eigenvalues of accessory parameter
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21: 29.15 Fourier Series and Chebyshev Series
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βΊA convenient way of constructing the coefficients, together with the eigenvalues, is as follows.
Equations (29.6.4), with , (29.6.3), and can be cast as an algebraic eigenvalue problem in the following way.
…Let the eigenvalues of be with
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29.15.7
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29.15.22
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22: 29.13 Graphics
23: 28.4 Fourier Series
24: 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 , , , , , , , .
The notation for the eigenvalues and functions is due to Erdélyi et al. (1955, §15.5.1) and that for the polynomials is due to Arscott (1964b, §9.3.2).
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βΊOther notations that have been used are as follows: Ince (1940a) interchanges with .
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25: 30.9 Asymptotic Approximations and Expansions
§30.9 Asymptotic Approximations and Expansions
… βΊFor uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … βΊThe asymptotic behavior of and as in descending powers of is derived in Meixner (1944). …The behavior of for complex and large is investigated in Hunter and Guerrieri (1982).26: 29.20 Methods of Computation
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§29.20(i) Lamé Functions
… βΊInitial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). … βΊA third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … βΊ§29.20(ii) Lamé Polynomials
βΊThe eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). …27: 30.7 Graphics
28: 28.29 Definitions and Basic Properties
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βΊA generalization of Mathieu’s equation (28.2.1) is Hill’s equation
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βΊiff is an eigenvalue of the matrix
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§28.29(iii) Discriminant and Eigenvalues in the Real Case
… βΊTo every equation (28.29.1), there belong two increasing infinite sequences of real eigenvalues: … βΊ
28.29.17
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29: 29.9 Stability
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βΊIf is not an integer, then (29.2.1) is unstable iff or lies in one of the closed intervals with endpoints and , .
If is a nonnegative integer, then (29.2.1) is unstable iff or for some .