Mathieu Functions Examples - Maple Help

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 $\mathrm{y\text{'}\text{'}}+\left(a-2q\mathrm{cos}\left(2z\right)\right)y=0$, 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.
 > $\mathrm{restart}$