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33 Coulomb FunctionsVariables r,ϵ

§33.21 Asymptotic Approximations for Large |r|


§33.21(i) Limiting Forms

We indicate here how to obtain the limiting forms of f(ϵ,;r), h(ϵ,;r), s(ϵ,;r), and c(ϵ,;r) as r±, with ϵ and fixed, in the following cases:

  1. (a)

    When r± with ϵ>0, Equations (33.16.4)–(33.16.7) are combined with (33.10.1).

  2. (b)

    When r± with ϵ<0, Equations (33.16.10)–(33.16.13) are combined with

    33.21.1 ζ(ν,r) e-r/ν(2r/ν)ν,
    ξ(ν,r) er/ν(2r/ν)-ν,
    33.21.2 ζ(-ν,r) er/ν(-2r/ν)-ν,
    ξ(-ν,r) e-r/ν(-2r/ν)ν,

    Corresponding approximations for s(ϵ,;r) and c(ϵ,;r) as r can be obtained via (33.16.17), and as r- via (33.16.18).

  3. (c)

    When r± with ϵ=0, combine (33.20.1), (33.20.2) with §§10.7(ii), 10.30(ii).

§33.21(ii) Asymptotic Expansions

For asymptotic expansions of f(ϵ,;r) and h(ϵ,;r) as r± with ϵ and fixed, see Curtis (1964a, §6).