Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-12T05:36:08.915Z Has data issue: false hasContentIssue false
  • English
  • Français

XVII.—On the Asymptotic Expansion of the Characteristic Numbers of the Mathieu Equation

Published online by Cambridge University Press:  15 September 2014

Sydney Goldstein
Sydney Goldstein
Affiliation:
St. John's College, Cambridge
Get access

Extract

An asymptotic formula has recently been given for the characteristic numbers of the Mathieu equation

From tabular values, it will be seen that the formula provides good numerical approximations to the characteristic numbers of integral order; but as pointed out by Ince, it provides better approximations to the characteristic numbers of order (m + ½), where m is a positive integer or zero. In this paper we shall first attempt to find out why this should be so, and then go on to show that the formula is probably an asymptotic expansion, in the Poincaré sense, for any characteristic number. A new asymptotic formula is then found for the difference between two characteristic numbers.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 49 , 1930 , pp. 210 - 223
Copyright
Copyright © Royal Society of Edinburgh 1930

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Ince, Proc. Roy. Soc. Edinburgh, vol. xlvi, pp. 20–29, 1925.Google Scholar
2.Ince, Journal London Math. Soc., vol. ii, pp. 46–51, 1927.Google Scholar
3.Ince, Proc. Roy. Soc. Edinburgh, vol. xlvi, pp. 316–322, 1926.Google Scholar
4.Goldstein, , Trans. Gamb. Phil. Soc., vol. xxiii, pp. 303–336, 1927.Google Scholar
5.Ince, Proc. Roy. Soc. Edinburgh, vol. xlvii, pp. 294–301, 1927.Google Scholar
6.Jeffreys, , Proc. London Math. Soc., (2), vol. xxiii, pp. 428–436, 1924.Google Scholar
7.Goldstein, , Proc. London Math. Soc., (2), vol. xxviii, pp. 81–91, 1928.CrossRefGoogle Scholar