arXiv:1808.00997v2 [math.CA] 13 Mar 2019 osaetecasclvrino hsterm(e,eg,[ e.g., (see, theorem this Cauchy of is version analysis classical complex the in state tools To prominent most the of One Introduction 1 1 o-nee audwnignmesadageneralized a and numbers winding valued Non- eateto ahmtc,EHZ¨urich, 809 R¨amistrasse ETH 101, , of Department 2 T/S riug E-OUiest fApidSine n Arts and Sciences Applied of University HES-SO Freiburg, HTA/HSW etr wteln,Peols8/hmnd u´e4 70Freibu 1700 Mus´ee 4, du P´erolles 80/Chemin Switzerland, Western 00MteaisSbetClassification Subject Mathematics 2010 words Key edfieagnrlzto ftewnignme fapiecewise a of number winding the of generalization a define We h ope ln hc a emti enn lofrpit hc lie which points for also meaning geometric a has which on complex the o hc h tnadtcnqewt h lsia eiu theore in residue improper classical c of the value the with the technique on calculate apply. standard lie to the used singularities which be re where for can generalized situation theorem a the residue establish also This to covers allows with which number an winding theorem via new version real The a in integrand. possible also is but value, principal h yl.Tecmuaino hswnignme eiso h Cauch the on relies number winding this of computation The cycle. the acypicplvle idn ubr eiu theorem residue number, winding value, principal Cauchy : obr Hungerb¨uhlerNorbert eiu Theorem Residue ac 4 2019 14, March Switzerland Switzerland Abstract 1 : 1 30E20 n ih Wasem Micha and 1 r[ or ] 8 sRsdeTheorem. Residue ’s hoe ,p 75]) p. 1, Theorem , 2 C osnot does m 1 bounded yl in cycle tegrals Z¨urich,2 sidue ycle. rg, y we briefly recall the following notions: A chain is a finite formal linear combination

k Γ= m γ , m Z, ℓ ℓ ℓ ∈ Xℓ=1 of continuous γ : [0, 1] C. A cycle C is a chain, where every a C ℓ → ∈ is, counted with the corresponding multiplicity mℓ, as often a starting point of a γ as it is an endpoint. A cycle C is null-homologous in D C, if its winding ℓ ⊂ number for all points in C D vanishes. Equivalently, C is null-homologous in D, \ if it can be written as a linear combination of closed curves wich are contractible in D. Then the can be expressed as follows:

Theorem (Classical Residue Theorem). Let U C be an open set and S U be ⊂ ⊂ a set without accumulation points in U such that f : U S C is holomorphic. \ → Furthermore, let C be a null-homologous cycle in U S. Then there holds \ 1 f(z)dz = ns(C)ress f(z), (1.1) 2πi C I Xs∈S where ns(C) denotes the winding number of C with respect to s.

Henrici considers in [4, Theorem 4.8f] a version of the residue theorem where C is the boundary of a semicircle in the upper half-plane with diameter [ R,R], and where − f is allowed to have poles on ( R,R) which involve odd powers only. The result is − 1 basically a version of the classical formula (1.1), but with winding number 2 for the singularities on the real axis, and where the integral on the left hand side of (1.1) is interpreted as a Cauchy Principal Value. Another very recent version of the residue theorem, where poles of order 1 on the piecewise C1 boundary curve γ of an open set are allowed, is discussed in [5, Theorem 1]. There, if a pole is sitting on a corner of γ, the winding number is replaced by the angle formed by γ in this point, divided by 2π. In[6] a version of the residue theorem for functions f with finitely many poles is presented where singularities in points z0 on a closed curve Γ are allowed provided f(z) = O( z z µ) near z , with 1 < µ 0. Further versions of generalizations | | | − 0| 0 − ≤ of the residue theorem are described in [10] and [11]: there, versions for unbounded multiply connected regions of the second class are applied to higher-order singular and transcendental singular integrals. In the present article, we introduce a generalized, non-integer winding number for piecewise C1 cycles C, and a general version of the residue theorem which covers all

2 cases of singularities on C. We will assume throughout the article that all curves are continuous. In particular, a piecewise C1 curve is a continuous curve which is piecewise C1. Recall that a closed piecewise C1 Λ : [a, b] C is a closed → curve such that there is a partition a = a < a < . . . < a = b such that Λ 0 1 n |[ak,ak+1] is of class C1 and such that Λ˙ = 0 for all k = 0,...,n 1. If Λ˙ is |[ak,ak+1] 6 − |[ak,ak+1] furthermore a Lipschitz for all k = 0,...,n 1, then Λ is called a closed − piecewise C1,1 immersion.

