Proceedings of the Edinburgh Mathematical Society
- Proceedings of the Edinburgh Mathematical Society / Volume 44 / February 1925, pp 57-71
- Copyright © Edinburgh Mathematical Society 1925
- DOI: http://dx.doi.org/10.1017/S0013091500034374 (About DOI), Published online: 20 January 2009
Research Article
The Solutions of Mathieu's Differential Equation: Representation by Contour Integrals, and Asymptotic Expansions
John Dougall
It is an obvious remark that the Mathieu functions, being the harmonic functions of the elliptic cylinder, must be closely related to the Bessel functions, the harmonic functions of the circular cylinder. Reference has been made to some aspects of this relationship in two earlier communications, to which the present paper may be regarded as a sequel.
(Received September 07 1925)
(Accepted May 02 1924)
Notes
page 57 note * The Solution of Mathieu's Differential Equation, Proc. Edin. Math. Soc., Vol. XXXIV (1915–1916); [OpenURL Query Data] [Google Scholar]
The Solutions of Mathieu's Differential Equation, and their Asymptotic Expansions, Proc. Edin. Math. Soc., Vol. XLI (1922–1923) [OpenURL Query Data] [Google Scholar].
These papers will be referred to as I and II; and 1(1), e.g., will be used for “Equation (1) of Paper I.”
page 57 note † Gray, Mathews and , MacRobert, Bessel Functions, 2nd Edition, Chap. V, §2;
Whittaker and , Watson, Modern Analysis, 3rd Edition, Chap. XVII, §17. 3.
page 59 note * Cf. Whittaker and Watson, Modern Analysis, Chap. XII, §§12. 22, 12. 43.
page 65 note * Whittaker and Watson, Modern Analysis; §12. 43.
page 66 note * It is easy to show from (12) that the integrals of e−ot F(t) over the two infinite paths, which were added to fig. 3 to give fig. 4, exist. Cf. Modern Analysis, § 5. 32
page 69 note * Cf. e.g. Forsyth, A. R., Theory of Differential Equations, Part III, Vol. IV, §§ 101–107.
