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.

Mathieu sine and cosine

File:MatheiuCosine.gif

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

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

1. is an even function,
2. takes the value ${\displaystyle MC(a,q,0)=1}$ .

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

1. is an odd function,
2. takes the value ${\displaystyle MS(a,q,0)=0}$ .

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

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