§29.10 Lamé Functions with Imaginary Periods

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: $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: 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		$$\frac{d^{2}w}{{d{z}^{\prime }}^2}+(h^{\prime }-\nu (\nu +1)k^{\prime }{}^2{\mathrm{sn}}^2(z^{\prime },k^{\prime }))w=0.$$
ⓘ Symbols: $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: 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		${\mathit{Ec}}_{\nu }^{2m}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k^{\prime }{}^2),$
		${\mathit{Ec}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k^{\prime }{}^2),$
		${\mathit{Es}}_{\nu }^{2m+1}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k^{\prime }{}^2),$
		${\mathit{Es}}_{\nu }^{2m+2}(\mathrm{i}(z-K-\mathrm{i}{K}^{\prime }),k^{\prime }{}^2),$
ⓘ Symbols: ${\mathit{Ec}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathit{Es}}_{\nu }^m(z,k^2)$: Lamé function, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: 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).