I am trying to prove the complex period of Jacobi elliptic functions are $2K + 2iK’$ and $4K + 4iK’$. The crucial step that I am missing is to show the following equality: $$\int_{0}^{1} \frac{\mathrm{d} t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{\prime 2} t^{2}\right)}}=\int_{1}^{1 / k} \frac{\mathrm{d} t}{\sqrt{\left(t^{2}-1\right)\left(1-k^{2} t^{2}\right)}} $$
where $k^2 + k’^2 = 1$.
I found the substitution $x = \frac{1-k}{k}t + 1$ changes the bounds correctly however by some algebraic manipulation I got in the denominator: $$\frac{k}{k-1} \sqrt{\left(\frac{k-1}{k}\right)^{2}-\left(1-k^{2}\right)(x-1)^{2}} \sqrt{\left(\frac{k-1}{k}\right)^{2}-(x-1)^{2}}$$
which is not the same. Any ideas?
Try $x = \bigl( 1-(k')^2 t^2 \bigr)^{-1/2}$ instead.