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COMPLEX ANALYSIS–Spring 2014 1 Winding Numbers

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COMPLEX ANALYSIS–Spring 2014 1 Winding Numbers COMPLEX ANALYSIS{Spring 2014 Cauchy and Runge Under the Same Roof. These notes can be used as an alternative to Section 5.5 of Chapter 2 in the textbook. They assume the theorem on winding numbers of the notes on Winding Numbers and Cauchy's formula, so I begin by repeating this theorem (and consequences) here. but first, some remarks on notation. A notation I'll be using on occasion is to write Z f(z) dz jz−z0j=r for the integral of f(z) over the positively oriented circle of radius r, center z0; that is, Z Z 2π it it f(z) dz = f z0 + re rie dt: jz−z0j=r 0 Another notation I'll use, to avoid confusion with complex conjugation is Cl(A) for the closure of a set A ⊆ C. If a; b 2 C, I will denote by λa;b the line segment from a to b; that is, λa;b(t) = a + t(b − a); 0 ≤ t ≤ 1: Triangles being so ubiquitous, if we have a triangle of vertices a; b; c 2 C,I will write T (a; b; c) for the closed triangle; that is, for the set T (a; b; c) = fra + sb + tc : r; s; t ≥ 0; r + s + t = 1g: Here the order of the vertices does not matter. But I will denote the boundary oriented in the order of the vertices by @T (a; b; c); that is, @T (a; b; c) = λa;b + λb;c + λc;a. If a; b; c are understood (or unimportant), I might just write T for the triangle, and @T for the boundary. 1 Winding Numbers In this section we assume that γ :[a; b] ! C is a closed path (recall that in this course,except if otherwise noted, all curves, paths are piecewise smooth). For z2 = γ∗ we define 1 Z 1 Wγ (z) = dζ: 2πi γ ζ − z By Theorem 1 in the notes on Winding Numbers and Cauchy's formula or, even γ0(s) better, by Theorem 5.4, Chapter 2, of our textbook∗ (with F (z; s) = ), γ(s) − z ∗To be precise, the Theorem does not apply in an immediate way, since γ0 is not necessarily continuous on all of [a; b]. However, the integral is a finite sum of integrals over intervals in which γ0 is continuous, hence a finite sum of holomorphic functions. 1 ∗ Wγ : Cnγ ! C is holomorphic; very much in particular continuous. One calls Wγ (z) the winding number of γ with respect to z; it measures how many times γ winds around z. It is also called the index of z with respect to γ. Its main properties are summarized in the following theorem. ∗ Theorem 1 Let γ :[a; b] ! C be a closed path and define Wγ : γ ! C as y above. Then Wγ is integer valued and constant on each connected component of the complement of γ∗. Moreover, it is 0 on the unbounded component of Cnγ∗. Proof. A continuous integer valued function on a connected open set has little choice but to be constant. So all that we need to prove is that Wγ is integer valued and zero on the unbounded component of the complement of the curve. For the former, let z2 = γ∗ (to be fixed for now) and consider the function F :[a; b] ! C defined by 0 − R t γ (s) ds F (t) = (γ(t) − z)e a γ(s)−z for t 2 [a; b]. It is easy to see that F is continuous and that F is differentiable wherever γ is differentiable with 0 R t γ0(s) ds 0 0 γ (t) − F (t) = γ (t) − (γ(t) − z) e a γ(s)−z = 0 γ(t) − z at such points. It follows that F is constant in all the intervals in which γ0 is defined. However since F is continuous in [a; b] it has to be constant in the whole interval; in particular F (a) = F (b); that is R b γ0(s) ds − −2πiW (z) γ(a) − z = F (a) = F (b) = (γ(b) − z)e a γ(s)−z = (γ(b) − z)e γ : Since γ(a) − z = γ(b) − z 6= 0 (because γ is closed and z2 = γ∗), we can cancel −2πiWγ (z) γ(a) − z to get e = 1. This implies that Wγ (z) 2 Z. As mentioned above, proving that Wγ (z) 2 Z also proves that