# §29.7 Asymptotic Expansions

## §29.7(i) Eigenvalues

As $\nu\to\infty$,

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

where

 29.7.2 $\displaystyle\kappa$ $\displaystyle=k(\nu(\nu+1))^{1/2},$ $\displaystyle p$ $\displaystyle=2m+1,$ ⓘ Defines: $\kappa$ (locally) Symbols: $m$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.7.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 29.7(i), 29.7 and 29
 29.7.3 $\displaystyle\tau_{0}$ $\displaystyle=\frac{1}{2^{3}}(1+k^{2})(1+p^{2}),$ ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter and $\tau_{j}$: coefficients Permalink: http://dlmf.nist.gov/29.7.E3 Encodings: TeX, pMML, png See also: Annotations for 29.7(i), 29.7 and 29 29.7.4 $\displaystyle\tau_{1}$ $\displaystyle=\frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k^{2}(p^{2}+5)).$ ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter and $\tau_{j}$: coefficients Permalink: http://dlmf.nist.gov/29.7.E4 Encodings: TeX, pMML, png See also: Annotations for 29.7(i), 29.7 and 29

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

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

 29.7.6 $\tau_{2}=\frac{1}{2^{10}}(1+k^{2})(1-k^{2})^{2}(5p^{4}+34p^{2}+9),$ ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter and $\tau_{j}$: coefficients Permalink: http://dlmf.nist.gov/29.7.E6 Encodings: TeX, pMML, png See also: Annotations for 29.7(i), 29.7 and 29
 29.7.7 $\tau_{3}=\frac{p}{2^{14}}((1+k^{2})^{4}(33p^{4}+410p^{2}+405)-24k^{2}(1+k^{2})% ^{2}(7p^{4}+90p^{2}+95)+16k^{4}(9p^{4}+130p^{2}+173)),$ ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter and $\tau_{j}$: coefficients Permalink: http://dlmf.nist.gov/29.7.E7 Encodings: TeX, pMML, png See also: Annotations for 29.7(i), 29.7 and 29
 29.7.8 $\tau_{4}=\frac{1}{2^{16}}((1+k^{2})^{5}(63p^{6}+1260p^{4}+2943p^{2}+486)-8k^{2% }(1+k^{2})^{3}(49p^{6}+1010p^{4}+2493p^{2}+432)+16k^{4}(1+k^{2})(35p^{6}+760p^% {4}+2043p^{2}+378)).$ ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter and $\tau_{j}$: coefficients Permalink: http://dlmf.nist.gov/29.7.E8 Encodings: TeX, pMML, png See also: Annotations for 29.7(i), 29.7 and 29
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 $\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)$ and $\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)$. Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.