relation to Lamé equation
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1: 29.2 Differential Equations
2: 31.8 Solutions via Quadratures
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►For , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form.
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3: 29.12 Definitions
§29.12 Definitions
►§29.12(i) Elliptic-Function Form
… ►There are eight types of Lamé polynomials, defined as follows: …In consequence they are doubly-periodic meromorphic functions of . … ►§29.12(ii) Algebraic Form
…4: Bibliography
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Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem.
Comm. Pure Appl. Math. 30 (1), pp. 95–148.
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Dirichlet series related to the Riemann zeta function.
J. Number Theory 19 (1), pp. 85–102.
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Tables of Lamé Polynomials.
Pergamon Press, The Macmillan Co., New York.
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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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5: 31.7 Relations to Other Functions
§31.7 Relations to Other Functions
►§31.7(i) Reductions to the Gauss Hypergeometric Function
… ►§31.7(ii) Relations to Lamé Functions
… ►equation (31.2.1) becomes Lamé’s equation with independent variable ; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …6: 29.6 Fourier Series
§29.6 Fourier Series
►§29.6(i) Function
… ►In addition, if satisfies (29.6.2), then (29.6.3) applies. … ►Consequently, reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). … ►§29.6(ii) Function
…7: Bibliography S
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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations.
SIAM J. Math. Anal. 10 (4), pp. 823–838.
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Lamé polynomial solutions to some elliptic crack and punch problems.
Internat. J. Engrg. Sci. 16 (8), pp. 551–563.
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Structure of avoided crossings for eigenvalues related to equations of Heun’s class.
J. Phys. A 30 (2), pp. 673–687.
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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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The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity.
Differ. Uravn. 23 (5), pp. 834–842 (Russian).
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8: 28.34 Methods of Computation
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(c)
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(d)
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(f)
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(d)
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§28.34(i) Characteristic Exponents
… ►Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).
9: 29.1 Special Notation
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►All derivatives are denoted by differentials, not by primes.
►The main functions treated in this chapter are the eigenvalues , , , , the Lamé functions , , , , and the Lamé polynomials , , , , , , , .
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►Other notations that have been used are as follows: Ince (1940a) interchanges with .
The relation to the Lamé functions , of Jansen (1977) is given by
…The relation to the Lamé functions , of Ince (1940b) is given by
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10: Bibliography W
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Asymptotics of orthogonal polynomials via recurrence relations.
Anal. Appl. (Singap.) 10 (2), pp. 215–235.
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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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Some useful integrals of
and related integrals.
Optica Acta 14 (3), pp. 317–322.
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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials.
Ph.D. Thesis, University of Wisconsin, Madison, WI.
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Computation with Recurrence Relations.
Pitman, Boston, MA.
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