Abstract
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\(\bar \partial \) 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.
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The work of both authors is partially supported by grants 2000SGR-00059, 2001SGR 00172 of Generalitat de Catalunya and BFM 2002-04072-C02-02 of Ministerio de Ciencia y Tecnologia