 Furthermore, holomorphy is a necessary condition for a function to have an antiderivative, since the derivative of any holomorphic function is holomorphic.Various versions of Cauchy integral theorem, an underpinning result of Cauchy function theory, which makes heavy use of path integrals, gives sufficient conditions under which, for a holomorphic "g",

vanishes for any closed path γ (which may be, for instance, that the domain of "g" be simply connected or star-convex).First we show that if "f" is an antiderivative of "g" on "U", then "g" has the path integral property given above.