No abstract

Abstract. 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.

The process of integration was a subject of significant development during the last century. Despite that the Lebesgue integral is complete and has many good properties, its inability to integrate all derivatives prompted the introduction of new approaches -Denjoy, Perron and others introduced new ways of integration aimed at preserving the good properties of the Lebesgue integral but extending the set of functions to which it could be applied. The goal was achieved but neither of the new approaches was elegant or simple or transparent. In the 50s a new integral was introduced, independently by Kurzweil and Henstock,in a very simple, Riemann like way, but it turned out that it was more powerful than the Lebesgue integral. There are many names attached to this integral, I will use here the name Henstock integral. The goal of this article is to introduce the generalized Riemann integral and the Lebesgue integral using generalized sequences.