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The Solutions of Mathieu's Differential Equation: Representation by Contour Integrals, and Asymptotic Expansions

Published online by Cambridge University Press:  20 January 2009

John Dougall
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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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 44 , February 1925 , pp. 57 - 71
Copyright
Copyright © Edinburgh Mathematical Society 1925

References

page 57 note * The Solution of Mathieu's Differential Equation, Proc. Edin. Math. Soc., Vol. XXXIV (1915–1916);Google Scholar

The Solutions of Mathieu's Differential Equation, and their Asymptotic Expansions, Proc. Edin. Math. Soc., Vol. XLI (1922–1923)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;Google Scholar

Whittaker, and , Watson, Modern Analysis, 3rd Edition, Chap. XVII, §17. 3Google Scholar.

page 59 note * Cf. Whittaker and Watson, Modern Analysis, Chap. XII, §§12. 22, 12. 43.

page 65 note * Whittaker, and Watson, , Modern Analysis; §12. 43Google Scholar.

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, §§ 101–107Google Scholar.