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1: 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 ) .
2: 33.15 Graphics
§33.15 Graphics
§33.15(i) Line Graphs of the Coulomb Functions f ( ϵ , ; r ) and h ( ϵ , ; r )
See accompanying text
Figure 33.15.1: f ( ϵ , ; r ) , h ( ϵ , ; r ) with = 0 , ϵ = 4 . Magnify
§33.15(ii) Surfaces of the Coulomb Functions f ( ϵ , ; r ) , h ( ϵ , ; r ) , s ( ϵ , ; r ) , and c ( ϵ , ; r )
See accompanying text
Figure 33.15.9: h ( ϵ , ; r ) with = 1 , 2 < ϵ < 2 , 15 < r < 15 . Magnify 3D Help
3: 33.2 Definitions and Basic Properties
§33.2(i) Coulomb Wave Equation
The functions H ± ( η , ρ ) are defined by … σ ( η ) is the Coulomb phase shift. …
§33.2(iv) Wronskians and Cross-Product
4: 33.3 Graphics
§33.3 Graphics
§33.3(i) Line Graphs of the Coulomb Radial Functions F ( η , ρ ) and G ( η , ρ )
See accompanying text
Figure 33.3.3: F ( η , ρ ) , G ( η , ρ ) with = 0 , η = 2 . The turning point is at ρ tp ( 2 , 0 ) = 4 . Magnify
§33.3(ii) Surfaces of the Coulomb Radial Functions F 0 ( η , ρ ) and G 0 ( η , ρ )
See accompanying text
Figure 33.3.8: G 0 ( η , ρ ) , 2 η 2 , 0 < ρ 5 . Magnify 3D Help
5: 33.18 Limiting Forms for Large
§33.18 Limiting Forms for Large
f ( ϵ , ; r ) ( 2 r ) + 1 ( 2 + 1 ) ! ,
h ( ϵ , ; r ) ( 2 ) ! π ( 2 r ) .
6: 33.24 Tables
§33.24 Tables
  • Abramowitz and Stegun (1964, Chapter 14) tabulates F 0 ( η , ρ ) , G 0 ( η , ρ ) , F 0 ( η , ρ ) , and G 0 ( η , ρ ) for η = 0.5 ( .5 ) 20 and ρ = 1 ( 1 ) 20 , 5S; C 0 ( η ) for η = 0 ( .05 ) 3 , 6S.

  • 7: 33.1 Special Notation
    The main functions treated in this chapter are first the Coulomb radial functions F ( η , ρ ) , G ( η , ρ ) , H ± ( η , ρ ) (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions f ( ϵ , ; r ) , h ( ϵ , ; r ) , s ( ϵ , ; r ) , c ( ϵ , ; r ) (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. …
  • Curtis (1964a):

    P ( ϵ , r ) = ( 2 + 1 ) ! f ( ϵ , ; r ) / 2 + 1 , Q ( ϵ , r ) = ( 2 + 1 ) ! h ( ϵ , ; r ) / ( 2 + 1 A ( ϵ , ) ) .

  • Greene et al. (1979):

    f ( 0 ) ( ϵ , ; r ) = f ( ϵ , ; r ) , f ( ϵ , ; r ) = s ( ϵ , ; r ) , g ( ϵ , ; r ) = c ( ϵ , ; r ) .

  • 8: 33.13 Complex Variable and Parameters
    §33.13 Complex Variable and Parameters
    The functions F ( η , ρ ) , G ( η , ρ ) , and H ± ( η , ρ ) may be extended to noninteger values of by generalizing ( 2 + 1 ) ! = Γ ( 2 + 2 ) , and supplementing (33.6.5) by a formula derived from (33.2.8) with U ( a , b , z ) expanded via (13.2.42). … The quantities C ( η ) , σ ( η ) , and R , given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that
    33.13.1 C ( η ) = 2 e i σ ( η ) ( π η / 2 ) Γ ( + 1 i η ) / Γ ( 2 + 2 ) ,
    33.13.2 R = ( 2 + 1 ) C ( η ) / C 1 ( η ) .
    9: 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
    10: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
    §33.5(i) Small ρ
    F ( η , ρ ) C ( η ) ρ + 1 ,
    §33.5(ii) η = 0
    §33.5(iii) Small | η |
    §33.5(iv) Large