Digital Library of Mathematical Functions
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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)=±θ(η,ρ)k=0(a)k(b)kk!(2ρ)k,

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

With arguments (η,ρ) suppressed, an equivalent formulation is given by

33.11.2 F =gcosθ+fsinθ,
G =fcosθ-gsinθ,
33.11.3 F =g^cosθ+f^sinθ,
G =f^cosθ-g^sinθ,
33.11.4 H±=±θ(f±g),

where

33.11.5 f k=0fk,
g k=0gk,
33.11.6 f^ k=0f^k,
g^ k=0g^k,
33.11.7 gf^-fg^=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),

where

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