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\frac{d^2y}{dx^2} + (a - 2q\cos(2x))y = 0,
</math>
where <math>a</math> and <math>q</math> are real-valued parameters. Since we may add <math>\pi</math> to <math>x</math> to change the sign of <math>q</math>, it is a usual convention to set <math>q \geq 0</math>.
They were first introduced by [[Émile Léonard Mathieu]], who encountered them while studying vibrating elliptical drumheads.<ref>Mathieu (1868).</ref><ref name="MorseandFeshbach">Morse and Feshbach (1953).</ref><ref name="BrimacombeCorlessZamir">Brimacombe, Corless and Zamir (2021)</ref> They have applications in many fields of the physical sciences, such as [[optics]], [[quantum mechanics]], and [[general relativity]]. They tend to occur in problems involving periodic motion, or in the analysis of [[partial differential equation]] [[boundary value problem]]s possessing [[elliptic]] symmetry.<ref name="Gutiérrez-Vega2015">Gutiérrez-Vega (2015).</ref> == Definition ==
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