Mathieu function

In mathematics, the Mathieu functions are solutions to the Mathieu differential equation, which is

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

The Mathieu functions are used in treating parametric resonance, and were introduced by Emile Mathieu.

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, ${\displaystyle a=1,\,q=0.2,\,\mu \approx 1+0.0995i}$ :

 

A Floquet solution F(1,0.2, x) to Mathieu's equation (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:

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

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

File:MathieuCosine.gif

Mathieu cosine C(1,0.2,x), in red, and cos(x) for comparison, in green.

Periodic solutions

For countably many special values of a (in terms of q), called eigenvalues, the Mathieu equation admits solutions which are periodic with period ${\displaystyle 2\pi }$ .

Symbolic computation engines

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

External links

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

References

• . LCCN 64016333. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help)