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Coulomb functions: variables r,ϵ

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1: 33.20 Expansions for Small | ϵ |
§33.20(i) Case ϵ = 0
§33.20(ii) Power-Series in ϵ for the Regular Solution
§33.20(iii) Asymptotic Expansion for the Irregular Solution
where A ( ϵ , ) is given by (33.14.11), (33.14.12), and …
§33.20(iv) Uniform Asymptotic Expansions
2: 33.17 Recurrence Relations and Derivatives
§33.17 Recurrence Relations and Derivatives
33.17.1 ( + 1 ) r f ( ϵ , - 1 ; r ) - ( 2 + 1 ) ( ( + 1 ) - r ) f ( ϵ , ; r ) + ( 1 + ( + 1 ) 2 ϵ ) r f ( ϵ , + 1 ; r ) = 0 ,
33.17.2 ( + 1 ) ( 1 + 2 ϵ ) r h ( ϵ , - 1 ; r ) - ( 2 + 1 ) ( ( + 1 ) - r ) h ( ϵ , ; r ) + r h ( ϵ , + 1 ; r ) = 0 ,
33.17.3 ( + 1 ) r f ( ϵ , ; r ) = ( ( + 1 ) 2 - r ) f ( ϵ , ; r ) - ( 1 + ( + 1 ) 2 ϵ ) r f ( ϵ , + 1 ; r ) ,
33.17.4 ( + 1 ) r h ( ϵ , ; r ) = ( ( + 1 ) 2 - r ) h ( ϵ , ; r ) - r h ( ϵ , + 1 ; r ) .
3: 33.14 Definitions and Basic Properties
§33.14(i) Coulomb Wave Equation
§33.14(ii) Regular Solution f ( ϵ , ; r )
§33.14(iii) Irregular Solution h ( ϵ , ; r )
§33.14(iv) Solutions s ( ϵ , ; r ) and c ( ϵ , ; r )
§33.14(v) Wronskians
4: 33.24 Tables
§33.24 Tables
5: 33.18 Limiting Forms for Large
§33.18 Limiting Forms for Large
6: 33.21 Asymptotic Approximations for Large | r |
§33.21(i) Limiting Forms
  • (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 ( 2 r / ν ) ν ,
    ξ ( ν , r ) e r / ν ( 2 r / ν ) - ν , r ,
    33.21.2
    ζ ( - ν , r ) e r / ν ( - 2 r / ν ) - ν ,
    ξ ( - ν , r ) e - r / ν ( - 2 r / ν ) ν , r - .

    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).

  • §33.21(ii) Asymptotic Expansions
    7: 33.22 Particle Scattering and Atomic and Molecular Spectra
    k Scaling
    Z Scaling
    i k Scaling
    §33.22(iii) Conversions Between Variables
    8: 33.15 Graphics
    §33.15 Graphics
    See accompanying text
    Figure 33.15.9: h ( ϵ , ; r ) with = 1 , - 2 < ϵ < 2 , - 15 < r < 15 . Magnify 3D Help
    9: 33.23 Methods of Computation
    §33.23 Methods of Computation
    10: 33.16 Connection Formulas
    §33.16(i) F and G in Terms of f and h
    §33.16(ii) f and h in Terms of F and G when ϵ > 0
    §33.16(iii) f and h in Terms of W κ , μ ( z ) when ϵ < 0
    §33.16(iv) s and c in Terms of F and G when ϵ > 0
    §33.16(v) s and c in Terms of W κ , μ ( z ) when ϵ < 0