Mathieu & Maple, forever

Maple `help' tells us about a number of Mathieu related functions in her tool box, including:

The Mathieu functions MathieuC(a, q, x) and MathieuS(a, q, x) are 
solutions of the Mathieu differential equation.

MathieuC and MathieuS are even and odd functions of x, respectively. 


MathieuFloquet(a, q, x) is a Floquet solution of Mathieu's equation.
where nu = MathieuExponent(a, q) is the characteristic exponent and 
P(x) is a Pi periodic function.

We present a few plots to demonstrate the usefullness of these functions below. This report has treated, in some detail, the mathematics behind the ideal rf Paul trap. Of course, the actual realization of the trap differs in many important ways from its ideal, but we may approach these realizations, in their many forms, with a fundamental understanding of their operational basis.

Figure 6.1: On the left, the ion is trapped with secular and micromotion; on the right, unbound orbit, the ion is lost forever.

Figure 6.2: A bound orbit, different formal solution ce, and a few of its components

Figure 6.3: The components of the ce function

Figure 6.4: Fascinating behavior of $\mu$ as q is varried, a transition from bound to unbound behavior