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