Chapter Five. Cauchy s Theorem

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1 hapter Five auchy s Theorem 5.. Homotopy. Suppose D is a connected subset of the plane such that every point of D is an interior point we call such a set a region and let and 2 be oriented closed curves in D. We say is homotopic to 2 in D if there is a continuous function H : S D, where S is the square S,s : s,t, such that H, describes and H, describes 2, and for each fixed s, the function H,s describes a closed curve s in D. The function H is called a homotopy between and 2. Note that if is homotopic to 2 in D, then 2 is homotopic to in D. Just observe that the function K,s H, s is a homotopy. It is convenient to consider a point to be a closed curve. The point c is a described by a constant function c. We thus speak of a closed curve being homotopic to a constant we sometimes say is contractible to a point. Emotionally, the fact that two closed curves are homotopic in D means that one can be continuously deformed into the other in D. Example Let D be the annular region D z : z 5. Suppose is the circle described by 2e i2,; and 2 is the circle described by 2 4e i2,. Then H,s 2 2se i2 is a homotopy in D between and 2. Suppose 3 is the same circle as 2 but with the opposite orientation; that is, a description is given by 3 4e i2,. A homotopy between and 3 is not too easy to construct in fact, it is not possible! The moral: orientation counts. From now on, the term closed curve will mean an oriented closed curve. 5.

2 Another Example Let D be the set obtained by removing the point z from the plane. Take a look at the picture. Meditate on it and convince yourself that and K are homotopic in D, but and are homotopic in D, while K and are not homotopic in D. Exercises. Suppose is homotopic to 2 in D, and 2 is homotopic to 3 in D. Prove that is homotopic to 3 in D. 2. Explain how you know that any two closed curves in the plane are homotopic in. 3. A region D is said to be simply connected if every closed curve in D is contractible to a point in D. Prove that any two closed curves in a simply connected region are homotopic in D. 5.2 auchy s Theorem. Suppose and 2 are closed curves in a region D that are homotopic in D, and suppose f is a function analytic on D. Let H,s be a homotopy between and 2. For each s, the function s describes a closed curve s in D. Let I be given by I s fzdz. Then, 5.2

3 I fh,s H,s. Now let s look at the derivative of I. We assume everything is nice enough to allow us to differentiate under the integral: I d ds fh,s H,s f H,s H,s f H,s H,s fh,s H,s H,s H,s fh,s 2 H,s fh,s 2 H,s fh,s H,s fh,s H,s. But we know each H,s describes a closed curve, and so H,s H,s for all s. Thus, I fh,s H,s fh,s H,s, which means I is constant! In particular, I I, or fzdz fzdz. 2 This is a big deal. We have shown that if and 2 are closed curves in a region D that are homotopic in D, and f is analytic on D, then fzdz 2 fzdz. An easy corollary of this result is the celebrated auchy s Theorem, which says that if f is analytic on a simply connected region D, then for any closed curve in D, 5.3

4 fzdz. In court testimony, one is admonished to tell the truth, the whole truth, and nothing but the truth. Well, so far in this chapter, we have told the truth and nothing but the truth, but we have not quite told the whole truth. We assumed all sorts of continuous derivatives in the preceding discussion. These are not always necessary specifically, the results can be proved true without all our smoothness assumptions think about approximation. Example Look at the picture below and convince your self that the path is homotopic to the closed path consisting of the two curves and 2 together with the line L. We traverse the line twice, once from to 2 and once from 2 to. Observe then that an integral over this closed path is simply the sum of the integrals over and 2, since the two integrals along L, being in opposite directions, would sum to zero. Thus, if f is analytic in the region bounded by these curves (the region with two holes in it), then we know that fzdz fzdz 2 fzdz. Exercises 4. Prove auchy s Theorem. 5. Let S be the square with sides x, and y with the counterclockwise orientation. Find 5.4

5 S z dz. dz, where is any circle centered at z with the usual counterclockwise z 6. a)find orientation: Ae 2it, t. b)find dz, where is any circle centered at z with the usual counterclockwise z orientation. c)find dz, where is the ellipse z 2 4x2 y 2 with the counterclockwise orientation. [Hint: partial fractions] d)find dz, where is the circle z 2 x2 x y 2 with the counterclockwise orientation. 8. Evaluate Log z 3dz, where is the circle z 2 oriented counterclockwise. 9. Evaluate dz where is the circle described by e 2it,, and n is an z n integer.. a)does the function fz z have an antiderivative on the set of all z? Explain. b)how about fz z n, n an integer?. Find as large a set D as you can so that the function fz e z2 have an antiderivative on D. 2. Explain how you know that every function analytic in a simply connected (cf. Exercise 3) region D is the derivative of a function analytic in D. 5.5