The Complete Elliptic Integral of the First Kind is defined as the following:
$$K(m)=\int_0^{\frac{\pi}{2}}\frac{dx}{\sqrt{1-m\sin^2(x)}}$$
Given this definition, I would like to solve the following question:
$$ K(i)=\int_0^{\frac{\pi}{2}}\frac{dx}{\sqrt{1-i\sin^2(x)}} $$
How should I start? I assume I’m missing some crucial identity that would allow me to find the answer immediately.
Furthermore, I would like to know if there is an identity for all imaginary moduli, such as $$K(m) = K(n\cdot i)$$
where $n$ is a real number.