§29.7 Asymptotic Expansions

As $\nu \to \mathrm{\infty }$,

29.7.1		$$a_{\nu }^m\left(k^2\right)\sim p\kappa -{\tau }_0-{\tau }_1{\kappa }^{-1}-{\tau }_2{\kappa }^{-2}-\mathrm{\cdots },$$
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $\sim$: Poincaré asymptotic expansion, $m$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, $\kappa$ and ${\tau }_j$: coefficients Permalink: http://dlmf.nist.gov/29.7.E1 Encodings: TeX, pMML, png See also: Annotations for §29.7(i), §29.7 and Ch.29

where

29.7.2		$\kappa$	$=k{(\nu (\nu +1))}^{1/2},$
		$p$	$=2m+1,$
ⓘ 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 Ch.29

29.7.3		${\tau }_0$	$={\displaystyle \frac{1}{2^3}}(1+k^2)(1+p^2),$
ⓘ 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 Ch.29
29.7.4		${\tau }_1$	$={\displaystyle \frac{p}{2^6}}({(1+k^2)}^2(p^2+3)-4k^2(p^2+5)).$
ⓘ 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 Ch.29

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

29.7.5		$$b_{\nu }^{m+1}\left(k^2\right)-a_{\nu }^m\left(k^2\right)=O\left({\nu }^{m+\frac{3}{2}}{\left(\frac{1-k}{1+k}\right)}^{\nu }\right),$$
$\nu \to \mathrm{\infty }$.
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Referenced by: §29.7(i) Permalink: http://dlmf.nist.gov/29.7.E5 Encodings: TeX, pMML, png See also: Annotations for §29.7(i), §29.7 and Ch.29

See also Volkmer (2004b).

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 Ch.29

29.7.7		$${\tau }_3=\frac{p}{2^{14}}(\begin{array}{l}{(1+k^2)}^4(33p^4+410p^2+405)-24k^2{(1+k^2)}^2(7p^4+90p^2+95)\\ +16k^4(9p^4+130p^2+173)),\end{array}$$
ⓘ 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 Ch.29

29.7.8		$${\tau }_4=\frac{1}{2^{16}}(\begin{array}{l}\begin{array}{l}{(1+k^2)}^5(63p^6+1260p^4+2943p^2+486)\\ -8k^2{(1+k^2)}^3(49p^6+1010p^4+2493p^2+432)\end{array}\\ +16k^4(1+k^2)(35p^6+760p^4+2043p^2+378)).\end{array}$$
ⓘ 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 Ch.29

Formulas for additional terms can be computed with the author’s Maple program LA5; see §29.22.

Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as $\nu \to \mathrm{\infty }$, 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 ${\mathit{Ec}}_{\nu }^m(z,k^2)$ and ${\mathit{Es}}_{\nu }^m(z,k^2)$. Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.