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Jacobi's Second Real Transformation

The second transformation is obtained by taking $ m = 0, m' = 1$ so that $ \omega = iK'/p$. On the face of it the transformation thus involves imaginary quantities, but will be shown later to be a real transformation only. In the case of this transformation we will use $ l_{1}$ in place of $ l$ and $ M_{1}$ in place of $ M$. Also we will keep the factor $ (-1)^{(p - 1)/2}$ with the multiplier $ M_{1}$. We thus obtain the following by putting $ \omega = iK'/p$ in the

general formulas in $ 2s\omega$:
\begin{align}\text{sn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \frac{\text{sn}\,u}{M_{1}}\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - \dfrac{\text{sn}^{2}\,u}{\text{sn}^{2}\,\dfrac{2siK'}{p}}}{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\,\dfrac{2siK'}{p}}\notag\\
&= \sqrt{\frac{k^{p}}{l_{1}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{sn}\left(u + \frac{2siK'}{p}\right)\notag\\
\text{cn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \text{cn}\,u\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - \dfrac{\text{sn}^{2}\,u}{\text{sn}^{2}\left(K - \dfrac{2siK'}{p}\right)}}{1 -
k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\,\dfrac{2siK'}{p}}\notag\\
&= \sqrt{\frac{l'_{1}k^{p}}{l_{1}k'^{p}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{cn}\left(u + \frac{2siK'}{p}\right)\notag\\
\text{dn}\left(\frac{u}{M_{1}}, l_{1}\right) &= \text{dn}\,u\prod_{s = 1}^{(p - 1)/2}\dfrac{1 - k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\left(K - \dfrac{2siK'}{p}\right)}{1 -
k^{2}\,\text{sn}^{2}\,u\,\text{sn}^{2}\,\dfrac{2siK'}{p}}\notag\\
&= \sqrt{\frac{l'_{1}}{k'^{p}}}\prod_{s = -(p - 1)/2}^{(p - 1)/2}\text{dn}\left(u + \frac{2siK'}{p}\right)\notag\\
M_{1} &= (-1)^{(p - 1)/2}\prod_{s = 1}^{(p - 1)/2}\left(\dfrac{\text{sn}\left(K - \dfrac{2siK'}{p}\right)}{\text{sn}\,\dfrac{2siK'}{p}}\right)^{2}\notag\\
l_{1} &= k^{p}\prod_{s = 1}^{(p - 1)/2}\text{sn}^{4}\left(K - \frac{2siK'}{p}\right)\notag\\
l'_{1} &= \dfrac{k'^{p}}{{\displaystyle\prod_{s = 1}^{(p - 1)/2}\text{dn}^{4}\,\frac{2siK'}{p}}}\notag\end{align}