29 Lamé Functions Lamé Functions 29.4 Graphics 29.6 Fourier Series

§29.5 Special Cases and Limiting Forms

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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 $a^{m}_{\nu}\left(0\right)=b^{m}_{\nu}\left(0\right)=m^{2},$

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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, : 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 $\mathit{Ec}^{0}_{\nu}\left(z,0\right)=2^{-\frac{1}{2}},$

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Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : 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

$\mathit{Ec}^{m}_{\nu}\left(z,0\right)=\cos\left(m(\tfrac{1}{2}\pi-z)\right),$ $m\geq 1$ ,

$\mathit{Es}^{m}_{\nu}\left(z,0\right)=\sin\left(m(\tfrac{1}{2}\pi-z)\right),$ $m\geq 1$ .

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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, : nonnegative integer, : 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 $\lim_{k\to 1-}a^{m}_{\nu}\left(k^{2}\right)=\lim_{k\to 1-}b^{m+1}_{\nu}\left(k% ^{2}\right)=\nu(\nu+1)-\mu^{2},$

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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, : nonnegative integer, : 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 ${\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\mathit{Ec}^{m% }_{\nu}\left(0,k^{2}\right)}=\lim_{k\to 1-}\frac{\mathit{Es}^{m+1}_{\nu}\left(% z,k^{2}\right)}{\mathit{Es}^{m+1}_{\nu}\left(0,k^{2}\right)}}=\frac{1}{(\cosh z% )^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu+\tfrac{1}{2}% \nu+\tfrac{1}{2}\atop\tfrac{1}{2}};{\tanh}^{2}z\right),$

even,

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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, : nonnegative integer, : complex variable, : 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 $\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\left.\ifrac{% \mathrm{d}\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\mathrm{d}z}\right|_{z=0}% }=\lim_{k\to 1-}\frac{\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\left.% \ifrac{\mathrm{d}\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\mathrm{d}z}% \right|_{z=0}}=\frac{\tanh z}{(\cosh z)^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1% }{2}\nu+\tfrac{1}{2},\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+1\atop\tfrac{3}{2}};{% \tanh}^{2}z\right),$

odd,

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