Mathieu function

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

vibrating elliptical drumheads,
the phenomenon of parametric resonance in forced oscillators,
exact plane wave solutions in general relativity.
the Stark effect for a rotating electric dipole.

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

1 Mathieu equation
2 Floquet solution
3 Mathieu sine and cosine
4 Periodic solutions
5 Symbolic computation engines
6 See also
7 References

[edit] Mathieu equation

The canonical form for Mathieu's differential equation is

$\frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0$

Closely related is Mathieu's modified differential equation

$\frac{d^2y}{du^2}-[a-2q\cosh (2u) ]y=0$

which follows on substitution $u = i x$ .

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

$(1-t^2)\frac{d^2y}{dt^2} - t\, \frac{d y}{dt} + (a + 2q (1- 2t^2)) \, y=0$

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.

[edit] Floquet solution

According to Floquet's theorem (or Bloch's theorem), for fixed values of a,q, Mathieu's equation admits a complex valued solution of form

$F(a,q,x) = \exp(i \mu \,x) \, P(a,q,x)$

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, $a=1, \, q=\frac{1}{5}, \, \mu \approx 1 + 0.0995 i$ (real part, red; imaginary part; green):

[edit] Mathieu sine and cosine

For fixed a,q, the Mathieu cosine $C (a, q,ξ)$ is a function of $ξ$ defined as the unique solution of the Mathieu equation which

takes the value $C (a, q,0) = 1$ ,
is an even function, or equivalently $C^\prime(a,q,0)=0$ .

Similarly, the Mathieu sine $S (a, q,ξ)$ is the unique solution which

takes the value $S (a, q,0) = 0$ ,
is an odd function, or equivalently $S^\prime(a,q,0)=1$ .

These are real-valued functions which are closely related to the Floquet solution:

$C(a,q,x) = \frac{\exp(i \mu)}{F(a,q,0)} \, \frac{P(a,q,x) + P(a,q,-x)}{2}$

$S(a,q,x) = \frac{\exp(i \mu)}{F(a,q,0)} \, \frac{P(a,q,x) - P(a,q,-x)}{2}$

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

$C(a,0,x) = \cos(\sqrt{a} x), \; S(a,0,x) = \frac{\sin(\sqrt{a} x)}{\sqrt{a}}$

In general, the Mathieu sine and cosine are aperiodic. Nonetheless, for small values of q, we have approximately

$C(a,q,x) \approx \cos(\sqrt(a) x), \; \; C^\prime(a,q,x) \approx \sqrt(a) \cos (\sqrt(a) x)$

For example:

Red: C(0.3,0.1,x).

Red: C'(0.3,0.1,x).

[edit] Periodic solutions

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 $a_n(q), \, b_n(q)$ , where n is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written $CE(n,q,x), \, SE(n,q,x)$ respectively, although they are traditionally given a different normalization (namely, that their L² norm equal $π$ ). Therefore, for positive q, we have