Mathieu function

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation

${\displaystyle {\frac {d^{2}y}{dx^{2}}}+(a-2q\cos(2x))y=0,}$

where a, q are real-valued parameters. Since we may add π/2 to x to change the sign of q, it is a usual convention to set q ≥ 0.

They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads.^[1][2][3] 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 (PDE) boundary value problems possessing elliptic symmetry.^[4]