Lamé equation

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1: 29.19 Physical Applications
§29.19(i) Lamé Functions
Simply-periodic Lamé functions ($\nu$ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …
2: 29.11 Lamé Wave Equation
§29.11 Lamé Wave Equation
The Lamé (or ellipsoidal) wave equation is given by …In the case $\omega=0$, (29.11.1) reduces to Lamé’s equation (29.2.1). …
3: 29.9 Stability
§29.9 Stability
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. …
4: 29.2 Differential Equations
§29.2(ii) Other Forms
we have …For the Weierstrass function $\wp$ see §23.2(ii). …
5: 29.3 Definitions and Basic Properties
29.3.2 $a^{m}_{\nu}\left(k^{2}\right)
satisfies the continued-fraction equation
6: 29.7 Asymptotic Expansions
29.7.1 $a^{m}_{\nu}\left(k^{2}\right)\sim p\kappa-\tau_{0}-\tau_{1}\kappa^{-1}-\tau_{2% }\kappa^{-2}-\cdots,$
29.7.5 $b^{m+1}_{\nu}\left(k^{2}\right)-a^{m}_{\nu}\left(k^{2}\right)=O\left(\nu^{m+% \frac{3}{2}}\left(\frac{1-k}{1+k}\right)^{\nu}\right),$ $\nu\to\infty$.
Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation28.29(i)) that are applicable to the Lamé equation.
8: 29.1 Special Notation
$(s_{\nu}^{m}(k^{2}))^{2}=\frac{4}{\pi}\int_{0}^{K}\left(\mathit{Es}^{m}_{\nu}% \left(x,k^{2}\right)\right)^{2}\mathrm{d}x.$
9: 29.5 Special Cases and Limiting Forms
29.5.1 $a^{m}_{\nu}\left(0\right)=b^{m}_{\nu}\left(0\right)=m^{2},$
29.5.4 $\lim_{k\to 1-}a^{m}_{\nu}\left(k^{2}\right)=\lim_{k\to 1-}b^{m+1}_{\nu}\left(k% ^{2}\right)=\nu(\nu+1)-\mu^{2},$