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XXVIII.—Researches into the Characteristic Numbers of the Mathieu Equation—(Second Paper)

Published online by Cambridge University Press:  15 September 2014

E. L. Ince
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Extract

In a paper recently communicated to the London Mathematical Society the present author showed that the characteristic numbers an and bn of the Mathieu equation—

may be developed, for large positive values of q, in the forms

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 46 , 1927 , pp. 316 - 322
Copyright
Copyright © Royal Society of Edinburgh 1927

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References

page 316 note * Read December 10, 1925.

page 316 note † For their definition, see the first paper under the above title.

page 318 note * If anyone had the courage to push the development on a stage or two further he would greatly enhance the value of an important expansion. But any reader who attempts to verify the results given above will realise that the work involved would be tremendous.

page 318 note † Note that a is unaltered if k is replaced by – k and m by – m – 1. This substitution interchanges (2.2) and (2.2a).

page 319 note * This difficult point is most easily dealt with by taking ζ=k cosx as a new (complex) independent variable, and by applying Birkhoff's theory to the equation so transformed. An account of this theory is given in the author's “Ordinary Differential Equations,” chap. xix.

page 319 note † The notation is new, but the functions in question are well known. See Poole, , Proc. Lond. Math. Soc. (2), xx (1922), p. 378Google Scholar; Ince, Proc. Camb. Phil. Soc., xxi (1922), p. 117Google Scholar. The functions reduce respectively to sin(m + ½)x; and cos(m + ½)x when q is zero. By a continuity argument it may be proved that they oscillate exactly m times in the open interval (0, π) for all real values of q. Cf. Ince, Proc. Lond. Math. Soc. (2), xxv (1924), p. 53Google Scholar.

page 320 note * This opportunity is taken of correcting a numerical mistake in the previous paper. At the bottom of p. 24 and the top of p. 25 read :

The table inserted in the first paper was calculated without the aid of a machine : it is not guaranteed to be, and in fact is not, free from errors which could not be detected by differencing. The present table was calculated with all possible care, and with the aid of a machine. It was not checked independently, but the nature of the method is such that each step forward serves to verify the foregoing work. The fact that the average time taken over each of the 120 entries of the table was very little under two hours, will make clear the amount of work put into the table. The author has, however, the satisfaction of knowing that this, the hardest part of the preliminary work of computing the Mathieu functions, will never have to be repeated with no basis to work upon.