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29 Lamé Functions Lamé Functions 29.9 Stability 29.11 Lamé Wave Equation

§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 $h=\nu(\nu+1)-h^{\prime},$

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Symbols:: : 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 $z^{\prime}=\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),$

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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, : complex variable and : 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 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z^{\prime}}^{2}}+(h^{\prime}-\nu(\nu+1){k^% {\prime}}^{2}{\operatorname{sn}}^{2}\left(z^{\prime},k^{\prime}\right))w=0.$

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : complex variable, : real parameter, : real parameter, $\nu$ : real parameter and : 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

$\mathit{Ec}^{2m}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{\prime}% }^{2}\right),$

$\mathit{Ec}^{2m+1}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$

$\mathit{Es}^{2m+1}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$

$\mathit{Es}^{2m+2}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$

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).

29.9 Stability 29.11 Lamé Wave Equation

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