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.
Consider now a closed piecewise immersion starting and ending in but such that for all and such that the (positively oriented) angle between and equals . By a suitable rotation we may assume, without loss of generality, that is a positive real number (see Figure 2). We assume that is homotopic to a model sector-curve with the same angle in the following sense: There is a continuous function such thatThen we claim thatFor small enough we have (see Figure 3 for the definition of ) by Cauchy’s integral theorem, and hence Since the claim (10) follows. Thus we get the geometrically reasonable result that the winding number of the curve with respect to is, as we have just seen, the angle divided by :Next, we consider a closed piecewise immersion with one zero and a positively oriented angle between and . Let be a closed piecewise curve which coincides with in a neighborhood of and which is homotopic in the sense of (9) to a model sector-curve with the same angle (see Figure 4). Then, we decompose by . By (14) we getFinally, if has more than one zero, we obtain in the same way the following proposition:
Proposition 3. Let be a closed piecewise immersion and . Then there exist at most finitely many points such that . Consider a decomposition , where coincides with outside of small neighborhoods of the points and avoids the point by driving around it on small circular arcs in clockwise direction. The closed curves are homotopic in the sense of (9) to a model sector-curve with oriented angle between and (see Figure 5). Then, the winding number of with respect to is
Proof. First we show that, for only finitely many points , we have . It suffices to consider a curve parametrized by arc length, and . Assume by contradiction that has infinitely many zeros . Then there is a subsequence, again denoted by , which converges to a point , and we may assume that is increasing. Then, by Rolle’s Theorem, since , there are points such that . But then, . Hence, cannot be continuous.
For the rest of the proof, observe that avoids the point . Thus, we have where we have used (14) in the last step. This completes the proof.
Proposition 3 generalizes immediately from curves to cycles. Definition 2 of a generalized winding number turns out to be useful as it allows to generalize the residue theorem (see Theorem 8 below). But before we turn our attention to this subject, let us briefly reformulate the formula (5) for the generalized winding number as an integral in the real plane. Interestingly, while the winding number as a complex integral requires an interpretation as a principal value, the real counterpart turns out to have a bounded integrand.
If is a closed piecewise curve, then decomposes as The considerations above imply that if is a closed piecewise immersion, then Hence only the imaginary part of is relevant for computing . We have the following proposition regarding its regularity:
Proposition 4. Let be a closed piecewise immersion. Then and the corresponding integrand is bounded. If is in a neighbourhood of a point with , then where is the signed curvature of in .
Proof. Let be a closed piecewise immersion. If avoids the origin, then the integrand is obviously bounded. On the other hand, as in the proof of Proposition 3, it follows that can have at most finitely many zeros, say . It suffices to concentrate on one of the zeros of and to simplify the notation we assume for the rest of the proof. We first show that the integrand is bounded provided is in a neighbourhood of . In this case, and are Lipschitz functions on andOn the other handTogether, (23) and (24) imply that the integrand in Proposition 4 is bounded in .
If is only on and , the claim follows in the same way by considering the unilateral intervals left and right of the zero.
Now we assume that is in a neighbourhood of the zero . It remains to show that the limit has the geometrical interpretation stated in the proposition. In fact, if we find and henceOn the other handFrom (27) and (28) we deduce This completes the proof.
It is worth noticing that Proposition 4 is more than just a technical remark. We will see in Section 3.1 an application of the observation that the imaginary part of the integrand is bounded.
The geometrical meaning of the winding number can be used to characterize the topological phases in one-dimensional chiral non-Hermitian systems. Chiral symmetry ensures that the winding numbers of Hermitian systems are integers, but non-Hermitian systems can take half integer values: see [9] for the corresponding physical interpretation of Proposition 4.
Example 5. Consider the curve given by which passes through the origin at (see Figure 6). According to Proposition 4, and the corresponding integrand is continuous.
3. A Generalized Residue Theorem
Let be an open neighborhood of zero and let be a holomorphic function on . Then there exists a Laurent series which represents in a punctured neighborhood of zero: For a closed piecewise curve with , we have by the Cauchy integral theorem, provided the principal value exists. If has only a pole of first order in , then the discussion in Section 2 shows that the principal value indeed exists. The general case however is more delicate: let us first consider a model sector curve with angle , and let . Then we haveOn an intuitive level it is clear that an angle condition decides whether the limit exists or not: Indeed, in order to compensate the purely real values on (see Figure 1), the integral along cannot have a nonreal singular part. Hence, the principal value in (34) exists (and is actually ) if and only if for some . Stated differently, if for some , , thenif for an integer ; otherwise the principal value (36) is infinite. Therefore we obtain the following:
Lemma 6. Let for some , . If the Laurent series of only contains terms for indices of the form for integers and if is a model sector-curve with angle and radius smaller than the radius of convergence of the Laurent series, then there holds
Proof. If has a pole in , then (37) follows directly from (36). If is an essential singularity of , we observe that the Laurent series converges locally uniformly to . Then we have for Now, for , we choose small enough, such that the absolute value of the second term on the right is smaller than . Note that by (34) this choice does not depend on . Then we can choose , depending on , large enough such that the absolute value of the first term is also smaller than , and the claim follows.
