§29.3 Definitions and Basic Properties

For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. They are denoted by $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, where $m=0,1,2,\mathrm{\dots }$; see Table 29.3.1.

The eigenvalues interlace according to

29.3.1		$a_{\nu }^m\left(k^2\right)$
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.3.E1 Encodings: TeX, pMML, png See also: Annotations for §29.3(ii), §29.3 and Ch.29
29.3.2		$a_{\nu }^m\left(k^2\right)$
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: 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 Permalink: http://dlmf.nist.gov/29.3.E2 Encodings: TeX, pMML, png See also: Annotations for §29.3(ii), §29.3 and Ch.29
29.3.3		$b_{\nu }^m\left(k^2\right)$
ⓘ Symbols: $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.3.E3 Encodings: TeX, pMML, png See also: Annotations for §29.3(ii), §29.3 and Ch.29
29.3.4		$b_{\nu }^m\left(k^2\right)$
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: 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 Permalink: http://dlmf.nist.gov/29.3.E4 Encodings: TeX, pMML, png See also: Annotations for §29.3(ii), §29.3 and Ch.29

The eigenvalues coalesce according to

29.3.5		$$a_{\nu }^m\left(k^2\right)=b_{\nu }^m\left(k^2\right),$$
$\nu =0,1,\mathrm{\dots },m-1$.
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: 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 Permalink: http://dlmf.nist.gov/29.3.E5 Encodings: TeX, pMML, png See also: Annotations for §29.3(ii), §29.3 and Ch.29

If $\nu$ is distinct from $0,1,\mathrm{\dots },m-1$, then

29.3.6		$$\left(a_{\nu }^m\left(k^2\right)-b_{\nu }^m\left(k^2\right)\right)\nu (\nu -1)\mathrm{\cdots }(\nu -m+1)>0.$$
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: 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: §29.3(ii) Permalink: http://dlmf.nist.gov/29.3.E6 Encodings: TeX, pMML, png See also: Annotations for §29.3(ii), §29.3 and Ch.29

If $\nu$ is a nonnegative integer, then

29.3.7		$$a_{\nu }^m\left(k^2\right)+a_{\nu }^{\nu -m}\left(1-k^2\right)=\nu (\nu +1),$$
$m=0,1,\mathrm{\dots },\nu$,
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.3.E7 Encodings: TeX, pMML, png See also: Annotations for §29.3(ii), §29.3 and Ch.29

29.3.8		$$b_{\nu }^m\left(k^2\right)+b_{\nu }^{\nu -m+1}\left(1-k^2\right)=\nu (\nu +1),$$
$m=1,2,\mathrm{\dots },\nu$.
ⓘ Symbols: $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.3.E8 Encodings: TeX, pMML, png See also: Annotations for §29.3(ii), §29.3 and Ch.29

For the special case $k=k^{\prime }=1/\sqrt{2}$ see Erdélyi et al. (1955, §15.5.2).

The quantity

29.3.9		$$H=2a_{\nu }^{2m}\left(k^2\right)-\nu (\nu +1)k^2$$
ⓘ Defines: $H$ (locally) Symbols: $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Referenced by: §29.3(iii) Permalink: http://dlmf.nist.gov/29.3.E9 Encodings: TeX, pMML, png See also: Annotations for §29.3(iii), §29.3 and Ch.29

satisfies the continued-fraction equation

29.3.10		$${\beta }_p-H-\frac{{\alpha }_{p-1}{\gamma }_p}{{\beta }_{p-1}-H-}\frac{{\alpha }_{p-2}{\gamma }_{p-1}}{{\beta }_{p-2}-H-}\mathrm{\cdots }=\frac{{\alpha }_p{\gamma }_{p+1}}{{\beta }_{p+1}-H-}\frac{{\alpha }_{p+1}{\gamma }_{p+2}}{{\beta }_{p+2}-H-}\mathrm{\cdots },$$
ⓘ Symbols: $p$: nonnegative integer, $H$, ${\alpha }_p$, ${\beta }_p$ and ${\gamma }_p$ Referenced by: §29.20(i), §29.3(iii), §29.3(iii), §29.3(iii), §29.3(iii) Permalink: http://dlmf.nist.gov/29.3.E10 Encodings: TeX, pMML, png See also: Annotations for §29.3(iii), §29.3 and Ch.29

where $p$ is any nonnegative integer, and

29.3.11
ⓘ Defines: ${\alpha }_p$ (locally) Symbols: $p$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Referenced by: §29.15(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.3.E11 Encodings: TeX, pMML, png See also: Annotations for §29.3(iii), §29.3 and Ch.29

29.3.12		${\beta }_p$	$=4p^2(2-k^2),$
		${\gamma }_p$	$=\frac{1}{2}(\nu -2p+2)(\nu +2p-1)k^2.$
ⓘ Defines: ${\beta }_p$ (locally) and ${\gamma }_p$ (locally) Symbols: $p$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Referenced by: §29.15(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.3.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §29.3(iii), §29.3 and Ch.29

The continued fraction following the second negative sign on the left-hand side of (29.3.10) is finite: it equals 0 if $p=0$, and if $p>0$, then the last denominator is ${\beta }_0-H$. If $\nu$ is a nonnegative integer and $2p\le \nu$, then the continued fraction on the right-hand side of (29.3.10) terminates, and (29.3.10) has only the solutions (29.3.9) with $2m\le \nu$. If $\nu$ is a nonnegative integer and $2p>\nu$, then (29.3.10) has only the solutions (29.3.9) with $2m>\nu$.

The quantity $H=2a_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with

29.3.13
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and ${\beta }_p$ Referenced by: §29.15(i), §29.6(ii) Permalink: http://dlmf.nist.gov/29.3.E13 Encodings: TeX, pMML, png See also: Annotations for §29.3(iii), §29.3 and Ch.29

29.3.14		${\alpha }_p$	$=\frac{1}{2}(\nu -2p-2)(\nu +2p+3)k^2,$
		${\gamma }_p$	$=\frac{1}{2}(\nu -2p+1)(\nu +2p)k^2.$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, ${\alpha }_p$ and ${\gamma }_p$ Referenced by: §29.15(i), §29.6(ii) Permalink: http://dlmf.nist.gov/29.3.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §29.3(iii), §29.3 and Ch.29

The quantity $H=2b_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with

29.3.15
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and ${\beta }_p$ Referenced by: §29.15(i), §29.6(iii) Permalink: http://dlmf.nist.gov/29.3.E15 Encodings: TeX, pMML, png See also: Annotations for §29.3(iii), §29.3 and Ch.29

29.3.16		${\alpha }_p$	$=\frac{1}{2}(\nu -2p-2)(\nu +2p+3)k^2,$
		${\gamma }_p$	$=\frac{1}{2}(\nu -2p+1)(\nu +2p)k^2.$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, ${\alpha }_p$ and ${\gamma }_p$ Referenced by: §29.15(i), §29.6(iii) Permalink: http://dlmf.nist.gov/29.3.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §29.3(iii), §29.3 and Ch.29

The quantity $H=2b_{\nu }^{2m+2}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with

29.3.17		${\alpha }_p$	$=\frac{1}{2}(\nu -2p-3)(\nu +2p+4)k^2,$
		${\beta }_p$	$={(2p+2)}^2(2-k^2),$
		${\gamma }_p$	$=\frac{1}{2}(\nu -2p)(\nu +2p+1)k^2.$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, ${\alpha }_p$, ${\beta }_p$ and ${\gamma }_p$ Referenced by: §29.15(i), §29.6(iv) Permalink: http://dlmf.nist.gov/29.3.E17 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §29.3(iii), §29.3 and Ch.29

The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by ${\mathit{Ec}}_{\nu }^{2m}(z,k^2)$, ${\mathit{Ec}}_{\nu }^{2m+1}(z,k^2)$, ${\mathit{Es}}_{\nu }^{2m+1}(z,k^2)$, ${\mathit{Es}}_{\nu }^{2m+2}(z,k^2)$. They are called Lamé functions with real periods and of order $\nu$, or more simply, Lamé functions. See Table 29.3.2. In this table the nonnegative integer $m$ corresponds to the number of zeros of each Lamé function in $(0,K)$, whereas the superscripts $2m$, $2m+1$, or $2m+2$ correspond to the number of zeros in $[0,2K)$.

29.3.18		${\displaystyle {\int }_0^K}\mathrm{dn}(x,k){\left({\mathit{Ec}}_{\nu }^{2m}(x,k^2)\right)}^{2}dx$	$={\displaystyle \frac{1}{4}}\pi ,$
		${\displaystyle {\int }_0^K}\mathrm{dn}(x,k){\left({\mathit{Ec}}_{\nu }^{2m+1}(x,k^2)\right)}^{2}dx$	$={\displaystyle \frac{1}{4}}\pi ,$
		${\displaystyle {\int }_0^K}\mathrm{dn}(x,k){\left({\mathit{Es}}_{\nu }^{2m+1}(x,k^2)\right)}^{2}dx$	$={\displaystyle \frac{1}{4}}\pi ,$
		${\displaystyle {\int }_0^K}\mathrm{dn}(x,k){\left({\mathit{Es}}_{\nu }^{2m+2}(x,k^2)\right)}^{2}dx$	$={\displaystyle \frac{1}{4}}\pi .$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathit{Ec}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathit{Es}}_{\nu }^m(z,k^2)$: Lamé function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$, $\int$: integral, $m$: nonnegative integer, $x$: real variable, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.3.E18 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §29.3(v), §29.3 and Ch.29

For $\mathrm{dn}(z,k)$ see §22.2.

To complete the definitions, ${\mathit{Ec}}_{\nu }^m(K,k^2)$ is positive and ${d{\mathit{Es}}_{\nu }^m(z,k^2)/dz|}_{z=K}$ is negative.

For $m\ne p$,

29.3.19		${\displaystyle {\int }_0^K}{\mathit{Ec}}_{\nu }^{2m}(x,k^2){\mathit{Ec}}_{\nu }^{2p}(x,k^2)dx$	$=0,$
		${\displaystyle {\int }_0^K}{\mathit{Ec}}_{\nu }^{2m+1}(x,k^2){\mathit{Ec}}_{\nu }^{2p+1}(x,k^2)dx$	$=0,$
		${\displaystyle {\int }_0^K}{\mathit{Es}}_{\nu }^{2m+1}(x,k^2){\mathit{Es}}_{\nu }^{2p+1}(x,k^2)dx$	$=0,$
		${\displaystyle {\int }_0^K}{\mathit{Es}}_{\nu }^{2m+2}(x,k^2){\mathit{Es}}_{\nu }^{2p+2}(x,k^2)dx$	$=0.$
ⓘ Symbols: ${\mathit{Ec}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathit{Es}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$, $\int$: integral, $m$: nonnegative integer, $p$: nonnegative integer, $x$: real variable, $k$: real parameter and $\nu$: real parameter Referenced by: §29.14 Permalink: http://dlmf.nist.gov/29.3.E19 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §29.3(vi), §29.3 and Ch.29

For the values of these integrals when $m=p$ see §29.6.

For power-series expansions of the eigenvalues see Volkmer (2004b).