I'm having trouble with this elliptic integral of first kind, the one denoted in Mathematica as EllipticK. This function should be defined for every value in the complex plane, however in Mathematica its domain in only x<1. How can I get its values for other points?

Moreover what is quite strange is that even if the ratio EllipticK[1-x]/EllipticK[x] is defined only in the domain [0, 1], if I ask to plot the exponential of this function, that is Exp[EllipticK[1-x] / EllipticK[x]] the plot actually covers the whole domain x<1, even though Mathematica is unable to give me the domain of this exponential function.

Can you help me in understanding what's going on? Thank you!