The double points of Mathieu’s differential equation
HTML articles powered by AMS MathViewer
- by G. Blanch and D. S. Clemm PDF
- Math. Comp. 23 (1969), 97-108 Request permission
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 ${a_r}(q)$, $r = 0$, $1$, $\dots$. 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
- Gertrude Blanch, Numerical aspects of Mathieu eigenvalues, Rend. Circ. Mat. Palermo (2) 15 (1966), 51–97. MR 229377, DOI 10.1007/BF02849408
- C. J. Bouwkamp, A note on Mathieu functions, Nederl. Akad. Wetensch., Proc. 51 (1948), 891–893=Indagationes Math. 10, 319–321 (1948). MR 29008 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. É. Mathieu, “Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique,” J. Math. Pures Appl., v. 13, 1868, pp. 137–203. H. P. Mulholland & S. Goldstein, “The characteristic numbers of the Mathieu equation with purely imaginary parameters,” Philos. Mag., v. 8, 1929, pp. 834–840.
- Hanan Rubin, Anecdote on power series expansions of Mathieu functions, J. Math. and Phys. 43 (1964), 339–341. MR 170046, DOI 10.1002/sapm1964431339
Additional Information
- © Copyright 1969 American Mathematical Society
- Journal: Math. Comp. 23 (1969), 97-108
- MSC: Primary 65.25
- DOI: https://doi.org/10.1090/S0025-5718-1969-0239727-4
- MathSciNet review: 0239727