COMPLEX ANALYSIS Spring 2014

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1 COMPLEX ANALYSIS Spring Preliminaries Homotopical topics Our textbook slides over a little problem when discussing homotopy. The standard definition of homotopy is for not necessarily piecewise smooth curves. In this section, by curve we will understand (I hope) a continuous map from an interval to a metric space (or a topological space). If the curve is to be piecewise smooth, I shall say so. Since one can assume that the parameter interval is the interval [0, 1], I will assume this. Just in case: In this section, if X is a topological or metric space (either one will do; you can even assume that X = C in all occurrences), a curve in X is a continuous map γ : [0, 1] X. It is closed iff γ(0) = γ(1). I ll begin with some definitions. A notation that could be useful, if X is a metric space, z, w X is C(X, a, w) = {γ : γ : [0, 1] X is a curve, γ(0) = z, γ(1) = w}. (Curves from z to w. I ll write C(X, z) for C(X, z, z). Then one defines: Definition 1 Let X be a metric space and assume that γ 0 C(X, z 0, w 0 ),γ 1 : [0, 1] C(X, z 1, w 1 ). Assume first z 0 w 0, z 1 w 1. We say the curves γ 0, γ 1 are homotopic, and we write γ 0 γ 1 iff there exists a continuous function F : [0.1] [0, 1] X such that F (s, 0) = γ 0 (s), F (s, 1) = γ 1 (s) for 0 s 1. The function F is said to realize the homotopy and we also write F : γ 0 γ 1 to indicate this. If z 0 = w 0 then we say γ 0, γ 1 are homotopic, and we write γ 0 γ 1 iff z 1 = w 1, they are homotopic by the definition so far, and the function F satisfies F (0, t) = F (1, t) for all t [0, 1]. The idea, of course, is that the curve γ 0 deforms to γ 1 as t goes from 0 to 1; for each t (0, 1) the map s F (s, t) is an intermediate curve that we could call γ t. If the curves are closed, the intermediate curves should also be closed. But there is a modified definition for curves sharing a common beginning and end. Definition 2 Let X be a metric space and assume that γ 0, γ 1 C(X, z 0, w 0 ). We say the curves γ 0, γ 1 are homotopic, and we write γ 0 = γ1 iff there exists a continuous function F : [0.1] [0, 1] X such that F (s, 0) = γ 0 (s), F (s, 1) = γ 1 (s) for 0 s 1, and F (0, t) = z 0, F (1, t) = w 0 for all t [0, 1]. We also write F : γ 0 = γ1 to indicate that F realizes this homotopy. Notice the slightly different symbol used to denote this homotopy! 1

2 For us, the most important case is the case of piecewise smooth curves. Stein-Shakarchi defines two such curves to be homotopic by adding, sort of in a hidden way, an extra hypothesis; namely, that for every t [0, 1], the map s F (s, t) is piecewise smooth; i.e., γ t is a piecewise smooth curve. I suppose it may be true that if two piecewise smooth curves are homotopic, then the homotopy can be realized by a function F such that s F (s, t) is piecewise smooth for all t [0, 1]. I do not know, but if wewant to integrate over any of these intermediate curves, we need to have some additional condition on the curve; continuity is not enough. We will try a more rigorous approach, but as warm-up exercise here is the first exercise on this topic. If you feel uncomfortable with general metric spaces, assume X = C with the usual topology. Exercise 1 Let X be a metric space; z, w X. Prove: Both form s of homotopy are equivalence relations in C(X, z, w). Reflexivity should be obvious, symmetry quite easy, and transitivity not too hard. Another easy exercise is Exercise 2 Assume X is a convex subset of C (or of R n, or of any vector space over R or C where a topology has been defined in which the vector operations are continuous). Let z, w X. Prove: γ 0 = γ1 for all γ 0, γ 1 C(X, z, w). In C(X, z) there is a particular curve I ll call δ (for now) defined by δ(t) = z for all t [0, 1]. Definition 3 Let X be a metric space, let z 0, z 1, z 2 X and let γ 0 C(X, z 0, z 1 ), γ 1 C(X, z 1, z 2 ). We define γ 0 γ 1 : [0, 1] X by { γ0 (2t), if 0 t 1/2, (γ 0 γ 1 )(t) = γ 1 (2t 1), if 1/2 < t 1. Exercise 3 Let X be a metric space, let z 0, z 1, z 2 X and let γ 0 C(X, z 0, z 1 ), γ 1 C(X, z 1, z 2 ). Verify that γ 1 γ 2 C(X, z 0, z 2 ). If z 0 = z 2 verify that γ 1 = γ2 if and only if γ 0 γ 1 = δ C(X, z0 ). The curve γ 0 γ 1 is just γ 0 followed by γ 1 ; for piecewise smooth curves it is equivalent to what we called γ 0 + γ 1. We changed notation here to emphasize that from a point of view of homotopy the operation is not commutative. We are not going to go deeply into homotopy; in fact, we barely skim the surface. But here are some basic facts that are not too hard to prove if one has the time. Let X be a metric space, z X. If γ C(X, z), denote by [γ] the equivalence class of γ with respect to the relation =: Then the following properties hold: [γ] = {γ C(X, z) : γ = γ}. If γ 0, γ 1, γ 0, γ 1 C(X, z) and γ 0 = γ 0, γ 1 = γ 1, then γ 0 γ 1 = γ 0 γ 1. It thus makes sense to define the product of [γ 0 ], [γ 1 ] C(X, z)/ = by [γ 0 ] [γ 1 ] = [γ 0 γ 1 ]. 2

