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spheroidal wave functions

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1: 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 . …Meixner and Schäfke (1954) use ps , qs , Ps , Qs for Ps , Qs , Ps , Qs , respectively.
Other Notations
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 …
2: 30.11 Radial Spheroidal Wave Functions
§30.11 Radial Spheroidal Wave Functions
§30.11(i) Definitions
Connection Formulas
See accompanying text
Figure 30.11.4: S n 1 ( 1 ) ( i y , 2 i ) , n = 1 , 2 , 0 y 10 . Magnify
§30.11(v) Connection with the Ps and Qs Functions
3: 30.10 Series and Integrals
4: 30 Spheroidal Wave Functions
Chapter 30 Spheroidal Wave Functions
5: 30.12 Generalized and Coulomb Spheroidal Functions
§30.12 Generalized and Coulomb Spheroidal Functions
Generalized spheroidal wave functions and Coulomb spheroidal functions are solutions of the differential equation …
6: 30.5 Functions of the Second Kind
§30.5 Functions of the Second Kind
30.5.1 Qs n m ( x , γ 2 ) , n = m , m + 1 , m + 2 , .
30.5.2 Qs n m ( - x , γ 2 ) = ( - 1 ) n - m + 1 Qs n m ( x , γ 2 ) ,
30.5.4 𝒲 { Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) } = ( n + m ) ! ( 1 - x 2 ) ( n - m ) ! A n m ( γ 2 ) A n - m ( γ 2 ) ( 0 ) ,
7: 30.6 Functions of Complex Argument
8: 30.15 Signal Analysis
§30.15 Signal Analysis
§30.15(i) Scaled Spheroidal Wave Functions
§30.15(ii) Integral Equation
9: 30.17 Tables
§30.17 Tables
  • Flammer (1957) includes 18 tables of eigenvalues, expansion coefficients, spheroidal wave functions, and other related quantities. Precision varies between 4S and 10S.

  • Zhang and Jin (1996) includes 24 tables of eigenvalues, spheroidal wave functions and their derivatives. Precision varies between 6S and 8S.

  • 10: 30.4 Functions of the First Kind
    §30.4 Functions of the First Kind
    §30.4(i) Definitions
    If γ = 0 , Ps n m ( x , 0 ) reduces to the Ferrers function P n m ( x ) : …
    §30.4(ii) Elementary Properties
    §30.4(iv) Orthogonality