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11: 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 .
12: 33.10 Limiting Forms for Large ρ or Large | η |
13: 30.13 Wave Equation in Prolate Spheroidal Coordinates
14: 28.22 Connection Formulas
The joining factors in the above formulas are given by …
28.22.9 f e , m ( h ) = π / 2 g e , m ( h ) Mc m ( 2 ) ( 0 , h ) ,
ge m ( 0 , h 2 ) = 1 2 π S m ( h 2 ) ( g o , m ( h ) ) 2 se m ( 0 , h 2 ) .
15: 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 .
16: 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 ) + ,
17: 30.16 Methods of Computation
§30.16(iii) Radial Spheroidal Wave Functions
The coefficients a n , k m ( γ 2 ) calculated in §30.16(ii) can be used to compute S n m ( j ) ( z , γ ) , j = 1 , 2 , 3 , 4 from (30.11.3) as well as the connection coefficients K n m ( γ ) from (30.11.10) and (30.11.11). …
18: 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 ) .
19: 28.23 Expansions in Series of Bessel Functions
§28.23 Expansions in Series of Bessel Functions
28.23.6 Mc 2 m ( j ) ( z , h ) = ( 1 ) m ( ce 2 m ( 0 , h 2 ) ) 1 = 0 ( 1 ) A 2 2 m ( h 2 ) 𝒞 2 ( j ) ( 2 h cosh z ) ,
28.23.7 Mc 2 m ( j ) ( z , h ) = ( 1 ) m ( ce 2 m ( 1 2 π , h 2 ) ) 1 = 0 A 2 2 m ( h 2 ) 𝒞 2 ( j ) ( 2 h sinh z ) ,
28.23.8 Mc 2 m + 1 ( j ) ( z , h ) = ( 1 ) m ( ce 2 m + 1 ( 0 , h 2 ) ) 1 = 0 ( 1 ) A 2 + 1 2 m + 1 ( h 2 ) 𝒞 2 + 1 ( j ) ( 2 h cosh z ) ,
20: 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 ) .