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According to [[Floquet's theorem]], for fixed values of a,q, Mathieu's equation admits a ''complex valued'' solution of form
:<math>F(a,q,x) = \exp(i \mu) \, P(a,q,x)</math>
where <math>\mu</math> is a complex number, the ''Mathieu exponent'', and P is a complex valued function which is ''periodic'' with period <math>\pi</math>. However, P is in general ''not'' sinusoidal. In the example plotted below, <math>a=1, \, q=0.2, \, \mu \approx 1 + 0.0995 i</math>:
[[Image:MathieuFloquet.gif|frame|center|A Floquet solution F(1,0.2, x) to Mathieu's equation (real part, red; imaginary part, green).]]
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