It appears that MMA produces unexpected evaluations for incomplete elliptic integrals of the first kind for complex-valued arguments. Naively speaking, one would define EllipticF for complex-valued arguments by analytically continuing its standard integral representation. However, numerical experiments (see attached file) reveal that EllipticF evaluations sometimes differ from this integral representation by an amount of 2EllipticK.
I have a hunch that MMA evaluates EllipticF for generic inputs by numerically inverting JacobiAmplitude (the latter function has period 2EllipticK, which explains the aforementioned offset), rather than adhering to a more straightforward integral representation. Is that the case?