The double points of Mathieu's differential equation
Authors:
G. Blanch and D. S. Clemm
Journal:
Math. Comp. 23 (1969), 97108
MSC:
Primary 65.25
MathSciNet review:
0239727
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Abstract: Mathieu's differential equation, , admits of solutions of period or for four countable sets of characteristic values, , which can be ordered as . The power series expansions for the converge up to the first double point for that order in the complex plane. [At a double point, .] The present work furnishes the double points for orders 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 and at the singular point. The method is quite general and can be used in other problems as well.
 [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. 137203.
 [5]
H. P. Mulholland & S. Goldstein, ``The characteristic numbers of the Mathieu equation with purely imaginary parameters,'' Philos. Mag., v. 8, 1929, pp. 834840.
 [6]
Hanan
Rubin, Anecdote on power series expansions of Mathieu
functions, J. Math. and Phys. 43 (1964),
339–341. MR 0170046
(30 #287)
 [1]
 G. Blanch, ``Numerical aspects of Mathieu eigenvalues,'' Rend. Circ. Mat. Palermo, v. 15, 1966, pp. 5197. MR 0229377 (37:4951)
 [2]
 C. J. Bouwkamp, ``A note on Mathieu functions,'' Nederl. Akad. Wetensch. Proc., v. 51, 1948, pp. 891893 = Indag. Math., v. 10, 1948, pp. 319321. MR 10, 533. 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. 137203.
 [5]
 H. P. Mulholland & S. Goldstein, ``The characteristic numbers of the Mathieu equation with purely imaginary parameters,'' Philos. Mag., v. 8, 1929, pp. 834840.
 [6]
 Hanan Rubin, ``Anecdote on power series expansions of Mathieu functions,'' J. Math. and Phys., v. 43, 1964, pp. 339341. MR 30 #287. MR 0170046 (30:287)
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DOI:
http://dx.doi.org/10.1090/S00255718196902397274
PII:
S 00255718(1969)02397274
Article copyright:
© Copyright 1969 American Mathematical Society
