Non-integer valued winding numbers and a generalized Residue Theorem

TL;DR

This paper introduces a generalized winding number that extends classical concepts to include points on the cycle, enabling a new residue theorem applicable to singularities on the cycle and aiding in evaluating certain improper integrals.

Contribution

It proposes a novel winding number definition that incorporates points on the cycle and establishes a generalized residue theorem for such cases.

Findings

Defines a generalized winding number with geometric meaning for points on the cycle.

Develops a residue theorem applicable when singularities lie on the cycle.

Enables calculation of improper integrals where classical residue theorem fails.

Abstract

We define a generalization of the winding number of a piecewise $C^1$ cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal value, but is also possible in a real version via an integral with bounded integrand. The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle. This residue theorem can be used to calculate the value of improper integrals for which the standard technique with the classical residue theorem does not apply.