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Coulomb radial functions

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1: 33.3 Graphics
§33.3(i) Line Graphs of the Coulomb Radial Functions F ( η , ρ ) and G ( η , ρ )
33.3.1 M ( η , ρ ) = ( F 2 ( η , ρ ) + G 2 ( η , ρ ) ) 1 / 2 = | H ± ( η , ρ ) | .
§33.3(ii) Surfaces of the Coulomb Radial Functions F 0 ( η , ρ ) and G 0 ( η , ρ )
2: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
33.5.6 C ( 0 ) = 2 ! ( 2 + 1 ) ! = 1 ( 2 + 1 ) !! .
33.5.9 C ( η ) e π η / 2 ( 2 + 1 ) !! e π η / 2 e 2 ( 2 ) + 1 .
3: 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. …
  • Greene et al. (1979):

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

  • 4: 33.2 Definitions and Basic Properties
    33.2.2 ρ tp ( η , ) = η + ( η 2 + ( + 1 ) ) 1 / 2 .
    33.2.3 F ( η , ρ ) = C ( η ) 2 1 ( i ) + 1 M ± i η , + 1 2 ( ± 2 i ρ ) ,
    33.2.5 C ( η ) = 2 e π η / 2 | Γ ( + 1 + i η ) | ( 2 + 1 ) ! .
    33.2.13 F 1 G F G 1 = / ( 2 + η 2 ) 1 / 2 , 1 .
    5: 33.10 Limiting Forms for Large ρ or Large | η |
    6: 33.13 Complex Variable and Parameters
    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 ) .
    7: 33.8 Continued Fractions
    33.8.1 F F = S + 1 R + 1 2 T + 1 R + 2 2 T + 2 .
    33.8.2 H ± H ± = c ± i ρ a b 2 ( ρ η ± i ) + ( a + 1 ) ( b + 1 ) 2 ( ρ η ± 2 i ) + ,
    8: 33.6 Power-Series Expansions in ρ
    33.6.1 F ( η , ρ ) = C ( η ) k = + 1 A k ( η ) ρ k ,
    33.6.2 F ( η , ρ ) = C ( η ) k = + 1 k A k ( η ) ρ k 1 ,
    33.6.5 H ± ( η , ρ ) = e ± i θ ( η , ρ ) ( 2 + 1 ) ! Γ ( ± i η ) ( k = 0 ( a ) k ( 2 + 2 ) k k ! ( 2 i ρ ) a + k ( ln ( 2 i ρ ) + ψ ( a + k ) ψ ( 1 + k ) ψ ( 2 + 2 + k ) ) k = 1 2 + 1 ( 2 + 1 ) ! ( k 1 ) ! ( 2 + 1 k ) ! ( 1 a ) k ( 2 i ρ ) a k ) ,
    9: 33.7 Integral Representations
    33.7.1 F ( η , ρ ) = ρ + 1 2 e i ρ ( π η / 2 ) | Γ ( + 1 + i η ) | 0 1 e 2 i ρ t t + i η ( 1 t ) i η d t ,
    33.7.2 H ( η , ρ ) = e i ρ ρ ( 2 + 1 ) ! C ( η ) 0 e t t i η ( t + 2 i ρ ) + i η d t ,
    33.7.3 H ( η , ρ ) = i e π η ρ + 1 ( 2 + 1 ) ! C ( η ) 0 ( exp ( i ( ρ tanh t 2 η t ) ) ( cosh t ) 2 + 2 + i ( 1 + t 2 ) exp ( ρ t + 2 η arctan t ) ) d t ,
    33.7.4 H + ( η , ρ ) = i e π η ρ + 1 ( 2 + 1 ) ! C ( η ) 1 i e i ρ t ( 1 t ) i η ( 1 + t ) + i η d t .
    10: 33.11 Asymptotic Expansions for Large ρ
    33.11.1 H ± ( η , ρ ) e ± i θ ( η , ρ ) k = 0 ( a ) k ( b ) k k ! ( ± 2 i ρ ) k ,
    33.11.4 H ± ( η , ρ ) = e ± i θ ( f ( η , ρ ) ± i g ( η , ρ ) ) ,