Mathieu - 3D plot of a mode with elliptical boundary conditions

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.[1]

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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