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IV.—Researches into the Characteristic Numbers of the Mathieu Equation

Published online by Cambridge University Press:  15 September 2014

E. L. Ince
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Extract

The characteristic numbers of the Mathieu equation

are those values of a for which, when q is given, the equation admits of a solution of period π or 2π. The periodic solutions, or Mathieu functions, may be developed as a Fourier-series convergent for all values of q,

multiplied by one or other of the factors

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 46 , 1927 , pp. 20 - 29
Copyright
Copyright © Royal Society of Edinburgh 1927

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References

page 20 note * Whittaker and Watson, Modern Analysis, chap. xix.

page 20 note † The present writer was the first to prove that, except when q = 0, there cannot be two solutions of period π or 2π. See Proc. Camb. Phil. Soc., xxi (1922), pp. 117–120Google Scholar; Proc. Lond. Math. Soc., (2) xxiii (1923), pp. 56–74.Google Scholar An independent proof was recently given by Einar Hille, ibid., p. 224.

page 20 note ‡ Proc. Edin. Math. Soc., xxxiv (1916), pp. 176–196Google Scholar; xli (1923), pp. 26–48.

page 21 note * Maclaurin, , Trans. Camb. Phil. Soc., xvii (1898), pp. 41–108Google Scholar ; Marshall, , Amer. J. Math., xxxi (1909), pp. 311–336CrossRefGoogle Scholar ; Jeffreys, , Proc. Lond. Math. Soc., (2) xxiii (1924), pp. 437–454Google Scholar.

page 22 note * Von, Koch, Acta Math., xvi (1892), p. 217Google Scholar.

page 23 note * Perron, Die Lehre von den Kettenbrüchen, § 56, Theorem 41.