Lamé equation
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21: 28.34 Methods of Computation
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(f)
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22: 29.20 Methods of Computation
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►The eigenvalues , , and the Lamé functions , , can be calculated by direct numerical methods applied to the differential equation (29.2.1); see §3.7.
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►Zeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii).
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23: Bibliography
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Integral equations and relations for Lamé functions.
Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
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Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions.
International Series of Monographs in Pure and Applied
Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.
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24: Bibliography W
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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25: 31.18 Methods of Computation
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►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 28–30.
26: Bibliography S
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Integral equations and relations for Lamé functions and ellipsoidal wave functions.
Proc. Cambridge Philos. Soc. 64, pp. 113–126.
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27: Bibliography R
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Some Applications of the Lamé Function Solutions of the Linearised Supersonic Flow Equations.
Technical Reports and Memoranda
Technical Report 2865, Aeronautical Research Council (Great Britain).
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28: 29.21 Tables
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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.
29: Hans Volkmer
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►Volkmer has published numerous papers on special functions, spectral theory, differential equations, and mathematical statistics.
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30: Bibliography E
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On Lamé functions.
Philos. Mag. (7) 31, pp. 123–130.
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On algebraic Lamé functions.
Philos. Mag. (7) 32, pp. 348–350.
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Integral equations for Heun functions.
Quart. J. Math., Oxford Ser. 13, pp. 107–112.
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The Fuchsian equation of second order with four singularities.
Duke Math. J. 9 (1), pp. 48–58.
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A catalogue of Sturm-Liouville differential equations.
In Sturm-Liouville theory,
pp. 271–331.
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