Mathieu function

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation 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. 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. In some usages, Mathieu function refers to solutions of the Mathieu differential equation for arbitrary values of and . When no confusion can arise, other authors use the term to refer specifically to - or -periodic solutions, which exist only for special values of and . More precisely, for given (real) such periodic solutions exist for an infinite number of values of , called characteristic numbers, conventionally indexed as two separate sequences and , for . The corresponding functions are denoted and , respectively. They are sometimes also referred to as cosine-elliptic and sine-elliptic, or Mathieu functions of the first kind. As a result of assuming that is real, both the characteristic numbers and associated functions are real-valued. and can be further classified by parity and periodicity (both with respect to ), as follows: {| class="wikitable"