XLIII. Solutions of Mathieu's Equation

Characteristic exponents of Mathieu functions

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The study of the properties and solutions of the Mathieu second-order differential equation has been very extensive in the last decades, due to the special interest presented by physical problems involving periodic media and problems separable in elliptic coordinate systems, On the other hand, numerical tables for the non-periodic Floquet-type solutions are scarce [5], [6], [7]; in addition, the increments between two successive tabulated values are relatively large for some of these tables and the accuracy is relatively low in the other tables. Consequently, the applicability of the available data is rather limited.

The regions of "stable" solutions of the Mathieu equation are of particular interest in applications involving the wave equation with periodic boundary conditions, since these regions correspond to the propagating waves in the corresponding media.

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Characteristic Exponents of Mathieu Functions

“…On the other hand, numerical tables for the non-periodic Floquet-type solutions are scarce [5], [6], [7]; in addition, the increments between two successive tabulated values are relatively large for some of these tables and the accuracy is relatively low in the other tables. Consequently, the applicability of the available data is rather limited.…”

“…For computation purposes, it is convenient to define (5) x -v -p so that equation 3will take the form…”