Given the definition of the complete elliptic integral of the 1st kind (see this link here), I am interested in finding a particular value of $k$ such that
$$K(k) \equiv \int_0^1\dfrac{\operatorname{d}t}{\sqrt{1-t^2}\sqrt{1-k^2t^2}} = d/2, $$
where $d$ is a known positive constant.
I can numerically find this value of $k$ using Mathematica for various values of $d$. However, I would like to know if it is possible to obtain a general solution for $k$ given some constant $d$; essentially inverting this relation. I have little experience with these elliptic functions; regardless I am not certain how I would begin to approach solving this implicit equation in general.
My instinct suggests that since this definition exists, it may not be possible...
Any ideas/solutions welcome!