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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)
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  • [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.
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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1969-0239727-4
PII: S 0025-5718(1969)0239727-4
Article copyright: © Copyright 1969 American Mathematical Society