3 With the product as defined, C(X, z)/ = becomes a group. To prove this, one has to prove that (γ 0 γ 1 ) γ 2 = γ0 (γ 1 γ 2 ) for all γ 0, γ 1, γ 2 C(X, z); γ δ = δ γ = γ for all γ C(X, z), and defining for γ C(X, z) the curve γ 1 by γ 1 (s) = γ(1 s) for s [0, 1], this definition is homotopically invariant (i.e., γ = γ implies γ 1 = γ 1, so that [γ] 1 := [γ 1 ] is well defined, and also that γ 1 γ = γ γ 1 = δ. This group is usually denoted by π 1 (X, z) ad could be called the fundamental group of X based at z. If X is path connected, z, w X, let Γ : [0, 1] X be a curve joining w to z; that is Γ C(X, w, z). If γ C(X, z) define γ sc(x, w) as the curve Γ, followed by γ, followed by the inverse of Γ. One way of doing this is to define it as (Γ γ) Γ 1. Just in case, Γ(4s), if 0 s 1/4, (Γ γ) Γ 1 (s) = γ(4s 1), if 1/4 < s 1/2, Γ(2 2s), if 1/2 < s 1. This map is again invariant under homotopies (including replacing Γ by a homotopically equivalent curve) so that it induces a map from π(x, w) to π(x, z). One sees that this map is a group isomorphism. Assume X is path connected. Because of the isomorphism between groups based at different points, one selects one such group as a representative, calls it the fundamental group of X, and denotes it by π(x). A pathwise connected metric space is said to be simply connected if π(x) is the trivial group. In other words, if taking z X, every curve in C(X, z) is homotopically equivalent to δ. Some examples in C are: C itself is, of course, simply connected. It is easy to see that all convex subsets of C, more generally, star shaped subsets, are simply connected. A star center can be used like a black hole; it sucks in all closed curves. Consider X = C\{0}. Take z = 1 (for example). One can show that every closed curve based at 1 is homotopically equivalent either to the constant curve at 1 (thus the unit element of the group) or equivalent to a circle gone through n times in either the positive or negative direction. Briefly: π(c\{0}) = Z. But I don t want to get into too deep waters here. Homotopy is a vast, important part of mathematics, best studied in an algebraic topology course. Eventually we would like to prove the following theorem. This may take a while. Theorem 1 Let Ω be an open, connected, subset of C. The following statements are equivalent. 3

