Mathieu function

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

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

The Mathieu equation and its solutions are used in treating parametric resonance. The equation and function are named after Emile Mathieu.

The Mathieu functions are analogous to sine and cosine, but have different periods.

Definition

The Mathieu cosine ${\displaystyle MC(a,b,\xi )}$  is the unique solution of the Mathieu equation which is

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

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

1. an odd function,
2. takes the value

Symbolic computation engines

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

External links

• Mathworld at Wolfram
• Emile Mathieu biography at St. Andrews University
• EqWorld offers a useful page on the Mathieu equation.

References

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