Mathieu functions

In mathematics, the Mathieu functions are certain special functions useful for treating a variety of problems in applied mathematics, including:

They were introduced by Émile Léonard Mathieu (1868) in the context of the first problem.

The canonical form for Mathieu's differential equation is

The Mathieu equation is a Hill equation with only 1 harmonic mode.

Closely related is Mathieu's modified differential equation

which follows on substitution ${\displaystyle u=ix}$.

The two above equations can be obtained from the Helmholtz equation in two dimensions, by expressing it in elliptical coordinates and then separating the two variables.[1] This is why they are also known as angular and radial Mathieu equation, respectively.

The substitution ${\displaystyle t=\cos(x)}$ transforms Mathieu's equation to the algebraic form

This has two regular singularities at ${\displaystyle t=-1,1}$ and one irregular singularity at infinity, which implies that in general (unlike many other special functions), the solutions of Mathieu's equation cannot be expressed in terms of hypergeometric functions.

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