# §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
 29.10.2 $z^{\prime}=\mathrm{i}(z-\!\mathop{K\/}\nolimits\!-\mathrm{i}\!\mathop{{K^{% \prime}}\/}\nolimits\!),$

transform (29.2.1) into

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

In consequence, the functions

 29.10.4 $\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!\left(\mathrm{i}(z-\!\mathop{K\/}% \nolimits\!-\mathrm{i}\!\mathop{{K^{\prime}}\/}\nolimits\!),{k^{\prime}}^{2}% \right),$ $\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!\left(\mathrm{i}(z-\!\mathop{K\/% }\nolimits\!-\mathrm{i}\!\mathop{{K^{\prime}}\/}\nolimits\!),{k^{\prime}}^{2}% \right),$ $\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!\left(\mathrm{i}(z-\!\mathop{K\/% }\nolimits\!-\mathrm{i}\!\mathop{{K^{\prime}}\/}\nolimits\!),{k^{\prime}}^{2}% \right),$ $\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!\left(\mathrm{i}(z-\!\mathop{K\/% }\nolimits\!-\mathrm{i}\!\mathop{{K^{\prime}}\/}\nolimits\!),{k^{\prime}}^{2}% \right),$

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

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