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spheroidal wave functions

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11: 30.9 Asymptotic Approximations and Expansions
§30.9(i) Prolate Spheroidal Wave Functions
§30.9(ii) Oblate Spheroidal Wave Functions
§30.9(iii) Other Approximations and Expansions
12: Hans Volkmer
13: 30.16 Methods of Computation
§30.16(ii) Spheroidal Wave Functions of the First Kind
30.16.9 Ps n m ( x , γ 2 ) = lim d j = 1 d ( - 1 ) j - p e j , d P n + 2 ( j - p ) m ( x ) .
§30.16(iii) Radial Spheroidal Wave Functions
14: 31.12 Confluent Forms of Heun’s Equation
This has regular singularities at z = 0 and 1 , and an irregular singularity of rank 1 at z = . Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions30.12) are special cases of solutions of the confluent Heun equation. …
15: 30.7 Graphics
§30.7(ii) Functions of the First Kind
See accompanying text
Figure 30.7.21: | Qs 0 0 ( x + i y , - 4 ) | , - 1.8 x 1.8 , - 2 y 2 . Magnify 3D Help
16: 31.18 Methods of Computation
The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 2830.
17: 30.8 Expansions in Series of Ferrers Functions
§30.8 Expansions in Series of Ferrers Functions
30.8.1 Ps n m ( x , γ 2 ) = k = - R ( - 1 ) k a n , k m ( γ 2 ) P n + 2 k m ( x ) ,
30.8.2 a n , k m ( γ 2 ) = ( - 1 ) k ( n + 2 k + 1 2 ) ( n - m + 2 k ) ! ( n + m + 2 k ) ! - 1 1 Ps n m ( x , γ 2 ) P n + 2 k m ( x ) d x .
30.8.6 a n , k - m ( γ 2 ) = ( n - m ) ! ( n + m + 2 k ) ! ( n + m ) ! ( n - m + 2 k ) ! a n , k m ( γ 2 ) .
18: 30.2 Differential Equations
§30.2(i) Spheroidal Differential Equation
19: 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
20: 30.18 Software
§30.18(iii) Spheroidal Wave Functions