Mathieu function

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

vibrating elliptical drumheads,
quadrupoles mass filters and quadrupole ion traps for mass spectrometry
wave motion in periodic media, such as ultracold atoms in an optical lattice
the phenomenon of parametric resonance in forced oscillators,
exact plane wave solutions in general relativity,
the Stark effect for a rotating electric dipole,
in general, the solution of differential equations that are separable in elliptic cylindrical coordinates.

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

1 Mathieu equation
2 Floquet solution
3 Mathieu sine and cosine
4 Periodic solutions
5 See also
6 References
7 External links

Mathieu equation

The canonical form for Mathieu's differential equation is

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

Closely related is Mathieu's modified differential equation

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

which follows on substitution $u=ix$ .

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

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

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 as models in many contexts, including the stability of railroad rails as trains drive over them, seasonally forced population dynamics, the four-dimensional wave equation, optimal paths for punt returns, and the Floquet theory of the stability of limit cycles.

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 $\mu$ is a complex number, the Mathieu exponent, and P is a complex valued function which is periodic in $x$ with period $\pi$ . However, P is in general not sinusoidal. In the example plotted below, $a=1, \, q=\frac{1}{5}, \, \mu \approx 1 %2B 0.0995 i$ (real part, red; imaginary part; green):

Mathieu sine and cosine

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

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

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

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

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

$C(a,q,x) = \frac{F(a,q,x) %2B F(a,q,-x)}{2 F(a,q,0)}$

$S(a,q,x) = \frac{F(a,q,x) - F(a,q,-x)}{2 F^\prime(a,q,0)}.$

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

$C(a,q,x) \approx \cos(\sqrt{a} x), \; \; S(a,q,x) \approx \frac{\sin (\sqrt{a} x)}{\sqrt{a}}.$

For example:

Periodic solutions

Given $q$ , for countably many special values of $a$ , called characteristic values, the Mathieu equation admits solutions which are periodic with period $2\pi$ . 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 $\pi$ ). Therefore, for positive q, we have