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

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Abstract
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.