Floquet's theorem asserts that any solution to the Mathieu equation (Eq. 
) is of the form:[4,29,28]
where 
is either real or imaginary and 
is a 
-periodic function
of 
. This assertion implies that solutions of Eq. can be separated into the product of a term of the form 
and a series of harmonics, 
, each of which is 
-periodic in 
. For values of 
with a non-zero imaginary component and 
, Eq. indicates that:
Thus, 
, yields an unbounded solution to the Mathieu equation. On the other hand, values of 
which are purely real yield bounded, oscillatory solutions for 
.
There are two types of bounded solutions to Mathieu's equation: functions of fractional order and functions of integral order. Functions of fractional order are obtained for 
. The oscillatory periods of the frequencies associated with fractional-order Mathieu functions are non-integer multiples of 
. The special case of 
corresponds to an 
-periodic (in terms of 
) solution. These 
-periodic solutions are Mathieu functions of integral order.
All bounded solutions to the Mathieu equation--- those of fractional as well as integral order--- are described (in accordance with Floquet theory) by an infinite series of harmonic oscillations whose amplitudes decrease with increasing frequency:
The 
argument of 
in Eq. represents harmonics of frequency 
, while the 
term represents a (secular) frequency which is a function of the trap specifications (or, equivalently, of
the Mathieu parameters a, q). The Mathieu parameters 
and 
also determine the coefficients 
, which approach zero as n increases.[36]
