Mathieu function

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

• problems parametric resonance,
• vibrating elliptical drumheads,
• gravitational waves in general relativity.

They were introduced by Emile Mathieu in 1868 in the context of the second problem.

Definition

Mathieu functions are, by definition, solutions to the Mathieu equation

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

The substitution ${\displaystyle x\rightarrow \arccos(x)}$  transforms Matheiu's equation to the rational form

${\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {x}{1-x^{2}}}\,{\frac {dy}{dx}}+{\frac {(4x^{2}-2)\,q-a}{1-x^{2}}}\,y}$ 

This has two regular singularities at x = -1,1 and one irregularity singularity at infinity, whic impleis that in general (unlike many other special functions), the solutions of Mathieu's equation cannot be expressed in terms of hypergeometric functions.

Floquet solution

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

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

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

 

Mathieu sine and cosine

For fixed a,q, the Mathieu cosine ${\displaystyle C(a,q,\xi )}$  is a function of ${\displaystyle \xi }$  defined as the unique solution of the Mathieu equation which

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

Similarly, the Mathieu sine ${\displaystyle S(a,q,\xi )}$  is the unique solution which

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

These are closely related to the Floquet solution:

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

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

For small values of q, these functions do resemble sine and cosine near the origin. In the example plotted below (same values of a,q as above), the Mathieu cosine is plotted in red and cosine in green:

File:MathieuCosine example.gif

For this example, the McLaurin series is

${\displaystyle C(1,{\frac {1}{5}},x)=1-{\frac {3}{10}}\,x^{2}-{\frac {31}{600}}\,x^{4}+{\frac {1613}{9000}}\,x^{6}+O(x^{8})}$ 

A noteworthy special case is

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

Periodic solutions

For countably many special values of a, called characteristic values, the Mathieu equation admits solutions which are periodic with period ${\displaystyle 2\pi }$ . The characteristic values of the Mathieu cosine, sine functions respectively are written ${\displaystyle a_{n}(q),\,b_{n}(q)}$ , where n in N. The periodic special cases of the Mathieu cosine and sine functions are often written ${\displaystyle CE(n,q,x),\,SE(n,q,x)}$  respectively. Here are the first few periodic Mathieu cosine functions:

 

Note that, for example, ${\displaystyle CE(1,1,x)}$  (green) resembles a cosine function, but with flatter hills and shallower valleys.

Symbolic computation engines

Various special functions related to the Mathieu functions are implemented in Maple and Mathematica.

See also

• Monochromatic electromagnetic plane wave, an example of an important exact plane wave solution to the Einstein field equation in general relativity which is expressed using Mathieu cosine functions.

External links

• Mathieu function at Mathworld (Wolfram Research).
• EqWorld offers a useful page on the Mathieu equation.

References

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