4 1. Ω is simply connected in the sense that every closed curve is homotopic to a constant curve; i.e., if z Ω, π 1 (Ω, z) = {[δ]}. 2. Every piecewise smooth closed curve γ satisfies W γ (z) = 0 for all z / Ω. 3. (C { })\Ω is connected. For the time being we will prove, a bit more correctly than our textbook perhaps, that Cauchy s Theorem is valid in a simply connected region. For this all we need to prove, given what we have proved already, is that part 1 of Theorem 1 implies part 2. The first step will be to extend the notion of winding number to include curves that are not necessarily piecewise smooth. 2 Winding Numbers for General Curves The purpose of this section is to extend the definition of W γ (z) so that it makes sense for all closed curves γ in C, z C, z / C. This is done by approximating a closed curve by piecewise smooth ones, and the most effective way is using Weierstrass Theorem, which allows you to even approximate by almost totally smooth curves. But, in a slightly more awkward way, it can also be done by polygonals. Lemma 2 Let γ : [0, 1] C be a closed curve; γ(0) = γ(1). There exists a sequence {γ n } of piecewise smooth closed curves, γ n : [0, 1] C, for all n, such that {γ n } converges uniformly to γ; i.e, for each ϵ > 0 there exists N such that γ n (t) γ(t) < ϵ for all m N, all t [0, 1]. The proof of this lemma is not terribly difficult. If you are familiar with the Stone-Weierstrass theorem (or just Weierstrass). Then you can show that γ can be uniformly approximated by functions that are finite linear combinations of the set {e 2πimt } m Z. These curves are not only piecewise smooth, they are everywhere as smooth as it can get. Otherwise you can use uniform continuity to partition given ϵ > 0 the interval [0, 1] into subintervals [t i 1, t i ] with t i t i 1 < δ, δ > 0 so chosen that γ(s) γ(t) < ϵ/2 if s t < δ. If then γ δ is the polygonal curve whose links are the line segments from γ(t i 1 to γ(t i ), then γ(t) γ δ (t) < ϵ for all t [0, 1]. We do not request that the approximating curves start and end at the same points where the approximated curve starts and end, but the would be easy to achieve. For notational convenience, if γ 0, γ 1 : [0, 1] C are curves, set γ 0 γ 1 = sup γ 0 (t) γ 1 (t). 0 t 1 It is now fairly easy to see how one can define W γ (z) if z / γ and γ is merely continuous (and closed). We have 4

5 Lemma 3 Let γ 0, γ 1 be closed, piecewise smooth curves in C and let ρ = γ 0 γ 1. If z C and d(z, γ 0 γ 1) 2ρ, then W γ0 (z) = W γ1 (z). Proof. It would be nice if we could assume that in some sense a map such as γ W γ (z) is continuous. It is, but one has to be careful. In principle, W γ (z) depends on γ, and γ can be as close as you wish to another curve while the derivative γ is in a different galaxy. So we have to imitate the proof of Theorem in Stein-Shakarchi. Let δ > 0 be such that if s 1 s 2 < δ, s 1, s 2 [0, 1], then γ 0 (s 1 ) γ 0 (s 2 ) < ρ, and γ 1 (s 1 ) γ 1 (s 2 ) < ρ. Partition [0, 1] by s 0 = 0 < s 1 < < s n = 1 where s i s i 1 < δ for i = 1, 2,..., n. For i = 1,..., n let σ i be the closed curve defined by σ i = γ 0 [si 1,s i ] + L[γ 0 (s i ), γ 1 (s i )] γ 1 [si 1,s 0 ] + L[γ 1 (s i 1, s i ], where the operations are used in the original (usual) sense and L[z, w] is the le segment from z to w. One sees that σ i is a closed, piecewise smooth curve and σ i D(γ 0(s i 1, 2ρ). Since z is in the complement of this disk, (1) W σi (z) = 0, i = 1, 2,..., n. Because the curves γ 0, γ 1 are closed, one has that L(γ 0 (1), γ 1 (1)) = L(γ 0 (0), γ 1 (0)) = L(γ 1 (0), γ 0 (0)) and it follows that (2) σ σ n = γ 0 γ 1 The lemma is an immediate consequence of (1), (2). In class I ll probably prove this by pictures, which make it all quite obvious. It should be clear now how to define W γ (z) for an arbitrary closed curve in C, z / γ. We let ρ = 1 4 d(z, γ ). Let γ 0 be a piecewise smooth curve such that γ γ 0 < ρ. Define W γ (z) = W γ0 (z). By Lemma 2, such a curve γ 0 exists. By Lemma 3, any two such curves have the same winding number with respect to z. Thus W γ (z) is well defined. Moreover, if γ happens to be piecewise smooth, the new definition coincides with the previous one. We also have Theorem 4 Let γ be a closed curve in C. Then 1. W γ (z) Z for all z C\γ. 2. z W γ (z) is constant in all connected components of C\γ. 3. W γ (z) = 0 for z in the unbounded connected component of γ. 4. Lemma 3 is valid for all closed curves; if γ 0, γ 1 are closed curves in C (not necessarily piecewise smooth), if z C and d(z, γ 0 γ 1) 2 γ 0 γ 1, then W γ0 (z) = W γ1 (z). 5

