Taking the directional limits will reveal *Mathematica*'s conventions:

    Assuming[x > 1, Limit[EllipticK[x + I ε], ε -> 0, Direction -> 1]]

> `EllipticK[x]`

    Assuming[x > 1, Limit[EllipticK[x + I ε], ε -> 0, Direction -> -1]]

> `2 I EllipticK[1 - x] + EllipticK[x]`