Digital Library of Mathematical Functions
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28 Mathieu Functions and Hill’s EquationMathieu Functions of Noninteger Order

§28.17 Stability as x±

If all solutions of (28.2.1) are bounded when x± 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 ν.

However, if ν0, then (a,q) always comprises an unstable pair. For example, as x+ one of the solutions meν(x,q) and meν(-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 .

For real a and q (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=an(q) and a=bn(q); compare Figure 28.2.1.

See accompanying text
Figure 28.17.1: Stability chart for eigenvalues of Mathieu’s equation (28.2.1). Magnify