§29.5 Special Cases and Limiting Forms

ⓘ

29.5.1		$a^{m}_{\nu}\left(0\right)=b^{m}_{\nu}\left(0\right)=m^{2},$
ⓘ Symbols: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : 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		$\mathit{Ec}^{0}_{\nu}\left(z,0\right)=2^{-\frac{1}{2}},$
ⓘ Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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		$\displaystyle\mathit{Ec}^{m}_{\nu}\left(z,0\right)$	$\displaystyle=\cos\left(m(\tfrac{1}{2}\pi-z)\right),$
	$m\geq 1$ ,
		$\displaystyle\mathit{Es}^{m}_{\nu}\left(z,0\right)$	$\displaystyle=\sin\left(m(\tfrac{1}{2}\pi-z)\right),$
	$m\geq 1$ .
ⓘ Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : 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		$\lim_{k\to 1-}a^{m}_{\nu}\left(k^{2}\right)=\lim_{k\to 1-}b^{m+1}_{\nu}\left(k% ^{2}\right)=\nu(\nu+1)-\mu^{2},$
ⓘ Symbols: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : 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		${\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\mathit{Ec}^{m% }_{\nu}\left(0,k^{2}\right)}=\lim_{k\to 1-}\frac{\mathit{Es}^{m+1}_{\nu}\left(% z,k^{2}\right)}{\mathit{Es}^{m+1}_{\nu}\left(0,k^{2}\right)}}=\frac{1}{(\cosh z% )^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu+\tfrac{1}{2}% \nu+\tfrac{1}{2}\atop\tfrac{1}{2}};{\tanh}^{2}z\right),$
$m$ even,
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : 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		$\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\left.\ifrac{% \mathrm{d}\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\mathrm{d}z}\right\|_{z=0}% }=\lim_{k\to 1-}\frac{\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\left.% \ifrac{\mathrm{d}\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\mathrm{d}z}% \right\|_{z=0}}=\frac{\tanh z}{(\cosh z)^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1% }{2}\nu+\tfrac{1}{2},\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+1\atop\tfrac{3}{2}};{% \tanh}^{2}z\right),$
$m$ odd,
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : 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 $\nu\to\infty$ in such a way that $k^{2}\nu(\nu+1)=4\theta$ (a positive constant), then

29.5.7	$\displaystyle\lim\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)$	$\displaystyle=\operatorname{ce}_{m}\left(\tfrac{1}{2}\pi-z,\theta\right),$
29.5.7	$\displaystyle\lim\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)$	$\displaystyle=\operatorname{se}_{m}\left(\tfrac{1}{2}\pi-z,\theta\right),$
ⓘ Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $\operatorname{ce}_{m}\left(z,\theta\right)$ and $\operatorname{se}_{m}\left(z,\theta\right)$ are Mathieu functions; see §28.2(vi).