They were introduced by Émile Léonard Mathieu in 1868 in the context of the first problem.
The canonical form for Mathieu's differential equation is
Closely related is Mathieu's modified differential equation
which follows on substitution u = ix.
The substitution t = cos(x) transforms Mathieu's equation to the algebraic form
This has two regular singularities at 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.
Mathieu's differential equations arise when the four-dimensional wave equation is written in elliptic cylinder coordinates, followed by a separation of variables. In the algebraic form, it can be seen to be a special case of the spheroidal wave equation.
where μ is a complex number, the Mathieu exponent, and P is a complex valued function which is periodic with period π. However, P is in general not sinusoidal. In the example plotted below, (real part, red; imaginary part; green):
For fixed a,q, the Mathieu cosine C(a,q,x) is a function of x defined as the unique solution of the Mathieu equation which
Similarly, the Mathieu sine S(a,q,x) is the unique solution which
These are real-valued functions which are closely related to the Floquet solution:
The general solution to the Mathieu equation (for fixed a,q) is a linear combination of the Mathieu cosine and Mathieu sine functions.
A noteworthy special case is
In general, the Mathieu sine and cosine are aperiodic. Nonetheless, for small values of q, we have approximately
Given q, for countably many special values of a, called characteristic values, the Mathieu equation admits solutions which are periodic with period 2π. The characteristic values of the Mathieu cosine, sine functions respectively are written , where n is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written respectively, although they are traditionally given a different normalization (namely, that their L2 norm equal π). Therefore, for positive q, we have
Here are the first few periodic Mathieu cosine functions for q=1:
Note that, for example, CE(1,1,x) (green) resembles a cosine function, but with flatter hills and shallower valleys.
Redirecting to Mathieu function