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]

Fri May 12 10:36:01 EDT 1995