Mathieu Sine and Cosine
For fixed a,q, the Mathieu cosine is a function of defined as the unique solution of the Mathieu equation which
- takes the value ,
- is an even function, hence .
Similarly, the Mathieu sine is the unique solution which
- takes the value ,
- is an odd function, hence .
These are real-valued functions which are closely related to the Floquet solution:
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
In general, the Mathieu sine and cosine are aperiodic. Nonetheless, for small values of q, we have approximately
For example:
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