ρ) ρ-(2+1)C(η),
G(η,ρ) -ρ--1(2+1)C(η),

§33.5(ii) η=0

33.5.3 F(0,ρ) =ρj(ρ),
G(0,ρ) =-ρy(ρ).


33.5.4 F(0,ρ) =(πρ/2)1/2J+12(ρ),
G(0,ρ) =-(πρ/2)1/2Y+12(ρ).

For the functions j, y, J, Y see §§10.47(ii), 10.2(ii).

33.5.5 F0(0,ρ) =sinρ,
G0(0,ρ) =cosρ,
H0±(0,ρ) =e±iρ.
33.5.6 C(0)=2!(2+1)!=1(2+1)!!.

§33.5(iii) Small |η|

33.5.7 σ0(η)-γη,

where γ is Euler’s constant (§5.2(ii)).

§33.5(iv) Large

As with η and ρ (0) fixed,

33.5.8 F(η,ρ) C(η)ρ+1,
G(η,ρ) ρ-(2+1)C(η),
33.5.9 C(η)e-πη/2(2+1)!!e-πη/2e2(2)+1.