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The Mathieu functions are used in treating problems involving [[parametric resonance]], vibrating elliptical drumheads, and gravitational waves in [[general relativity]], among other applications. They were introduced by [[Emile Mathieu]] in [[1868]] in the context of the second problem.
==Definition==
==Floquet solution==▼
The substitution <math>x \rightarrow \arccos(x)</math> transforms Matheiu's equation to the ''rational form''
:<math> y^{\prime \prime} = \frac{x}{1-x^2} \, y^\prime + \frac{( 4x^2-2) \, q -a}{1-x^2} \, y</math>
This has two regular singularities at x = -1,1 and one irregularity singularity at infinity, whic impleis that in general (unlike many other [[special functions]]), the solutions of Mathieu's equation ''cannot'' be expressed in terms of [[hypergeometric functions]].
According to [[Floquet's theorem]], for fixed values of a,q, Mathieu's equation admits a ''complex valued'' solution of form
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