A variant of Cauchy's argument principle for analytic functions which applies to curves containing zeroes

TL;DR

This paper generalizes Cauchy's argument principle to include cases where the holomorphic function has zeros on the integration curve, expanding its applicability and providing an illustrative application.

Contribution

It introduces a new variant of Cauchy's argument principle that applies even when zeros lie on the curve of integration, a case not covered by the classical version.

Findings

Generalized argument principle for zeros on the curve

Applicable to broader classes of holomorphic functions

Includes an illustrative application

Abstract

It is known that the Cauchy's argument principle, applied to an holomorphic function $f$, requires that $f$ has no zeros on the curve of integration. In this short note, we give a generalization of such a principle which covers the case when $f$ has zeros on the curve, as well as an application.