# §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 $z^{\prime}=\mathrm{i}(z-\!K\!-\mathrm{i}\!{K^{\prime}}\!),$

transform (29.2.1) into

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

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).