2 A Generalized Winding Number

The aim of this section is to generalize the winding number to piecewise C1 cycles with respect to points sitting on the cycle itself. The usual standard situation is the following: The winding number of a closed piecewise C1 curve γ : [a, b] C 0 around z = 0 is given by → \ { } 1 dz n (γ)= Z, 0 2πi z ∈ Iγ see for example [8, p. 70] or [2, p. 75]. More generally, one can replace the curve γ 1 k by a piecewise C cycle C = ℓ=1 mℓγℓ. An integral over the cycle is then

P k f(z)dz := mℓ f(z)dz. C γℓ I Xℓ=1 I In order to make sense of the winding number also for points on the curve, we use the Cauchy principal value:

Definition 2.1. The winding number of a piecewise C1 cycle C : [a, b] C with → respect to z C is 0 ∈ 1 dz 1 dz nz0 (C) := PV = lim . (2.1) 2πi C z z0 εց0 2πi z z0 I − |C−Zz0|>ε −

It 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

3 value we can easily compute the winding number with respect to z = 0 of the model sector-curve γ = γ1 + γ2 + γ3, where γ : [0, r] C, t t, γ′ (t) = 1 1 → 7→ 1 γ : [0, α] C, t reit, γ′ (t)= rieit 2 → 7→ 2 γ : [0, r] C, t (r t)eiα, γ′ (t)= eiα 3 → 7→ − 3 − for α [0, 2π] (see Figure 1). Since ∈ reiα

γ3 γ2

α 0 γ1 r

Figure 1: The model sector-curve γ = γ1 + γ2 + γ3.

dz r dt α rieit r−ε eiα PV = lim + dt + − dt = z εց0 t reit (r t)eiα Iγ Zε Z0 Z0 −  = lim(ln r ln ε + iα + ln ε ln r) = iα, εց0 − − we obtain 1 dz α n (γ)=PV = 0 2πi z 2π Iγ with a meaningful geometrical interpretation. Consider now a closed piecewise C1 immersion Γ : [a, b] C starting and ending in → 0 but such that Γ(t) = 0 for all t = a, b and such that the (positively oriented) angle 6 6 between lim Γ(˙ t) and lim Γ(˙ t) equals α [0, 2π]. By a suitable rotation we tցa − tրb ∈ may assume, without loss of generality, that limtցa Γ(˙ t) is a positive (see Figure 2). We assume that Γ is homotopic to a model sector-curve γ with the same angle α in the following sense: There is a H : [a, b] [0, 1] C × → such that H(t, 0) = Γ(t) for all t [a, b], ∈ H(t, 1) = γ(t) for all t [a, b], ∈ (2.2) 0= H(a, s) = H(b, s) for all s [0, 1], ∈ H(t,s) = 0 for all t (a, b) and all s [0, 1]. 6 ∈ ∈ 4 Γ

α 0

Figure 2: Winding number with respect to the origin sitting on the curve Γ.

Then we claim that dz dz lim = lim . (2.3) εց0 z εց0 z Z|Γ|>ε Z|γ|>ε

Γ

α1

β

α2 0 ε r

Figure 3: The curves Γ and γ.

For 0 < ε < r small enough we have (see Figure 3 for the definition of β) dz dz = z z Z|Γ|>ε Zβ by Cauchy’s integral theorem, and hence dz dz dz dz = = z − z z − z Z|Γ|>ε Z|γ|>ε Zβ Z|γ|>ε dz dz dz = + = z − z z Zβ Z|γ|>ε Z(|γ|>ε)+α1−β+α2 dz = . z Zα1+α2 5 Since dz 1 Length(α + α ) 0 for ε 0, z ≤ ε 1 2 −→ ց Zα1+α2 =o(ε) the claim (2.3) follows. Thus we| get the{z geometrically} reasonable result that the winding number of the curve Γ with respect to z = 0 is, as we have just seen, the angle α divided by 2π: 1 dz α n (Γ) = PV = . (2.4) 0 2πi z 2π IΓ Next, we consider an closed piecewise C1 immersion Λ : [a, b] C with one zero x , → 1

Λ

Γ2

α Γ1 Γ3

Figure 4: Winding number of Λ.

