Lamé polynomials
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1: 29.14 Orthogonality
§29.14 Orthogonality
βΊLamé polynomials are orthogonal in two ways. … βΊ
29.14.4
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29.14.5
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29.14.6
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2: 29.12 Definitions
§29.12 Definitions
βΊ§29.12(i) Elliptic-Function Form
… βΊThere are eight types of Lamé polynomials, defined as follows: … βΊ …3: 29.19 Physical Applications
4: 29.13 Graphics
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§29.13(i) Eigenvalues for Lamé Polynomials
… βΊ βΊ§29.13(ii) Lamé Polynomials: Real Variable
… βΊ§29.13(iii) Lamé Polynomials: Complex Variable
… βΊ5: 29.15 Fourier Series and Chebyshev Series
6: 29.20 Methods of Computation
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βΊThese matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that has to be chosen sufficiently large.
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§29.20(ii) Lamé Polynomials
… βΊThe corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. … βΊ§29.20(iii) Zeros
βΊZeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). …7: 29.16 Asymptotic Expansions
§29.16 Asymptotic Expansions
βΊHargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. The approximations for Lamé polynomials hold uniformly on the rectangle , , when and assume large real values. …8: 29.1 Special Notation
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βΊAll derivatives are denoted by differentials, not by primes.
βΊThe main functions treated in this chapter are the eigenvalues , , , , the Lamé functions , , , , and the Lamé polynomials
, , , , , , , .
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9: 29.22 Software
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§29.22(ii) Lamé Polynomials
… βΊLA3: Eigenvalues for Lamé polynomials.
LA4: Lamé polynomials.
10: 29.21 Tables
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Ince (1940a) tabulates the eigenvalues , (with and interchanged) for , , and . Precision is 4D.
Arscott and Khabaza (1962) tabulates the coefficients of the polynomials in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues for , . Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.