Generalizations of the Cauchy Integral Theorems

2011/01/02 by Jenny Harrison, Harrison, Jenny, Harrison Pugh +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Mathematical Analysis and Transform Methods #Polynomial and algebraic computation #math.CV #math.FA #msc:30E20 #msc:46F05

paper · pdf · doi:10.48550/arxiv.1101.0385

10 pages, 3 figures

arxiv created 2011/07/22 · arxiv updated 2011/07/26

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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