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basic solutions of Mathieu equation

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1: 28.2 Definitions and Basic Properties
§28.2(ii) Basic Solutions w I , w II
(28.2.1) possesses a fundamental pair of solutions w I ( z ; a , q ) , w II ( z ; a , q ) called basic solutions with …
§28.2(vi) Eigenfunctions
2: 28.29 Definitions and Basic Properties
A generalization of Mathieu’s equation (28.2.1) is Hill’s equation
§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 q
For the basic solutions w I and w II see §28.2(ii).
4: 28.20 Definitions and Basic Properties
§28.20(i) Modified Mathieu’s Equation
28.20.1 w ′′ ( a 2 q cosh ( 2 z ) ) w = 0 ,
§28.20(ii) Solutions Ce ν , Se ν , Me ν , Fe n , Ge n
It follows that (28.20.1) has independent and unique solutions M ν ( 3 ) ( z , h ) , M ν ( 4 ) ( z , h ) such that …
§28.20(iv) Radial Mathieu Functions Mc n ( j ) , Ms n ( j )
5: 28.12 Definitions and Basic Properties
§28.12 Definitions and Basic Properties
§28.12(ii) Eigenfunctions me ν ( z , q )
They have the following pseudoperiodic and orthogonality properties: … When ν = s / m is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period 2 m π . …
6: Bibliography
  • A. S. Abdullaev (1985) 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).
  • H. Airault, H. P. McKean, and J. Moser (1977) 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.
  • H. Airault (1979) Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1), pp. 31–53.
  • F. M. Arscott (1956) Perturbation solutions of the ellipsoidal wave equation. Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.
  • U. M. Ascher, R. M. M. Mattheij, and R. D. Russell (1995) 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.
  • 7: Bibliography G
  • G. Gasper and M. Rahman (1990) Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.
  • G. Gasper and M. Rahman (2004) Basic Hypergeometric Series. Second edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge.
  • V. I. Gromak and N. A. Lukaševič (1982) Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).
  • V. I. Gromak (1976) The solutions of Painlevé’s fifth equation. Differ. Uravn. 12 (4), pp. 740–742 (Russian).
  • V. I. Gromak (1978) One-parameter systems of solutions of Painlevé equations. Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).
  • 8: Bibliography F
  • N. J. Fine (1988) Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.
  • A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.
  • A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
  • F. N. Fritsch, R. E. Shafer, and W. P. Crowley (1973) Solution of the transcendental equation w e w = x . Comm. ACM 16 (2), pp. 123–124.
  • Y. Fukui and T. Horiguchi (1992) 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.