it is constant in the connected components of the complement of γ∗. Let U be the unbounded one of these components. Let z 2 U and let R = dist(z; γ∗). Then 1 1 ≤ ζ − z R for all ζ 2 γ∗ and 1 1 jW (z)j ≤ Λ(γ): γ 2π R In U there is z at arbitrary large distance from the curve; we can find z with R as large as we please, for example R > Λ(γ)/π for which jWgm(z)j < 1=2. One may do some searching, but the only integer of absolute value less than 1=2 that one can find is 0, so Wγ (z) = 0 for z large enough, thus for all z 2 U. yWith some perversity, I believe some textbooks refer to the components of the complement of γ∗ as the components of γ. 2 Proposition 2 Let C be the positively oriented circle of radius r, centered at z 2 Ω. Then 0 1; if z 2 Dr(z0); WC (z) = 0 if jz − z0j > r: Proof. It is clear that the complement of C∗ has exactly two connected com- ponents; Dr(z0) and U = fz 2 C : jz − z0j > rg. It is also clear that U is the unbounded component, thus WC (z) = 0 for all z 2 U. We also have, writing it C(t) = z0 + ie , 0 ≤ t ≤ 2π, 1 Z 2π rieit dt 1 rieit 1 Z 2 WC (z0) = it = it = πi dt = 1: 2πi 0 (z0 + ie ) − z0 2πi re 2πi 0 Thus WC (z) = 1 for all z 2 Dr(z0). 2 A slightly stronger version of the Cauchy The- orem for a triangle Theorem 3 Let Ω be an open subset of C, let p 2 Ω and assume f :Ω ! C is continuous, and f is holomorphic at all z 2 Ω, z 6= p. Let T be a closed triangle entirely contained in Ω, then Z f(z) dz = 0: @T Proof. Let T = T (A; B; C) ⊂ Ω. We consider four cases. Case 1. p2 = T . Then the previous version of the theorem is valid. Case 2. p is a vertex of the triangle, say p = C. Select two points Q; R on the sides AC, BC,respectively, and consider the boundaries of T1;T2;T3 of the triangles of vertices RCQ, ABR, ARQ, respectively. If we integrate over @T1 = @T (B; C; Q), @T2 = @T (A; B; R) and over @T3 = @T (A; R:Q), we see that Z Z Z Z f(z) dz = f(z) dz + f(z) dz + f(z) dz: @T @T1 @T2 @T3 3 Z Z By Case 1, f(z) dz = f(z) dz = 0 so that @T2 @T3 Z Z f(z) dz = f(z) dz ≤ sup jf(z)jΛ(@T1): ∗ @T @T1 z2@T1 The sup in the last inequality above is finite, since f is continuous everywhere, while Λ(@T1) ! 0 as Q; R ! C. The result follows. Case 3. p is on a side of T , but not a vertex. Suppose p is on the side AC. We then split T into the triangles T1 = T (A; B; p) and T2 = T (p; B; C). Z Z By case 2, f(z) dz = f(z) dz = 0, thus @T1 @T2 Z Z Z f(z) dz = f(z) dz + f(z) dz = 0: @T @T1 @T2 Case 4. The point p is in the interior of T . The picture shows how to split T so we are in case 3 The following theorem is now proved EXACTLY as in the case of a function holomorphic everywhere, so the proof is omitted. Theorem 4 Let Ω be an open star shaped set and assume f :Ω ! C is holo- morphic at all points of Ω except, perhaps at one point p 2 Ω. Assume, however, f is continuous at p. Then f has a primitive in Ω; i.e., there is F :Ω ! C holomorphic at all points of Ω such that F 0(z) = f(z) for all z 2 Ω. In particu- lar, Z f(z) dz = 0 γ 4 for all closed curves in Ω. Theorem 5 A somewhat more general Cauchy formula. Let Ω be a star shaped region. Let f :Ω ! C be holomorphic. Let γ :[a; b] ! Ω be a closed path in Ω. Then, for every z 2 Ωnγ∗ we have 1 Z f(ζ) Wγ (z)f(z) = dζ: 2πi γ ζ − z In particular if Cl(Dr(z0)) = fz 2 C : jz − z0j ≤ rg ⊂ Ω (r > 0), then 1 Z f(ζ) f(z) = C dζ 2πi jz−z0j=r ζ − z for all z 2 Dr(z0). Proof. The second formula is an immediate consequence of the first since ∗ WC (z) = 1 for all z 2 Dr(z0). For the first one let z 2 Ωnγ and define g :Ω ! C by ( f(ζ)−f(z) ; if ζ 2 Ωnfzg; g(ζ) = ζ−z f 0(z); if ζ = z: This function is continuous in Ω and holomorphic everywhere except perhaps at z.
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