˙ ˙ and a positively oriented angle α [0, 2π] between limtցx1 Λ(t) and limtրx1 Λ(t). ∈ 1 − Let Γ = Γ1 +Γ2 +Γ3 be a closed piecewise C curve which coincides with Λ in a neighborhood of x1 and which is homotopic in the sense of (2.2) to a model sector- curve with the same angle α (see Figure 4). Then, we decompose Λ by Λ = Λ+Γ.˜

6 By (2.4) we get 1 dz α n (Λ) = PV = n (Λ)+˜ . (2.5) 0 2πi z 0 2π IΛ Finally, if Λ has more than one zero, we obtain in the same way the following proposition:

Proposition 2.2. Let Λ : [a, b] C be a closed piecewise C1 immersion and z C. → 0 ∈ Then there exist at most finitely many points x ,...,x [a, b] such that Λ(x )= z . 1 n ∈ ℓ 0 Consider a decomposition Λ= Λ+Γ˜ 1 +. . .+Γn, where Λ˜ coincides with Λ outside of small neighborhoods of the points xℓ and avoids the point z0 by driving around it on small circular arcs in clockwise direction. The closed curves Γℓ are homotopic in the ˙ sense of (2.2) to a model sector-curve with oriented angle αℓ between limtցxℓ Λ(t) and lim Λ(˙ t) (see Figure 5). Then, the winding number of Λ with respect to − tրxℓ z0 is n 1 dz ˜ αℓ nz0 (Λ) = PV = nz0 (Λ)+ . 2πi Λ z z0 2π I − Xℓ=1

Λ Λ˜ Γ1

z0 z0 Γ2

Figure 5: Decomposition of Λ = Λ+Γ˜ 1 +Γ2.

Proof. First we show that for only finitely many points xℓ, we have Λ(xℓ) = z0. It suffices to consider a C1 curve Λ : [a, b] R2 parametrized by arc length, and → z0 = (0, 0). Assume by contradiction that Λ has infinitely many zeros xℓ. Then there is a subsequence, again denoted by x , which converges to a point x [a, b], ℓ ∈ and we may assume, that xℓ is increasing. Then, by Rolle’s Theorem, since Λ1(xℓ)= ′ ′ Λ2(xℓ) = 0, there are points uℓ, vℓ (xℓ,xℓ+1) such that Λ1(uℓ) = Λ2(vℓ) = 0. But ′ ′ ∈′ then, Λ1(vℓ) = Λ2(uℓ) = 1. Hence, Λ cannot be continuous.

7 For the rest of the proof, observe, that Λ˜ avoids the point z0. Thus, we have 1 dz nz0 (Λ) = PV = 2πi Λ z z0 I − n 1 dz 1 dz = + PV = 2πi ˜ z z0 2πi z z0 IΛ − ℓ=1 IΓℓ − n X α = n (Λ)+˜ ℓ , z0 2π Xℓ=1 where we have used (2.4) in the last step.

Proposition 2.2 generalizes immediately from curves to cycles. The Definition 2.1 of a generalized winding number turns out to be useful as it allows to generalize the residue theorem (see Theorem 3.3 below). But before we our attention this subject, let us briefly reformulate the formula (2.1) 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 Λ= x + iy : [a, b] C is a closed piecewise C1 curve, then dz/z decomposes as → dz xx˙ + yy˙ xy˙ yx˙ = dt + i − dt. z x2 + y2 x2 + y2

The considerations above imply that if Λ is a closed piecewise C1 immersion, then x dx + y dy PV = 0, x2 + y2 IΛ hence only the imaginary part of dz/z is relevant for computing n0(Λ). We have the following proposition regarding its regularity:

Proposition 2.3. Let Λ= x + iy : [a, b] C be a closed piecewise C1,1 immersion. → Then 1 b x(t)y ˙(t) y(t)x ˙(t) n (Λ) = − dt 0 2π x(t)2 + y(t)2 Za and the corresponding integrand is bounded. If Λ is C2 in a neighbourhood of a point t˜ (a, b) with Λ(t˜) = 0, then ∈ x(t)y ˙(t) y(t)x ˙(t) 1 ˜ ˙ ˜ lim 2 − 2 = kΛ(t) Λ(t) , t→t˜ x(t) + y(t) 2 | |

8 where x˙(t˜)¨y(t˜) y˙(t˜)¨x(t˜) kΛ(t˜)= − (x ˙(t˜)2 +y ˙(t˜)2)3/2 is the signed curvature of Λ in t˜.

