I'm doing some computations in a specific case of a problem about ellipses, and I was wondering if there was a nice solution for the following product:
$$E( 1-b^2/a^2)E( 1-a^2/b^2)$$
Where $E(m)$ is the complete elliptic integral of the second kind with parameter $m=k^2$.
Geometric considerations suggest that this should be $\pi^2/4$, and while that looks spiritually related to the product of the infinite series representations of $E$, I haven't the faintest on how to get to this evaluation, or even if it is correct.
This is equivalent to showing that:
$$_2F_1\left(\frac{1}{2},-\frac{1}{2};1;\left(1-\frac{b^2}{a^2}\right)\right){}_2F_1\left(\frac{1}{2},-\frac{1}{2};1;\left(1-\frac{a^2}{b^2}\right)\right) = 1$$
So this question can be rephrased in terms of known formulae for products of Gauss's hypergeometric function.
Please let me know if the notation is wrong/confusing-- the ways that elliptic integrals are written is still a little fuzzy to me.