Journal of Mathematics

Volume 2019, Article ID 6130464, 9 pages

https://doi.org/10.1155/2019/6130464

## Non-Integer Valued Winding Numbers and a Generalized Residue Theorem

^{1}Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland^{2}HTA/HSW Freiburg, HES-SO University of Applied Sciences and Arts Western Switzerland, Pérolles 80/Chemin du Musée 4, 1700 Freiburg, Switzerland

Correspondence should be addressed to Norbert Hungerbühler; hc.zhte.htam@relheubregnuh.trebron

Received 8 January 2019; Accepted 18 February 2019; Published 11 March 2019

Academic Editor: Mike Tsionas

Copyright © 2019 Norbert Hungerbühler and Micha Wasem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We define a generalization of the winding number of a piecewise 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.

#### 1. Introduction

One of the most prominent tools in complex analysis is Cauchy’s Residue Theorem. To state the classical version of this theorem (see, e.g., [1] or [2, Theorem 1, p. 75]) we briefly recall the following notions: A* chain* is a finite formal linear combination, of continuous curves . A* cycle* is a chain, where every point is, counted with the corresponding multiplicity , as often a starting point of a curve as it is an endpoint. A cycle is* null-homologous* in , if its winding number for all points in vanishes. Equivalently, is null-homologous in , if it can be written as a linear combination of closed curves which are contractible in . Then the residue theorem can be expressed as follows:

Theorem 1 (Classical Residue Theorem). *Let be an open set and let be a set without accumulation points in such that is holomorphic. Furthermore, let be a null-homologous cycle in . Then there holdswhere denotes the winding number of with respect to .*

Henrici considers in [3, Theorem 4.8f] a version of the residue theorem where is the boundary of a semicircle in the upper half-plane with diameter and where is allowed to have poles on which involve odd powers only. The result is basically a version of the classical formula (2), but with winding number for the singularities on the real axis and where the integral on the left-hand side of (2) is interpreted as a Cauchy principal value. Another very recent version of the residue theorem, where poles of order 1 on the piecewise boundary curve of an open set are allowed, is discussed in [4, Theorem 1]. There, if a pole is sitting on a corner of , the winding number is replaced by the angle formed by in this point, divided by . In [5] a version of the residue theorem for functions with finitely many poles is presented where singularities in points on a closed curve are allowed provided near , with . Further versions of generalizations of the residue theorem are described in [6, 7]: there, versions for unbounded multiply connected regions of the second class are applied to higher-order singular integrals and transcendental singular integrals.

In the present article, we introduce a generalized, non-integer winding number for piecewise cycles and a general version of the residue theorem which covers all cases of singularities on . We will assume throughout the article that all curves are continuous. In particular, a piecewise curve is a continuous curve which is piecewise . Recall that a closed piecewise immersion is a closed continuous curve such that there is a partition such that is of class and such that for all . If is furthermore a Lipschitz function for all , then is called a closed piecewise immersion.

#### 2. A Generalized Winding Number

The aim of this section is to generalize the winding number to piecewise cycles with respect to points sitting on the cycle itself.

The usual standard situation is the following: The winding number of a closed piecewise curve around is given by See, for example, [2, p. 70] or [8, p. 75]. More generally, one can replace the curve by a piecewise cycle . An integral over the cycle is then In order to make sense of the winding number also for points* on* the curve, we use the Cauchy Principal Value:

*Definition 2. *The winding number of a piecewise cycle with respect to isIt is not a priori clear whether this limit exists and what its geometric meaning is. So, we start by looking at the following model case: Using the Cauchy principal value we can easily compute the winding number with respect to of the* model sector-curve *, where for (see Figure 1). Since we obtain with a meaningful geometrical interpretation.