About the Project
33 Coulomb FunctionsVariables ρ,η

§33.10 Limiting Forms for Large ρ or Large |η|


§33.10(i) Large ρ

As ρ with η fixed,

33.10.1 F(η,ρ) =sin(θ(η,ρ))+o(1),
G(η,ρ) =cos(θ(η,ρ))+o(1),
33.10.2 H±(η,ρ)exp(±iθ(η,ρ)),

where θ(η,ρ) is defined by (33.2.9).

§33.10(ii) Large Positive η

As η with ηρ fixed,

33.10.3 F(η,ρ) (2+1)!C(η)(2η)+1(2ηρ)1/2I2+1((8ηρ)1/2),
G(η,ρ) 2(2η)(2+1)!C(η)(2ηρ)1/2K2+1((8ηρ)1/2).

In particular, for =0,

33.10.4 F0(η,ρ) e-πη(πρ)1/2I1((8ηρ)1/2),
G0(η,ρ) 2eπη(ρ/π)1/2K1((8ηρ)1/2),
33.10.5 F0(η,ρ) e-πη(2πη)1/2I0((8ηρ)1/2),
G0(η,ρ) -2eπη(2η/π)1/2K0((8ηρ)1/2).

§33.10(iii) Large Negative η

As η- with ηρ fixed,

33.10.7 F(η,ρ) =(2+1)!C(η)(-2η)+1((-2ηρ)1/2J2+1((-8ηρ)1/2)+o(|η|1/4)),
G(η,ρ) =-π(-2η)(2+1)!C(η)((-2ηρ)1/2Y2+1((-8ηρ)1/2)+o(|η|1/4)).

In particular, for =0,

33.10.8 F0(η,ρ) =(πρ)1/2J1((-8ηρ)1/2)+o(|η|-1/4),
G0(η,ρ) =-(πρ)1/2Y1((-8ηρ)1/2)+o(|η|-1/4).
33.10.9 F0(η,ρ) =(-2πη)1/2J0((-8ηρ)1/2)+o(|η|1/4),
G0(η,ρ) =-(-2πη)1/2Y0((-8ηρ)1/2)+o(|η|1/4).