Proof. Let Λ be a closed piecewise C1,1 immersion. If Λ avoids the origin, then the integrand is obviously bounded. On the other hand, as in the proof of Proposition 2.2, it follows that Λ can have at most finitely many zeros, say a

x(t)2 + y(t)2 = tx˙(0) + o(t) 2 + ty˙(0) + o(t) 2 = t2 x˙(0)2 +y ˙(0)2 + o(t2). (2.7)

Together, (2.6) and (2.7), imply that the integrand in Proposition 2.3 is bounded in U. If Λ is only C1,1 on U − = ( ε, 0] and U + = [0, ε), the claim follows in the same way − by considering the unilateral intervals left and right of the zero. Now we assume that Λ is C2 in a neighbourhood U = ( ε, ε) of the zero t˜ = 0. It − remains to show that the limit x(t)y ˙(t) y(t)x ˙(t) lim − t→0 x(t)2 + y(t)2 has the geometrical interpretation stated in the proposition. In fact, if Λ(0) = 0 we find t2 x(t)y ˙(t)= tx˙(0) + x¨(0) + o(t2) (y ˙(0) + ty¨(0) + o(t)) 2   1 = tx˙(0)y ˙(0) + t2 x˙(0)¨y(0) + x¨(0)y ˙(0) + o(t2) 2   9 and hence 1 x(t)y ˙(t) x˙(t)y(t)= t2 x˙(0)¨y(0) y˙(0)¨x(0) + (¨x(0)y ˙(0) y¨(0)x ˙(0)) + o(t2) − − 2 −   t2 = x˙ (0)¨y(0) y˙(0)¨x(0) + o(t2). 2 −   (2.8) On the other hand

t2 2 t2 2 x2(t)+ y2(t)= tx˙(0) + x¨(0) + o(t2) + ty˙(0) + y¨(0) + o(t2) 2 2 (2.9)     = t2(x ˙(0)2 +y ˙(0)2)+ t3(x ˙(0)¨x(0) +y ˙(0)¨y(0)) + o(t3).

From (2.8) and (2.9) we deduce

x(t)y ˙(t) y(t)x ˙(t) 1 ˜ ˙ ˜ lim 2 − 2 = kΛ(t) Λ(t) . t→t˜ x(t) + y(t) 2 | |

It is worth noticing, that Proposition 2.3 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 symme- try ensures that the winding number of Hermitian systems are , but non- Hermitian systems can take half integer values: see [9] for the corresponding physical interpretation of Proposition 2.3.

Example 2.4. Consider the curve Λ : [0, 2π] C given by → Λ(t)= x(t) + iy(t) := cos(t) + cos(2t) + i sin(2t) which passes through the origin at t = π (see Figure 6). According to Proposition 2.3,

1 b x(t)y ˙(t) y(t)x ˙(t) 3 n (Λ) = − dt = 0 2π x(t)2 + y(t)2 2 Za and the corresponding integrand is continuous.

10 y

x

3 Figure 6: The (solid) curve from Example 2.4 has winding number 2 with respect to the origin, and the dashed and dotted one have winding number 2 and 1 respectively.

3 A Generalized Residue Theorem

Let U C be an open neighborhood of zero and let f be a ⊂ on U 0 . Then there exists a which represents f in a punctured \ { } neighborhood z C : 0 < z < R of zero: { ∈ | | } a f(z)= . . . + −1 + a + a z + . . . . z 0 1 =h(z) =g(z) | {z } For a closed piecewise C1 curve |γ with{z γ} < R, we have | | PV f(z)dz = PV (g(z)+ h(z)) dz = PV g(z)dz Iγ Iγ Iγ by the Cauchy integral theorem, provided the principal value exists. If f has only a pole of first order in 0, then the discussion in Section 2 shows, that the principal value indeed exists. The general case however is more delicate: let us first consider

