28 Mathieu Functions and Hill’s EquationMathieu Functions of Noninteger Order28.16 Asymptotic Expansions for Large $q$28.18 Integrals and Integral Equations

If all solutions of (28.2.1) are bounded when $x\to \pm \mathrm{\infty}$
along the real axis, then the corresponding pair of parameters $\left(a,q\right)$ is
called *stable*. All other pairs are *unstable*.

For example, positive real values of $a$ with $q=0$ comprise stable pairs, as do values of $a$ and $q$ that correspond to real, but noninteger, values of $\nu $.

However, if $\mathrm{\Im}\nu \ne 0$, then $\left(a,q\right)$ always comprises an unstable pair. For example, as $x\to +\mathrm{\infty}$ one of the solutions ${\mathrm{me}}_{\nu}\left(x,q\right)$ and ${\mathrm{me}}_{\nu}\left(-x,q\right)$ tends to $0$ and the other is unbounded (compare Figure 28.13.5). Also, all nontrivial solutions of (28.2.1) are unbounded on $\mathrm{\mathbb{R}}$.