Generalizations of the Cauchy Integral Theorems

TL;DR

This paper generalizes the classical Cauchy integral theorems to broader classes of domains, including divergence-free vector fields and non-Lipschitz curves, by extending the residue theorem and winding number concepts.

Contribution

It introduces a generalized framework for Cauchy theorems applicable to differential chains and non-smooth curves, expanding their applicability.

Findings

Classical Cauchy theorems hold for more general domains.

Winding number is extended and retains expected properties.

The framework includes divergence-free vector fields and non-Lipschitz curves.

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.