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.

Mathieu equation

The canonical form for Mathieu's differential equation is

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

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

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 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. This is why they are also known as angular and radial Mathieu equation, respectively.

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 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, 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 + 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

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

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

  1. takes the value S^\prime(a,q,0)=1,
  2. 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) + 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}},

i.e. when the corresponding Helmholtz equation problem has circular symmetry.

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:

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


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

C \left( a_n(q),q,x \right) = \frac{CE(n,q,x)}{CE(n,q,0)}
S \left( b_n(q),q,x \right) = \frac{SE(n,q,x)}{SE^\prime(n,q,0)}.

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.

See also

References

External links