In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation where and are parameters. 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 boundary value problems possessing elliptic symmetry. Definition Mathieu functions 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" ! Function !! Parity !! Period |- | | even | |- | | even | |- | | odd | |- | | odd | |} The indexing with the integer , besides serving to arrange the characteristic numbers in ascending order, is convenient in that and become proportional to and as . With being an integer, this gives rise to the classification of and as Mathieu functions (of the first kind) of integral order. For general and , solutions besides these can be defined, including Mathieu functions of fractional order as well as non-periodic solutions. Modified Mathieu functions Closely related are the modified Mathieu functions, also known as radial Mathieu functions, which are solutions of Mathieu's modified differential equation which can be related to the original Mathieu equation by taking . Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by and , are defined from These functions are real-valued when is real. Normalization A common normalization, which will be adopted throughout this article, is to demand as well as require and as . Floquet theory Many properties of