The index of a plane curve and green's formula

In this paper we compute the line integral of a complex function on a rectifiable cycle homologous to zero obtaining a Green's formula with multiplicities that involves the ∂̄, of the function and the index of the cycle. We consider this formula in several settings and we obtain a sharp version in terms of the Lebesgue integrability properties of the partial derivatives of the function. This result depends on the proven fact that the index of a rectifiable cycle is square integrable with respect to the planar Lebesgue measure. © 2004 Springer.