§29.10 Lamé Functions with Imaginary Periods

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Keywords:: Lamé functions, with imaginary periods
Permalink:: http://dlmf.nist.gov/29.10
See also:: Annotations for Ch.29

The substitutions

29.10.1		$h=\nu(\nu+1)-h^{\prime},$
ⓘ Symbols: $h$ : real parameter and $\nu$ : real parameter Permalink: http://dlmf.nist.gov/29.10.E1 Encodings: TeX, pMML, png See also: Annotations for §29.10 and Ch.29

29.10.2		$z^{\prime}=\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),$
ⓘ Symbols: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $k$ : real parameter Permalink: http://dlmf.nist.gov/29.10.E2 Encodings: TeX, pMML, png See also: Annotations for §29.10 and Ch.29

transform (29.2.1) into

29.10.3		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z^{\prime}}^{2}}+(h^{\prime}-\nu(\nu+1){k^% {\prime}}^{2}{\operatorname{sn}}^{2}\left(z^{\prime},k^{\prime}\right))w=0.$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : solution Permalink: http://dlmf.nist.gov/29.10.E3 Encodings: TeX, pMML, png See also: Annotations for §29.10 and Ch.29

In consequence, the functions

29.10.4		$\mathit{Ec}^{2m}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{\prime}% }^{2}\right),$
		$\mathit{Ec}^{2m+1}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$
		$\mathit{Es}^{2m+1}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$
		$\mathit{Es}^{2m+2}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$
ⓘ 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, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $m$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $\nu$ : real parameter Permalink: http://dlmf.nist.gov/29.10.E4 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §29.10 and Ch.29

are solutions of (29.2.1). The first and the fourth functions have period $2\mathrm{i}{K^{\prime}}$ ; the second and the third have period $4\mathrm{i}{K^{\prime}}$ .

For these results and further information see Erdélyi et al. (1955, §15.5.2).