Periodic Solutions
Given, for countably many special values of, called characteristic values, the Mathieu equation admits solutions which are periodic with period . The characteristic values of the Mathieu cosine, sine functions respectively are written, where n is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written respectively, although they are traditionally given a different normalization (namely, that their L2 norm equal ). Therefore, for positive q, we have
Here are the first few periodic Mathieu cosine functions for q = 1:
Note that, for example, (green) resembles a cosine function, but with flatter hills and shallower valleys.
Read more about this topic: Mathieu Function
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