On the convergence of WKB approximations of the damped Mathieu equation

Abstract

The form of the fundamental set of solutions of the damped Mathieu equation is determined by Floquet theory. In the limit as m → 0, we can apply WKB theory to get first order approximations of the fundamental set. WKB theory states that this approximation gets better as m → 0 and T is fixed. However, convergence of the periodic part and characteristic exponent is not addressed. We show that they converge to those predicted by WKB theory. We also provide a rate of convergence that is not dependent on T.