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1: 30.11 Radial Spheroidal Wave Functions
§30.11 Radial Spheroidal Wave Functions
§30.11(i) Definitions
Connection Formulas
§30.11(ii) Graphics
§30.11(iv) Wronskian
2: 28.20 Definitions and Basic Properties
§28.20(iv) Radial Mathieu Functions Mc n ( j ) , Ms n ( j )
28.20.15 Mc n ( j ) ( z , h ) = M n ( j ) ( z , h ) , n = 0 , 1 , ,
28.20.16 Ms n ( j ) ( z , h ) = ( - 1 ) n M - n ( j ) ( z , h ) , n = 1 , 2 , .
§28.20(vii) Shift of Variable
And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).
3: 30.17 Tables
§30.17 Tables
  • Hanish et al. (1970) gives λ n m ( γ 2 ) and S n m ( j ) ( z , γ ) , j = 1 , 2 , and their first derivatives, for 0 m 2 , m n m + 49 , - 1600 γ 2 1600 . The range of z is given by 1 z 10 if γ 2 > 0 , or z = - i ξ , 0 ξ 2 if γ 2 < 0 . Precision is 18S.

  • EraŠevskaja et al. (1973, 1976) gives S m ( j ) ( i y , - i c ) , S m ( j ) ( z , γ ) and their first derivatives for j = 1 , 2 , 0.5 c 8 , y = 0 , 0.5 , 1 , 1.5 , 0.5 γ 8 , z = 1.01 , 1.1 , 1.4 , 1.8 . Precision is 15S.

  • 4: 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
    See accompanying text
    Figure 33.3.4: F ( η , ρ ) , G ( η , ρ ) with = 0 , η = 10 . The turning point is at ρ tp ( 10 , 0 ) = 20 . Magnify
    See accompanying text
    Figure 33.3.6: F ( η , ρ ) , G ( η , ρ ) , and M ( η , ρ ) with = 5 , η = 0 . The turning point is at ρ tp ( 0 , 5 ) = 30 (as in Figure 33.3.5). Magnify
    §33.3(ii) Surfaces of the Coulomb Radial Functions F 0 ( η , ρ ) and G 0 ( η , ρ )
    5: 28.21 Graphics
    Radial Mathieu Functions: Surfaces
    See accompanying text
    Figure 28.21.6: Ms 1 ( 2 ) ( x , h ) for 0.2 h 2 , 0 x 2 . Magnify 3D Help
    6: 30.1 Special Notation
    The main functions treated in this chapter are the eigenvalues λ n m ( γ 2 ) and the spheroidal wave functions Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) , Ps n m ( z , γ 2 ) , Qs n m ( z , γ 2 ) , and S n m ( j ) ( z , γ ) , j = 1 , 2 , 3 , 4 . … Flammer (1957) and Abramowitz and Stegun (1964) use λ m n ( γ ) for λ n m ( γ 2 ) + γ 2 , R m n ( j ) ( γ , z ) for S n m ( j ) ( z , γ ) , and …
    7: 30.18 Software
  • SWF4: S n m ( j ) ( z , γ ) , j = 1 , 2 .

  • 8: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
    33.5.6 C ( 0 ) = 2 ! ( 2 + 1 ) ! = 1 ( 2 + 1 ) !! .
    9: 33.23 Methods of Computation
    Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … Inside the turning points, that is, when ρ < ρ tp ( η , ) , there can be a loss of precision by a factor of approximately | G | 2 . … Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … WKBJ approximations (§2.7(iii)) for ρ > ρ tp ( η , ) are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. … Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for F 0 and G 0 in the region inside the turning point: ρ < ρ tp ( η , ) .
    10: 30.14 Wave Equation in Oblate Spheroidal Coordinates
    30.14.8 w 1 ( ξ ) = a 1 S n m ( 1 ) ( i ξ , γ ) + b 1 S n m ( 2 ) ( i ξ , γ ) .
    §30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids