 So "g" fails the condition cited above.This is similar to the existence of potential functions for conservative vector fields, in that Green's theorem is only able to guarantee path independence when the function in question is defined on a "simply connected" region, as in the case of the Cauchy integral theorem.In fact, holomorphy is characterized by having an antiderivative "locally", that is, "g" is holomorphic if for every "z" in its domain, there is some neighborhood "U" of "z" such that "g" has an antiderivative on "U".