About the Project
33 Coulomb FunctionsVariables r,Ο΅

Β§33.21 Asymptotic Approximations for Large |r|

  1. Β§33.21(i) Limiting Forms
  2. Β§33.21(ii) Asymptotic Expansions

Β§33.21(i) Limiting Forms

We indicate here how to obtain the limiting forms of f⁑(Ο΅,β„“;r), h⁑(Ο΅,β„“;r), s⁑(Ο΅,β„“;r), and c⁑(Ο΅,β„“;r) as rβ†’Β±βˆž, with Ο΅ and β„“ fixed, in the following cases:

  1. (a)

    When rβ†’Β±βˆž with Ο΅>0, Equations (33.16.4)–(33.16.7) are combined with (33.10.1).

  2. (b)

    When rβ†’Β±βˆž with Ο΅<0, Equations (33.16.10)–(33.16.13) are combined with

    33.21.1 ΢ℓ⁑(Ξ½,r) ∼eβˆ’r/ν⁒(2⁒r/Ξ½)Ξ½,
    ξℓ⁑(Ξ½,r) ∼er/ν⁒(2⁒r/Ξ½)βˆ’Ξ½,
    33.21.2 ΢ℓ⁑(βˆ’Ξ½,r) ∼er/ν⁒(βˆ’2⁒r/Ξ½)βˆ’Ξ½,
    ξℓ⁑(βˆ’Ξ½,r) ∼eβˆ’r/ν⁒(βˆ’2⁒r/Ξ½)Ξ½,

    Corresponding approximations for s⁑(Ο΅,β„“;r) and c⁑(Ο΅,β„“;r) as rβ†’βˆž can be obtained via (33.16.17), and as rβ†’βˆ’βˆž via (33.16.18).

  3. (c)

    When rβ†’Β±βˆž with Ο΅=0, combine (33.20.1), (33.20.2) with §§10.7(ii), 10.30(ii).

Β§33.21(ii) Asymptotic Expansions

For asymptotic expansions of f⁑(Ο΅,β„“;r) and h⁑(Ο΅,β„“;r) as rβ†’Β±βˆž with Ο΅ and β„“ fixed, see Curtis (1964a, Β§6).