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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


The double points of Mathieu's differential equation

Authors: G. Blanch and D. S. Clemm
Journal: Math. Comp. 23 (1969), 97-108
MSC: Primary 65.25
MathSciNet review: 0239727
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Abstract: Mathieu's differential equation, $ y'' + (a - 2q\cos 2x)y = 0$, admits of solutions of period $ \pi $ or $ 2\pi $ for four countable sets of characteristic values, $ a(q)$, which can be ordered as $ {a_r}(q),r = 0,1, \cdots $. The power series expansions for the $ {a_r}(q)$ converge up to the first double point for that order in the complex plane. [At a double point, $ {a_r}(q) = {a_r} + 2(q)$.] The present work furnishes the double points for orders $ r$ up to and including 15. These double points are singular points, and the usual methods of determining the characteristic values break down at a singular point. However, it was possible to determine two smooth functions in which one could interpolate for both $ q$ and $ {a_r}(q)$ at the singular point. The method is quite general and can be used in other problems as well.

References [Enhancements On Off] (What's this?)

  • [1] Gertrude Blanch, Numerical aspects of Mathieu eigenvalues, Rend. Circ. Mat. Palermo (2) 15 (1966), 51–97. MR 0229377 (37 #4951)
  • [2] C. J. Bouwkamp, A note on Mathieu functions, Nederl. Akad. Wetensch., Proc. 51 (1948), 891–893=Indagationes Math. 10, 319–321 (1948). MR 0029008 (10,533b)
  • [3] D. S. Clemm, A Comprehensive Code for Mathieu's Equation, to be published in a forthcoming A.R.L. Report. A transcript of the code can be made available on request to the author.
  • [4] É. Mathieu, ``Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique,'' J. Math. Pures Appl., v. 13, 1868, pp. 137-203.
  • [5] H. P. Mulholland & S. Goldstein, ``The characteristic numbers of the Mathieu equation with purely imaginary parameters,'' Philos. Mag., v. 8, 1929, pp. 834-840.
  • [6] Hanan Rubin, Anecdote on power series expansions of Mathieu functions, J. Math. and Phys. 43 (1964), 339–341. MR 0170046 (30 #287)

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PII: S 0025-5718(1969)0239727-4
Article copyright: © Copyright 1969 American Mathematical Society

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