I am reading Ahlfors' complex analysis.
He let $0<k<1$ and defined $$K=\int_{-1}^1 \frac1{\sqrt{1-t^2} \sqrt{1-k^2t^2}}dz$$ $$K'=\int_{1}^{1/k} \frac1{\sqrt{t^2-1} \sqrt{1-k^2t^2}}dz$$ An exercise requires me to prove that $K=K'$ iff $k=(\sqrt{2}-1)^2$. So these are elliptic integral and I read online that in general, they cannot be computed. And since Ahlfors did not say much about it, I don't think we can compute $K$ and $K'$. So how do we approach?