Free potential functions

Abstract

We show that the derivative of a noncommutative free analytic map must be free-curl free -an analog of having zero curl. Moreover, under the assumption that the free domain is connected, this necessary condition is sufficient. Specifically, if T is analytic free demilinear (linear in half its variables) map defined on a connected free domain then DT (X, H)[K, 0] = DT (X, K)[H, 0] if and only if there exists an analytic free map f such that Df (X)[H] = T (X, H).