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Mathieu function


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 .

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 transforms Mathieu's equation to the algebraic form

This has two regular singularities at 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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