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spheroidal differential equation

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1: 30.2 Differential Equations
§30.2(i) Spheroidal Differential Equation
The Liouville normal form of equation (30.2.1) is …
§30.2(iii) Special Cases
2: 30.3 Eigenvalues
§30.3 Eigenvalues
With μ = m = 0 , 1 , 2 , , the spheroidal wave functions Ps n m ( x , γ 2 ) are solutions of Equation (30.2.1) which are bounded on ( - 1 , 1 ) , or equivalently, which are of the form ( 1 - x 2 ) 1 2 m g ( x ) where g ( z ) is an entire function of z . …
§30.3(iii) Transcendental Equation
§30.3(iv) Power-Series Expansion
Further coefficients can be found with the Maple program SWF9; see §30.18(i).
3: 30.17 Tables
§30.17 Tables
4: 30.9 Asymptotic Approximations and Expansions
§30.9 Asymptotic Approximations and Expansions
30.9.1 λ n m ( γ 2 ) - γ 2 + γ q + β 0 + β 1 γ - 1 + β 2 γ - 2 + ,
30.9.4 λ n m ( γ 2 ) 2 q | γ | + c 0 + c 1 | γ | - 1 + c 2 | γ | - 2 + ,
5: 30.16 Methods of Computation
§30.16(i) Eigenvalues
30.16.4 α p , d - λ n m ( γ 2 ) = O ( γ 4 d 4 2 d + 1 ( ( m + 2 d - 1 ) ! ( m + 2 d + 1 ) ! ) 2 ) , d .
If λ n m ( γ 2 ) is known, then we can compute Ps n m ( x , γ 2 ) (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions w ( 0 ) = 1 , w ( 0 ) = 0 if n - m is even, or w ( 0 ) = 0 , w ( 0 ) = 1 if n - m is odd. …
6: 30.7 Graphics
§30.7(i) Eigenvalues
See accompanying text
Figure 30.7.4: Eigenvalues λ n 10 ( γ 2 ) , n = 10 , 11 , 12 , 13 , - 50 γ 2 150 . Magnify
7: 30.6 Functions of Complex Argument
8: 30.8 Expansions in Series of Ferrers Functions
30.8.4 A k f k - 1 + ( B k - λ n m ( γ 2 ) ) f k + C k f k + 1 = 0 ,
30.8.8 λ n m ( γ 2 ) - B k A k a n , k m ( γ 2 ) a n , k - 1 m ( γ 2 ) = 1 + O ( 1 k 4 ) .
30.8.10 A - N - 1 a n , - N - 2 m ( γ 2 ) + ( B - N - 1 - λ n m ( γ 2 ) ) a n , - N - 1 m ( γ 2 ) + C a n , - N m ( γ 2 ) = 0 ,
9: 30.1 Special Notation
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 …
10: 30.4 Functions of the First Kind
30.4.5 α k g k + 2 + ( β k - λ n m ( γ 2 ) ) g k + γ k g k - 2 = 0