§29.7 Asymptotic Expansions

Permalink:: http://dlmf.nist.gov/29.7

§29.7(i) Eigenvalues
§29.7(ii) Lamé Functions

§29.7(i) Eigenvalues

Notes:: See Ince (1940a) and Müller (1966a). For (29.7.5) see Volkmer (2004b).
Keywords:: Lamé functions
Referenced by:: §29.20(i), §29.22(i)
Permalink:: http://dlmf.nist.gov/29.7.i

As $\nu\to\infty$ ,

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

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\sim$ : asymptotic equality, $m$ : 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

where

29.7.2

$\kappa=k(\nu(\nu+1))^{{1/2}},$

$p=2m+1,$

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 $\tau _{0}=\frac{1}{2^{3}}(1+k^{2})(1+p^{2}),$

Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $\tau _{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.7.E3
Encodings:: TeX, pMML, png

29.7.4 $\tau _{1}=\frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k^{2}(p^{2}+5)).$

Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $\tau _{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.7.E4
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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 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

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

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

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

§29.7(ii) Lamé Functions

Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.7.ii

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.

§29.7 Asymptotic Expansions

Contents

§29.7(i) Eigenvalues

§29.7(ii) Lamé Functions