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41: 30.3 Eigenvalues
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βΊWith , the spheroidal wave functions are solutions of Equation (30.2.1) which are bounded on , or equivalently, which are of the form where is an entire function of .
These solutions exist only for eigenvalues , , of the parameter .
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βΊThe eigenvalues are analytic functions of the real variable and satisfy
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βΊhas the solutions , .
If is an odd positive integer, then Equation (30.3.5) has the solutions , .
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42: 29.6 Fourier Series
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βΊ
§29.6(i) Function
… βΊIn the special case , , there is a unique nontrivial solution with the property , . … βΊ§29.6(ii) Function
… βΊ§29.6(iii) Function
… βΊ§29.6(iv) Function
…43: 29.12 Definitions
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βΊThe Lamé functions , , and , , are called the Lamé
polynomials.
…where , .
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βΊwhere , , are either or .
The polynomial is of degree and has zeros (all simple) in and zeros (all simple) in .
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βΊdefined for with
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44: 34.3 Basic Properties: Symbol
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βΊWhen any one of is equal to , or , the symbol has a simple algebraic form.
…For these and other results, and also cases in which any one of is or , see Edmonds (1974, pp. 125–127).
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βΊEven permutations of columns of a symbol leave it unchanged; odd permutations of columns produce a phase factor , for example,
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βΊ
34.3.13
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βΊ
34.3.15
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45: 28.14 Fourier Series
46: 30.16 Methods of Computation
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βΊand real eigenvalues , , , , arranged in ascending order of magnitude.
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βΊFor , , ,
…which yields .
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βΊIf is known, then can be found by summing (30.8.1).
The coefficients are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5).
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47: 28.4 Fourier Series
48: 19.3 Graphics
49: 10.53 Power Series
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βΊ
10.53.1
βΊ
10.53.2
βΊ
10.53.3
βΊ
10.53.4
βΊFor and combine (10.47.10), (10.53.1), and (10.53.2).
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