11 a model sector curve γ with angle α [0, 2π], and let n> 1. Then we have ∈ dz r dt α rieit r−ε eiα PV = lim + dt + − dt zn εց0 tn rneint (r t)neinα Iγ Zε Z0 Z0 −  rnε rεn e−α(n−1)i 1 rεn rnε = lim − − + − e−α(n−1)i εց0 (rε)n(n 1) − rn−1(n 1) (rε)n(n 1) − − − ! 1 e−i(n−1)α 0 if α(n−1) Z, = lim − = 2π ∈ (3.1) εց0 (n 1)εn−1 complex infinity otherwise. − ( On 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 γ1 (see Figure 1), the integral along γ3 cannot have a non-real singular part. Hence, the principal value in (3.1) exists (and is actually 0) if and only if 2kπ α = n 1 − for some k Z. Stated differently: If α = p π for some p,q N, q = 0, then ∈ q ∈ 6 dz PV = 0 (3.2) zn Iγ if n = 2kq + 1 for an integer k 0, otherwise the principal value (3.2) is infinite. p ≥ Therefore we obtain: Lemma 3.1. Let α = p π for some p,q N, q = 0. If the Laurent series of f q ∈ 6 only contains terms a /zn for indices of the form n = 2kq + 1 for integers k 0, n p ≥ and if γ is a model sector-curve with angle α and radius smaller than the radius of convergence of the Laurent series, then there holds 1 PV f(z)dz = n (γ)res f(z). (3.3) 2πi 0 0 Iγ Proof. If f has a pole in 0, then (3.3) follows directly from (3.2). If 0 is an es- k sential singularity of f = k∈Z akz , we observe that the Laurent series fn(z) = ∞ a zk converges locally uniformly to f(z). Then we have for ε> 0 k=−n k P P 1 1 f(z)dz = f(z) f (z) dz+ 2πi 2πi − n Z|γ|>ε Z|γ|>ε 1  + fn(z)dz n0(γ)res0 f(z) + n0(γ)res0 f(z) =: I + II + III. 2πi |γ|>ε −  Z 

12 Now, for δ > 0, we choose ε > 0 small enough, such that the absolute value of the second term II on the right is smaller than δ. Note that by (3.1) this choice does not depend on n. Then we can choose n, depending on ε, large enough such that the absolute value of the first term I is also smaller than δ, and the claim follows.

For a more general curve than a model sector curve, we need the following definition:

Definition 3.2. Let Γ : (a, b) C be a piecewise C1 curve and Γ(x ) =: z . Let → 1 1 t+ and t− be the tangents in z in the direction lim Γ(˙ x) and lim Γ(˙ x) 1 xցx1 − xրx1 respectively. We say that Γ is flat of order n in x1, if

Γ(x) P +(Γ(x)) = o( Γ(x) z n) for x x and | − | | − 1| ց 1 Γ(x) P −(Γ(x)) = o( Γ(x) z n) for x x | − | | − 1| ր 1 where P + and P − denote the orthogonal projection to t+ and t− respectively (see Figure 7).

Notice, that a piecewise C1 curve is always flat of order 1 in all of its points.

P −(Γ(x))

Γ(x) − t Γ

z1 Γ(x) t+

P +(Γ(x))

Figure 7: Γ is flat of order n in x1.

Now, let us consider a closed piecewise C1 immersion Γ : [a, b] C starting and → ending in 0 but such that Γ(t) = 0 for all t = a, b and such that the (positively 6 6 oriented) angle between lim Γ(˙ t) and lim Γ(˙ t) equals α [0, 2π]. We assume tցa − tրb ∈ that, after a suitable rotation, limtցa Γ(˙ t) is a positive real number and that Γ is homotopic in the sense of (2.2) to a model sector-curve γ with the same angle α.

13 Moreover, we assume that Γ is flat of order n in 0. Then, as in Section 2 (see Figure 3), we have for n> 1

dz dz dz dz n n = n n = |Γ|>ε z − |γ|>ε z β z − |γ|>ε z Z Z Z Z dz = zn Zα1+α2 1 Length( α + α ) 0 for ε 0. ≤ εn 1 2 −→ ց =o(εn)

Hence, we only have a finite principal value| if α{z= p π for} some p,q N, q = 0, and q ∈ 6 dz PV = 0 (3.4) zn IΓ if n = 2kq + 1 for an integer k 0, otherwise the principal value is infinite. This p ≥ leads to the main Theorem:

Theorem 3.3. Let U C be an open set, and S = z , z ,... U be a set without ⊂ { 1 2 }⊂ accumulation points in U such that f : U S C is holomorphic. Moreover, let C \ → be a null-homologous immersed piecewise C1 cycle in U such that C only contains singularities of f which are poles of order 1. Then 1 PV f(z)dz = nzℓ (C)reszℓ f(z). (3.5) 2πi C I Xℓ The formula remains correct for poles of higher order on C if the following two conditions hold:

(A) If z0 is a pole on C of order n, then C is flat of order n in z0, or, if z0 is an + − essential singularity, C coincides near z0 locally with the tangents t and t in z0.

