basic solutions of Mathieu equation
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1: 28.2 Definitions and Basic Properties
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§28.2(ii) Basic Solutions ,
… ►(28.2.1) possesses a fundamental pair of solutions called basic solutions with … ►§28.2(vi) Eigenfunctions
…2: 28.29 Definitions and Basic Properties
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►A generalization of Mathieu’s equation (28.2.1) is Hill’s equation
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§28.29(ii) Floquet’s Theorem and the Characteristic Exponent
… ► … ► … ► …3: 28.4 Fourier Series
§28.4 Fourier Series
… ►§28.4(ii) Recurrence Relations
… ►§28.4(iii) Normalization
… ►§28.4(v) Change of Sign of
… ►For the basic solutions and see §28.2(ii).4: 28.20 Definitions and Basic Properties
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§28.20(i) Modified Mathieu’s Equation
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28.20.1
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§28.20(ii) Solutions , , , ,
… ►It follows that (28.20.1) has independent and unique solutions , such that … ►§28.20(iv) Radial Mathieu Functions ,
…5: 28.12 Definitions and Basic Properties
§28.12 Definitions and Basic Properties
… ► … ►§28.12(ii) Eigenfunctions
… ►They have the following pseudoperiodic and orthogonality properties: … ►When is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period . …6: Bibliography
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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation.
Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).
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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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Rational solutions of Painlevé equations.
Stud. Appl. Math. 61 (1), pp. 31–53.
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Perturbation solutions of the ellipsoidal wave equation.
Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.
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Numerical Solution of Boundary Value Problems for Ordinary Differential Equations.
Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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7: Bibliography G
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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals.
Ann. of Math. (2) 48 (4), pp. 785–826.
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Basic Hypergeometric Series.
Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.
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Special classes of solutions of Painlevé equations.
Differ. Uravn. 18 (3), pp. 419–429 (Russian).
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The solutions of Painlevé’s fifth equation.
Differ. Uravn. 12 (4), pp. 740–742 (Russian).
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One-parameter systems of solutions of Painlevé equations.
Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).
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8: Bibliography F
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Basic Hypergeometric Series and Applications.
Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.
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On a unified approach to transformations and elementary solutions of Painlevé equations.
J. Math. Phys. 23 (11), pp. 2033–2042.
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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions.
Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
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Solution of the transcendental equation
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Comm. ACM 16 (2), pp. 123–124.
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Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice.
Phys. A 190 (3-4), pp. 346–362.
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