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Mathieu Functions

 

Mathieu functions appear frequently in physical problems involving elliptical shapes or periodic potentials. These functions were first introduced by Mathieu (1868) when analyzing the solutions to the equation y''+a2qcos2zy=0, which arises from the separation of the 2-D or 3-D wave equation modeling the motion of an elliptic membrane. The rational form of Mathieu's equation has two regular singularities and one irregular singularity; hence, Mathieu functions are perhaps the simplest class of special functions (Heun type), which are not essentially hypergeometric. The Maple implementation of Mathieu functions includes: MathieuC and MathieuS, representing the solution to Mathieu's equation; MathieuCE and MathieuSE representing the periodic cases; MathieuFloquet representing the Floquet type solutions, and the set of auxiliary functions MathieuA, MathieuB, and MathieuExponent relating the parameters (a,q) entering Mathieu's equation.

 

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