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33 Coulomb FunctionsVariables ρ,η

§33.11 Asymptotic Expansions for Large ρ

For large ρ, with and η fixed,

33.11.1 H±(η,ρ)e±iθ(η,ρ)k=0(a)k(b)kk!(±2iρ)k,

where θ(η,ρ) is defined by (33.2.9), and a and b are defined by (33.8.3).

An equivalent formulation is given by

33.11.2 F(η,ρ) =g(η,ρ)cosθ+f(η,ρ)sinθ,
G(η,ρ) =f(η,ρ)cosθ-g(η,ρ)sinθ,
33.11.3 F(η,ρ) =g^(η,ρ)cosθ+f^(η,ρ)sinθ,
G(η,ρ) =f^(η,ρ)cosθ-g^(η,ρ)sinθ,
33.11.4 H±(η,ρ)=e±iθ(f(η,ρ)±ig(η,ρ)),


33.11.5 f(η,ρ) k=0fk,
g(η,ρ) k=0gk,
33.11.6 f^(η,ρ) k=0f^k,
g^(η,ρ) k=0g^k,
33.11.7 g(η,ρ)f^(η,ρ)-f(η,ρ)g^(η,ρ)=1.

Here f0=1, g0=0, f^0=0, g^0=1-(η/ρ), and for k=0,1,2,,

33.11.8 fk+1 =λkfk-μkgk,
gk+1 =λkgk+μkfk,
f^k+1 =λkf^k-μkg^k-(fk+1/ρ),
g^k+1 =λkg^k+μkf^k-(gk+1/ρ),


33.11.9 λk =(2k+1)η(2k+2)ρ,
μk =(+1)-k(k+1)+η2(2k+2)ρ.