In mathematics, the Mathieu functions are certain special functions useful for treating a variety of problems in applied mathematics, including:
- vibrating elliptical drumheads,
- quadrupole mass analyzers and quadrupole ion traps for mass spectrometry
- wave motion in periodic media, such as ultracold atoms in an optical lattice
- the phenomenon of parametric resonance in forced oscillators,
- exact plane wave solutions in general relativity,
- the Stark effect for a rotating electric dipole,
- in general, the solution of differential equations that are separable in elliptic cylindrical coordinates.
They were introduced by Émile Léonard Mathieu (1868) in the context of the first problem.
The canonical form for Mathieu’s differential equation is
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