# §28.17 Stability as $x\to\pm\infty$

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

For real $a$ and $q$ $(\neq 0)$ the stable regions are the open regions indicated in color in Figure 28.17.1. The boundary of each region comprises the characteristic curves $a=\mathop{a_{n}\/}\nolimits\!\left(q\right)$ and $a=\mathop{b_{n}\/}\nolimits\!\left(q\right)$; compare Figure 28.2.1.