§29.5 Special Cases and Limiting Forms

Notes:: See Erdélyi et al. (1955, §15.5.4) and Volkmer (2004b).
Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.5

29.5.1 $\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(0\right)=\mathop{b^{{m}}_{{\nu}}\/}\nolimits\!\left(0\right)=m^{2},$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.5.E1
Encodings:: TeX, pMML, png

29.5.2 $\mathop{\mathit{Ec}^{{0}}_{{\nu}}\/}\nolimits\!\left(z,0\right)=2^{{-\frac{1}{2}}},$

Symbols:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $z$ : complex variable and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.5.E2
Encodings:: TeX, pMML, png

29.5.3

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

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

Symbols:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

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

29.5.4 $\lim _{{k\to 1-}}\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)=\lim _{{k\to 1-}}\mathop{b^{{m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)=\nu(\nu+1)-\mu^{2},$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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, $\nu$ : real parameter and $\mu$ : parameter
Permalink:: http://dlmf.nist.gov/29.5.E4
Encodings:: TeX, pMML, png

29.5.5 ${\lim _{{k\to 1-}}\frac{\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)}{\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(0,k^{2}\right)}=\lim _{{k\to 1-}}\frac{\mathop{\mathit{Es}^{{m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)}{\mathop{\mathit{Es}^{{m+1}}_{{\nu}}\/}\nolimits\!\left(0,k^{2}\right)}}=\frac{1}{(\mathop{\cosh\/}\nolimits z)^{{\mu}}}\mathop{F\/}\nolimits\!\left({\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\atop\tfrac{1}{2}};{\mathop{\tanh\/}\nolimits^{{2}}}z\right),$ $m$ even,

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\tanh\/}\nolimits z$ : 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

29.5.6 $\lim _{{k\to 1-}}\frac{\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)}{\left.\ifrac{d\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)}{dz}\right|_{{z=0}}}=\lim _{{k\to 1-}}\frac{\mathop{\mathit{Es}^{{m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)}{\left.\ifrac{d\mathop{\mathit{Es}^{{m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)}{dz}\right|_{{z=0}}}=\frac{\mathop{\tanh\/}\nolimits z}{(\mathop{\cosh\/}\nolimits z)^{\mu}}\mathop{F\/}\nolimits\!\left({\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2},\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+1\atop\tfrac{3}{2}};{\mathop{\tanh\/}\nolimits^{{2}}}z\right),$ $m$ odd,

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\tanh\/}\nolimits z$ : 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

where $\mathop{F\/}\nolimits$ 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

$\lim\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)=\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(\tfrac{1}{2}\pi-z,\theta\right),$

$\lim\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)=\mathop{\mathrm{se}_{{m}}\/}\nolimits\!\left(\tfrac{1}{2}\pi-z,\theta\right),$

Symbols:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $m$ : 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

where $\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(z,\theta\right)$ and $\mathop{\mathrm{se}_{{m}}\/}\nolimits\!\left(z,\theta\right)$ are Mathieu functions; see §28.2(vi).