In this paper we prove that cocycles of holomorphic flows on domains in the complex plane are automatically differentiable with respect to the flow parameter, and their derivatives are holomorphic functions. We use this result to show that, on simply connected domains, an additive cocycle is a coboundary if and only if this cocycle vanishes at the fixed point of the flow.
Proceedings of the American Mathematical Society
Jafari, Farhad; Tonev, T.; and Toneva, E. (2005). "Automatic Differentiability and Characterization of Cocycles of Holomorphic Flows." Proceedings of the American Mathematical Society 133.11, 3389-3394.