For a more general curve than a model sector curve, we need the following definition:
Definition 7. Let be a piecewise curve and . Let and be the tangents in in the direction and , respectively. We say that is flat of order in , if where and denote the orthogonal projection to and , respectively (see Figure 7).
Notice that a piecewise curve is always flat of order 1 in all of its points.
Now, let us consider a closed piecewise immersion starting and ending in but such that for all and such that the (positively oriented) angle between and equals . We assume that, after a suitable rotation, is a positive real number and that is homotopic in the sense of (9) to a model sector-curve with the same angle . Moreover, we assume that is flat of order in . Then, as in Section 2 (see Figure 3), we have for Hence, we only have a finite principal value if for some , , andif for an integer ; otherwise the principal value is infinite. This leads to the main theorem:
Theorem 8. Let be an open set, and let be a set without accumulation points in such that is holomorphic. Moreover, let be a null-homologous immersed piecewise cycle in such that only contains singularities of which are poles of order 1. ThenThe formula remains correct for poles of higher order on if the following two conditions hold: (A)If is a pole on of order , then is flat of order in , or, if is an essential singularity, coincides near locally with the tangents and in .(B)If is a singularity of on and is the angle between the tangents and in , then , , and the Laurent series of in contains only terms with for indices of the form , an integer.
Proof. Let with and where are closed piecewise immersions. Then, there are at most finitely many points such that . For each consider a decomposition , where coincides with outside of small neighborhoods of the points and avoids the singularity at by driving around it on small circular arcs in clockwise direction. The closed curves are homotopic in the sense of (9) to a model sector-curve with oriented angle between the tangents and . The circular arcs are chosen small enough such that no singularity lies in the interior of the sectors whose boundaries are the curves and such that these sectors are contained in . Observe that the cycle avoids the singularities of and is null-homologous in . Hence, in the sequel we may apply the classical residue theorem to .
Now, suppose that the two conditions (A) and (B) hold. This covers in particular the case when only poles of first order lie on . Then, we have, by the classical residue theorem applied with , by Lemma 6, and (41) The first sum in (43) runs over (I)the singularities which are not lying on , with winding number ,(II)the singularities on . Thus, the summands in (I) appear exactly also in the sum in (42) since for singularities not on we have . The summands in (II) coming from a singularity on together with the corresponding terms in the double sum in (43) give and we are done.
As corollaries of Theorem 8 we obtain the residue theorems [4, Theorem 1] and [3, Theorem 4.8f].
3.1. Application
In [4], the version of the residue theorem is used to calculate principal values of integrals. At first sight it seems that this is the only advantage of Theorem 8 over the classical residue theorem. After all, poles on the curve necessarily mean that one is forced to consider principal values. However, Proposition 4 shows that it is possible to use Theorem 8 to compute integrals with bounded integrand.
Example 9. We want to compute the integral The current computer algebra systems give up on this integral after giving it some thought. To determine the integral we interpret the integrand as follows: for and , where Note that has a pole of order in on with residue . The winding number of with respect to is . By Theorem 8 we get The integral along converges to as tends to infinity, and Hence, we find the value of this improper integral:
3.2. Connection to the Sokhotskiĭ-Plemelj Theorem
In this section, we want to briefly show that the above-mentioned version of the residue theorem in [4, Theorem 1] can be obtained as a corollary of the Sokhotskiĭ-Plemelj Theorem.
Let be an open set, let be a set without accumulation points, and let be a holomorphic function. Let furthermore be a bounded domain with piecewise -boundary , consisting of finitely many components, such that . As usual, we assume that is oriented such that lies always on the left with respect to the direction of the parametrization. If only has first order poles on , we may use a decomposition where is holomorphic away from a single first order pole and has only singularities in . Then is holomorphic. Let By Cauchy’s integral formula we have provided . Furthermore and According to the Sokhotskiĭ-Plemelj formula (see [10, p. 385, (3)] or [11, Chapter 3]) we find where is the interior angle of in and hence and after rearranging Therefore we find
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Conflicts of Interest
The authors declare that they have no conflicts of interest.