# §29.7(i) Eigenvalues

As $\nu\to\infty$,

 29.7.1 $\mathop{a^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)\sim p\kappa-\tau_{0}-\tau_% {1}\kappa^{-1}-\tau_{2}\kappa^{-2}-\cdots,$

where

 29.7.2 $\displaystyle\kappa$ $\displaystyle=k(\nu(\nu+1))^{1/2},$ $\displaystyle p$ $\displaystyle=2m+1,$ Symbols: $m$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $\kappa$ Permalink: http://dlmf.nist.gov/29.7.E2 Encodings: TeX, TeX, pMML, pMML, png, png
 29.7.3 $\displaystyle\tau_{0}$ $\displaystyle=\frac{1}{2^{3}}(1+k^{2})(1+p^{2}),$ 29.7.4 $\displaystyle\tau_{1}$ $\displaystyle=\frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k^{2}(p^{2}+5)).$

The same Poincaré expansion holds for $\mathop{b^{m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, since

 29.7.5 $\mathop{b^{m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\mathop{a^{m}_{\nu}\/}% \nolimits\!\left(k^{2}\right)=\mathop{O\/}\nolimits\!\left(\nu^{m+\frac{3}{2}}% \left(\frac{1-k}{1+k}\right)^{\nu}\right),$ $\nu\to\infty$.

 29.7.6 $\tau_{2}=\frac{1}{2^{10}}(1+k^{2})(1-k^{2})^{2}(5p^{4}+34p^{2}+9),$
 29.7.7 $\tau_{3}=\frac{p}{2^{14}}((1+k^{2})^{4}(33p^{4}+410p^{2}+405)-24k^{2}(1+k^{2})% ^{2}(7p^{4}+90p^{2}+95)+16k^{4}(9p^{4}+130p^{2}+173)),$
 29.7.8 $\tau_{4}=\frac{1}{2^{16}}((1+k^{2})^{5}(63p^{6}+1260p^{4}+2943p^{2}+486)-8k^{2% }(1+k^{2})^{3}(49p^{6}+1010p^{4}+2493p^{2}+432)+16k^{4}(1+k^{2})(35p^{6}+760p^% {4}+2043p^{2}+378)).$
Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as $\nu\to\infty$, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions $\mathop{\mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ and $\mathop{\mathit{Es}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$. Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.