Mathieu Function - Mathieu Sine and Cosine

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

1. takes the value ,
2. is an even function, hence .

Similarly, the Mathieu sine is the unique solution which

1. takes the value ,
2. 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: