29 Lamé FunctionsLamé Functions29.4 Graphics29.6 Fourier Series

§29.5 Special Cases and Limiting Forms

ⓘ

Keywords:

Lamé functions, eigenvalues, limiting forms, special cases

Notes:

See Erdélyi et al. (1955, §15.5.4) and Volkmer (2004b).

Permalink:

http://dlmf.nist.gov/29.5

See also:

Annotations for Ch.29

29.5.1		$$a_{\nu }^m\left(0\right)=b_{\nu }^m\left(0\right)=m^2,$$
ⓘ 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 and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.5.E1 Encodings: TeX, pMML, png See also: Annotations for §29.5 and Ch.29

29.5.2		$${\mathrm{𝐸𝑐}}_{\nu }^0(z,0)=2^{-\frac{1}{2}},$$
ⓘ Symbols: ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $z$: complex variable and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.5.E2 Encodings: TeX, pMML, png See also: Annotations for §29.5 and Ch.29

29.5.3		${\mathrm{𝐸𝑐}}_{\nu }^m(z,0)$	$=\cos \left(m(\frac{1}{2}\pi -z)\right),$
	$m\ge 1$,
		${\mathrm{𝐸𝑠}}_{\nu }^m(z,0)$	$=\sin \left(m(\frac{1}{2}\pi -z)\right),$
	$m\ge 1$.
ⓘ Symbols: ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $m$: nonnegative integer, $z$: complex variable and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.5.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §29.5 and Ch.29

Let $\mu =\max (\nu -m,0)$. Then

29.5.4		$$\underset{k\to 1-}{lim}{a}_{\nu }^m\left(k^2\right)=\underset{k\to 1-}{lim}{b}_{\nu }^{m+1}\left(k^2\right)=\nu (\nu +1)-{\mu }^2,$$
ⓘ 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, $\nu$: real parameter and $\mu$: parameter Permalink: http://dlmf.nist.gov/29.5.E4 Encodings: TeX, pMML, png See also: Annotations for §29.5 and Ch.29

29.5.5		$$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$
$m$ even,
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter Permalink: http://dlmf.nist.gov/29.5.E5 Encodings: TeX, pMML, png See also: Annotations for §29.5 and Ch.29

29.5.6		$$\begin{array}{l}\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}\\ =\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),\end{array}$$
$m$ odd,
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter Permalink: http://dlmf.nist.gov/29.5.E6 Encodings: TeX, pMML, png See also: Annotations for §29.5 and Ch.29

where $F$ is the hypergeometric function; see §15.2(i).

If $k\to 0+$ and $\nu \to \mathrm{\infty }$ in such a way that $k^2\nu (\nu +1)=4\theta$ (a positive constant), then

29.5.7		$lim{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$	$={\mathrm{ce}}_m(\frac{1}{2}\pi -z,\theta ),$
		$lim{\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$	$={\mathrm{se}}_m(\frac{1}{2}\pi -z,\theta ),$
ⓘ Symbols: ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\theta$: positive constant Permalink: http://dlmf.nist.gov/29.5.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §29.5 and Ch.29

where ${\mathrm{ce}}_m(z,\theta )$ and ${\mathrm{se}}_m(z,\theta )$ are Mathieu functions; see §28.2(vi).

29.4 Graphics29.6 Fourier Series

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