Generalizations of the Cauchy Integral Theorems

Jenny Harrison; Harrison Pugh

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arXiv · 1101.0385

Generalizations of the Cauchy Integral Theorems

Jenny Harrison·Harrison Pugh

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Abstract

We extend the Cauchy residue theorem to a large class of domains including differential chains that represent, via canonical embedding into a space of currents, divergence free vector fields and non-Lipschitz curves. That is, while the classical Cauchy theorems involve integrals over piecewise smooth parameterized curves, these classical theorems actually hold for far more general notions of "curve." We also extend the definition of winding number to these domains and show that it behaves as expected.

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Jenny Harrison, Harrison Pugh. 2011-07-22. Generalizations of the Cauchy Integral Theorems. https://arxiv.org/abs/1101.0385

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