+ (B) If z0 is a singularity of f on C and α is the angle between the tangents t and t− in z , then α = p π, p,q N,q = 0, and the Laurent series of f in z 0 q ∈ 6 0 contains only terms a /(z z )n with a = 0 for indices of the form n = 2kq +1, n − 0 n 6 p k 0 an integer. ≥

14 Proof. Let C = k m γ with m Z and where γ : [0, 1] C are closed ℓ=1 ℓ ℓ ℓ ∈ ℓ → piecewise C1 immersions. Then, there are at most finitely many points x ,...,x P ℓ1 ℓkℓ such that γ (x ) = z S. For each ℓ consider a decomposition γ =γ ˜ +Γ + ℓ ℓj ℓj ∈ ℓ ℓ ℓ1 . . . +Γℓkℓ , whereγ ˜ℓ coincides with γℓ outside of small neighborhoods of the points xℓj and avoids the singularity at γℓ(xℓj) by driving around it on small circular arcs in clockwise direction. The closed curves Γℓj are homotopic in the sense of (2.2) to a model sector-curve with oriented angle αℓj between the tangents limtցxℓj γ˙ℓ and lim γ˙ . The circular arcs are chosen small enough such that no singularity − tրxℓj ℓ lies in the interior of the sectors whose boundary are the curves Γℓj, and such that ˜ k these sectors are contained in U. Observe that the cycle C := ℓ=1 mℓγ˜ℓ avoids the singularities of f and is null-homologous in U. Hence, in the sequel we may apply P the classical residue theorem to C˜. Now, suppose that the two conditions (A) and (B) hold. This covers in particular the case when only poles of first order lie on C. Then, we have, by the classical residue theorem applied with C˜, by Lemma 3.1, and (3.4) k k 1 1 ℓ 1 PV f(z)dz = PV f(z)dz + mℓ PV f(z)dz = 2πi C 2πi C˜ 2πi Γℓj I I Xℓ=1 Xj=1 I k kℓ ˜ = nz(C)resz f(z)+ mℓ nzℓj (Γℓj)reszℓj f(z). (3.6) Xz∈S Xℓ=1 Xj=1 The first sum in (3.6) runs over

(I) the singularities which are not lying on C, with winding number = 0. 6 (II) the singularities on C.

Thus, the summands in (I) appear exactly also in the sum in (3.5) since for singu- larities z not on C we have nz(C) = nz(C˜). The summands in (II) coming from a singularity zℓj on C together with the corresponding terms in the double sum in (3.6) give k kℓ ˜ nzℓj (C)+ mℓ nzℓj (Γℓj) reszℓj f(z)= nzℓj (C)reszℓj f(z)  Xℓ=1 Xj=1  and we are done.

As corollaries of Theorem 3.3 we obtain the residue theorems [5, Theorem 1] and [4, Theorem 4.8f].

15 3.1 Application

In [5], 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 3.3 over the classical residue theorem. After all, poles on the curve γ necessarily mean that one is forced to consider principal values. However, Proposition 2.3 shows that it is possible to use Theorem 3.3 to compute integrals with bounded integrand.

Example 3.4. We want to compute the integral ∞ sinc(t) sinh(t) dt. cos(t) + cosh(t) Z0 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 sinc(t) sinh(t) = im f(γ (t))γ ˙ (t) cos(t) + cosh(t) 3 3 for cos(z/2) f(z)= −z cosh(z/2) and γ = γ + γ γ , where 1 2 − 3 γ : [0, r] C, t t it, 1 → 7→ − γ : [ π , π ] C, t √2reit, 2 − 4 4 → 7→ γ : [0, r] C, t t + it. 3 → 7→ Note that f has a pole of order 1 in z = 0 on γ with residue 1. The winding 1 − number of γ with respect to z =0 is 4 . By Theorem 3.3 we get: 1 π im f(z)dz = im f(z)dz+ f(z)dz f(z)dz = im ( 1) 2πi = . − 4 · − · − 2 Iγ Zγ1 Zγ2 Zγ3    The integral along γ2 converges to 0 as r tends to infinity, and r sinc(t) sinh(t) im f(z)dz = im f(z)dz = dt. − − cos(t) + cosh(t) Zγ1 Zγ3 Z0 Hence, we find the value of this improper integral: ∞ sinc(t) sinh(t) π dt = . cos(t) + cosh(t) 4 Z0 16 3.2 Connection to the Sokhotski˘ı-Plemelj Theorem

