29 Lamé Functions Lamé Functions 29.9 Stability 29.11 Lamé Wave Equation

§29.10 Lamé Functions with Imaginary Periods

Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.10

The substitutions

29.10.1 $h=\nu(\nu+1)-h^{{\prime}},$

Symbols:: : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.10.E1
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29.10.2 $z^{{\prime}}=i(z-\!\mathop{K\/}\nolimits\!-i\!\mathop{{K^{{\prime}}}\/}% \nolimits\!),$

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : complex variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.10.E2
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transform (29.2.1) into

29.10.3 $\frac{{d}^{2}w}{{dz^{{\prime}}}^{2}}+(h^{{\prime}}-\nu(\nu+1){k^{{\prime}}}^{2% }{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z^{{\prime}},k^{{\prime}}\right)% )w=0.$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : complex variable, : real parameter, : real parameter, $\nu$ : real parameter and : solution
Permalink:: http://dlmf.nist.gov/29.10.E3
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In consequence, the functions

29.10.4

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

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

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

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

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

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

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