6 Exercise 4 Prove Theorem 4. The proof of Lemma 3 was not too bad, quite straightforward. In Real and Complex Analysis, Rudin provides what can be considered a very elegant proof of this lemma. Elegant proofs are sometimes best appreciated once one has seen the brute force proof. Here is, slightly adapted, Rudin s argument. Assume γ 0, γ 1, z, ρ are as in Lemma 3. Define γ : [0, 1] C by γ = γ 1 z γ 0 z, which makes sense since the denominator is never 0. Then γ is a closed piecewise smooth curve and (3) We have γ (s) γ(s) = γ 1(s) γ 1 (s) z γ 0(s) γ 0 (s) z. γ(s) 1 = γ 1 (s) γ 0 (s) γ 0 (s) z γ 1 γ 0 γ 0 (s) z ρ 2ρ = 1 2 < 1 for all s [0, 1], implying that γ D 1 (1); in particular 0 is in the unbounded component of C\γ hence W γ (0) = 0. By (??), 0 = W γ (0) = 1 dζ 2πi γ ζ = 1 1 γ (s) 1 2πi 0 γ(s) ds = γ 1 1(s) 0 γ 1 (s) z ds γ 0(s) 0 γ 0 (s) z ds = W γ1 (0) W γ0 (0), providing another proof of Lemma 3. 3 Homotopy, once again The homotopy we consider here is the one of the first type, where we do not assume the curves start and end at the same place. Even so, in a simply connected open subset of C, any two curves are homotopic. In fact, assume Ω is simply connected, open and γ 0, γ 1 are curves in Ω. Let λ, µ be curves in Ω; λ from γ 1 (0) to γ 0 (0), µ from γ 0 (1) to γ 1 (1) then, in the notation of section 1, Γ = γ 0 µ γ1 1 λ is a closed curve in Ω. With δ C(Ω, γ 0 (0)) the constant curve, let F : Γ = δ (second type of homotopy). It is not too hard to see how one can fashion G : γ 0 γ 1 out of F ; I may add an explicit definition later. Or not. At this point, all the heavy lifting has been done. It is smooth sailing, at least for a while. The following lemma is an immediate consequence of Part 4 of Theorem 4. Lemma 5 Let Ω be open in C and let γ 0, γ 1 be closed curves in Ω such that γ 0 γ 1 in Ω. Then W γ0 (z) = W γ1 (z). for all z / Ω. This argument works in every simply connected topological space. 6

7 The proof of Theorem 1 can begin. We essentially showed that 1 implies 2. In fact, if Ω is simply connected and γ is a closed piecewise smooth curve in Ω, then (say γ(0) = γ(1) = z), then γ δ and it follows from Lemma 5 that W γ (z) = W δ (z) = 0 for all z / Ω. Appealing now to the general form of Cauchy s Theorem proved in the notes Cauchy and Runge under the same roof, we proved: Theorem 6 Let Ω be a simply connected open subset of C. If f : Ω C is holomorphic, then f(z) dz = 0 for all closed piecewise smooth curves γ in Ω. The two following corollaries are also immediate. γ Corollary 7 Let Ω be a simply connected open subset of C and let f : Ω C be holomorphic. There exists F : Ω C holomorphic such that f = F. Corollary 8 Let Ω be a simply connected open subset of C and let f : Ω C be holomorphic. Assume f(z) 0 for all z Ω. Then there exists an analytic determination of log f in Ω; i.e., there exists g : Ω C holomorphic such that f(z) = e g(z) for all z Ω. We can also prove that 2 of Theorem 1 implies 3. The hypothesis is that every closed piecewise smooth curve gm in Ω satisfies W γ (z) = 0 for all z / Ω. The conclusion should be, (C { }\Ω is connected. Another way of phrasing this is to say C\Ω is either empty (so Ω = C), or C\Ω is an unbounded, connected set. Assume the conclusion is false. Then C\Ω = F K, F, K closed, F K, K compact and not not empty. (In this situation, either F = or F is unbounded). Let W = C\F, so W is open. We have W = Ω K. Now K is a compact subset of the open set W and I refer to my previous notes, Theorem 2, applying it to the holomorphic function f(z) = 1 for all z Ω, to conclude that there is a cycle Γ = m j=1 γ j such that Γ W \K = Ω and W Γ (z) = 1 for all z K; in particular there is z K, j with W γj (z) 0. Since z K implies z / Ω (and there is such a z since K is not empty), we have a contradiction. But closing the circle will take time. 7