Proceedings of the Edinburgh Mathematical Society

Research Article

Determination of the Arbitrary Constants which appear in the Asymptotic Expansions for the Functions of the Elliptic Cylinder

William Marshall

In an article, published some time since, the author of the present paper deduced an asymptotic expression for the functions of the elliptic cylinder, which expression took the following form:


Here P and Q are certain asymptotic series, and C and a arbitrary constants. General expressions for these constants were not determined in the aforementioned article on account of the difficulties there set forth, though it was pointed out that their numerical calculation for any particular problem was simply a matter of arithmetic computation. It is the object of the present paper to deduce general expressions for these constants C and a in terms of the two parameters which appear in the defining equation for U.

(Received December 09 1921)


page 2 note * Am. Jour, of Math. Vol. XXXI., (1909), p. 311. [OpenURL Query Data]  [Google Scholar]

page 2 note † Whittaker, E. T.. International Congress of Mathematicians, Cambridge, 1912.

page 3 note * Am. Journal of Math. Vol. XXXI. (1909) p. 311, et seq. [OpenURL Query Data]  [Google Scholar]

page 4 note * S. Dannacher. Diss. Zürich, 1906.

page 5 note * See Dannacher, l. c., for a study of this function in the case of the related equation.

page 7 note * E. Heine. Handbuch der Kugelfunktionen. 2te Auf. Band I. S. 414.

page 8 note * Nielson. Handbuch d Theorie d. Cylinderfunktionen, S. 156.