In this section, we want to briefly show that the above-mentioned version of the residue theorem in [5, Theorem 1] can be obtained as a corollary of the Sokhotski˘ı- Plemelj Theorem. Let U C be an open set, let S = z ,... U be a set without accumulation points ⊂ { 1 }⊂ and let f : U S C be a holomorphic function. Let furthermore D be a bounded \ → domain with piecewise C1-boundary C, consisting of finitely many components, such that D¯ U. As usual, we assume that C is oriented such that D lies always on the ⊂ left with respect to the direction of the parametrization. If f only has first order poles on C, we may use a decomposition m f = g + fk, Xk=1 where f is holomorphic away from a single first order pole z C and g has only k k ∈ singularities in z , z ,... / C. Then ϕ (z) := f (z)(z z ) is holomorphic. m+1 m+2 ∈ k k − k Let 1 ϕ (ξ) F (z) := k dξ. k 2πi ξ z ZC − By Cauchy’s integral formula we have F (z)= ϕ (z) provided z D. Furthermore k k ∈ + Fk (zk) := lim Fk(z) = lim ϕk(z) = lim fk(z)(z zk) D∋z→zk D∋z→zk D∋z→zk −

: = reszk fk(z) = reszk f(z) and 1 ϕ (ξ) 1 F (z )=PV k dξ = PV f (ξ)dξ. k k 2πi ξ z 2πi k ZC − k ZC According to the Sokhotski˘ı-Plemelj formula (see [3, p. 385, (3)] or [7, Chapter 3]) we find α F +(z )= F (z )+ 1 k ϕ (z ), k k k k − 2π k k   where αk is the interior angle of C in zk and hence

1 αk reszk f(z)=PV fk(ξ)dξ + 1 reszk f(z) 2πi C − 2π Z   and after rearranging 1 α PV f (ξ)dξ = k res f(z). 2πi k 2π zk ZC 17 Therefore we find

m 1 1 PV f(ξ)dξ = PV g(ξ)+ f (ξ) dξ 2πi 2πi k ZC ZC k=1 ! Xm α = res f(z)+ k res f(z) zk 2π zk > k Xm+1 Xk=1

= nzk (C)reszk f(z). > Xk 1

Acknowledgement

We would like to thank the referees for their valuable remarks which greatly helped to improve this article.

References

[1] Lars V. Ahlfors. . McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics.

[2] John A. Baker. Plane curves, polar coordinates and winding numbers. Mathe- matics Magazine, 64(2):75–91, 1991.

[3] Michiel Hazewinkel. Encyclopaedia of Mathematics: Reaction-Diffusion Equa- tion – Stirling Interpolation Formula. Encyclopaedia of Mathematics. Springer Netherlands, 2012.

[4] Peter Henrici. Applied and computational complex analysis. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series— integration—conformal mapping—location of zeros, Pure and Applied Mathe- matics.

[5] Matilde Legua and Luis M. S´anchez-Ruiz. Cauchy principal value contour in- tegral with applications. Entropy, 19(5), 2017.

[6] Jian Ke Lu. A generalized residue theorem and its applications. Wuhan Daxue Xuebao, (3):1–8, 1978.

18 [7] Mihail A. Lavrentьev i Boris V. Xabat. Metody Teorii Funkci˘i Kompleksnogo Peremennogo. Izdatelьstvo Nauka, Moskva, 1951.

[8] Raghavan Narasimhan and Yves Nievergelt. Complex Analysis in One Variable. Birkh¨auser Boston, 2000.

[9] Chuanhao Yin, Hui Jiang, Linhu Li, Rong L¨u, and Shu Chen. Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-hermitian systems. Phys. Rev. A, 97:052115, 2018.

[10] Shou Guo Zhong. A generalized residue theorem for unbounded multiply con- nected regions of the second class. Acta Math. Sci., 14(2):163–167, 1994.

[11] Shou Guo Zhong. Transcendental singular integrals and a generalized residue theorem. Acta Math. Sci., 15(1):43–48, 1995.

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