29 Lamé Functions Lamé Functions 29.8 Integral Equations 29.10 Lamé Functions with Imaginary Periods

§29.9 Stability

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Keywords:: Lamé’s equation, stability
Notes:: See Magnus and Winkler (1966, §2.1).
Permalink:: http://dlmf.nist.gov/29.9
See also:: Annotations for Ch.29

The Lamé equation (29.2.1) with specified values of $k,h,\nu$ is called stable if all of its solutions are bounded on $\mathbb{R}$ ; otherwise the equation is called unstable. If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\leq a^{0}_{\nu}\left(k^{2}\right)$ or lies in one of the closed intervals with endpoints $a^{m}_{\nu}\left(k^{2}\right)$ and $b^{m}_{\nu}\left(k^{2}\right)$ , $m=1,2,\dots$ . If $\nu$ is a nonnegative integer, then (29.2.1) is unstable iff $h\leq a^{0}_{\nu}\left(k^{2}\right)$ or $h\in[b^{m}_{\nu}\left(k^{2}\right),a^{m}_{\nu}\left(k^{2}\right)]$ for some $m=1,2,\dots,\nu$ .

29.8 Integral Equations 29.10 Lamé Functions with Imaginary Periods

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