CHAPTER 5 Complex Integration BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail:com 1 Module-4: Winding Number 1 Introduction We now prove an important result which is quite different from the theory of functions of a real variable. Theorem 1. If f(z) is analytic in a domain D then all the derivatives of f(z) exists and are analytic in D. Proof. Let z0 be an arbitrary point in D and we consider the circle C having center at z0 and radius r, so small that C lies entirely within D (see Fig. 1). Then for n = 0; 1; 2;:::; Fig. 1: and for any point α inside C, we have Z (n) n! f(z) f (α) = n+1 dz: 2πi C (z − α) Thus f(z) has a derivative of all order in a neighbourhood of z0. Since z0 is arbitrary, we can conclude that all the derivatives of f(z) exists and are analytic functions in D. This proves the theorem. 2 Note 1. In basic calculus we have learnt that the existence of the derivative of f(x) does not guarantee the continuity of the derivative f 0(x). We consider the function f(x) = x5=3 which has a first derivative for all x 2 R and 5 f 0(x) = x2=3: 3 See that f 0(x) does not have a first derivative at x = 0 and therefore f(x) does not have a second derivative at the origin. Similarly, f(x) = x7=3 has a first and second derivative in R but has no third derivative at x = 0: Thus Theorem 1 does not hold in the case of functions of a real variable. This result gives an essential difference between functions of a real variable and func- tions of a complex variable. Corollary 1. If f = u + iv is analytic in a domain D, then all partial derivatives of u and v exist and are continuous in D. Proof. Let f = u + iv be analytic in D. Then 0 f (z) = ux + ivx = vy − iuy: f 0(z) being analytic in D, it is continuous in D and hence each of the first order partial derivatives of u and v exist and are continuous in D. Also, because of analyticity of f 0(z) in D, we obtain from above equation 00 f (z) = uxx + ivxx = (vx)y − i(ux)y = (vy)x − i(uy)x: This process may be continued to conclude that u and v have continuous partial deriva- tives of all orders at each point where the function f = u + iv is analytic. R log z Example 1. Evaluate C (z−1)3 dz; where C is the circle j z − 2 j= 3=2: Solution. The center of the circle C : j z − 2 j= 3=2 is z = 2 and its radius is 3=2. The log z function f(z) = (z−1)3 is not analytic at the point z = 1 which lies inside C. Therefore, 3 from Cauchy's integral formula for nth derivative (here n = 2) we obtain Z 00 2! log z f (1) = 3 dz; where f(z) = log z 2πi C (z − 1) Z log z 00 i:e: 3 dz = πif (1) = −πi: C (z − 1) We might expect a function f(z) to be analytic at a point α if Z f(z)dz = 0 C along every simple closed rectifiable curve in which α is an interior point. Unfortunately, this converse of Cauchy's fundamental theorem is not true. As for example, Z 1 2 dz = 0 C z along every simple closed rectifiable curve C having the origin as an interior point. This is 1 because f(z) = z2 is analytic in the region between C and some circle j z j= " contained in C. Therefore by Cauchy's theorem for multiply connected regions, we have Z 1 Z 1 Z 2π i"eiθ 2 dz = 2 dz = 2 2iθ dθ C z jzj=" z 0 " e i Z 2π = e−iθdθ = 0: " 0 But we do have a partial converse to Cauchy's fundamental theorem even when the domain D is not simply connected. Theorem 2. (Morera's Theorem) R Let f(z) be continuous in a simply connected domain D. If Γ f(z)dz = 0 along every simple closed rectifiable curve Γ contained in D, then f(z) is analytic in D. Proof. Let α be a fixed point and z be a variable point in D. Then the value of the R z integral α f(w)dw is independent of the path so long the path lies in D (see Fig. 2). Let Z z g(z) = f(w)dw: α Let h be a complex number such that z + h lies inside D. Then Z z+h Z z Z z+h g(z + h) − g(z) = f(w)dw − f(w)dw = f(w)dw: α α z 4 Fig. 2: Hence g(z + h) − g(z) 1 Z z+h j − f(z) j = j f(w)dw − f(z) j h h z 1 Z z+h ≤ j f(w) − f(z) jj dw j : (1) j h j z Since f(z) is continuous at z, given " > 0 we can find a δ > 0 such that j f(w)−f(z) j< " whenever j w − z j< δ: Choosing j h j< δ we see that j f(w) − f(z) j< ", for all point w lying on the straight line joining z to z + h. Therefore using ML-formula we obtain from (1) g(z + h) − g(z) 1 j − f(z) j ≤ · "· j h j = ": h j h j Proceeding to the limit as h ! 0 we get g(z + h) − g(z) lim = f(z) h!0 h i:e: g0(z) = f(z): This shows that g(z) is analytic in D. Thus f(z), being the derivative of an analytic function, is itself analytic in D. This proves the theorem. Note 2. We note that even if f(z) is analytic in a domain D, we are not guaranteed that R C f(z)dz = 0 along every simple closed rectifiable curve C contained in D. The function 5 1 f(z) = z is analytic in the annulus bounded by the circles j z j= 1=3 and j z j= 3=2. But R jzj=1(1=z)dz = 2πi even though the circle is contained in the annulus. Note, however, that the interior of j z j= 1 is not contained in the annulus, so that Cauchy's theorem is not applicable. For a simply connected domain, it is true that the integral around every simple closed contour in the domain is zero. In view of Morera's theorem, we can say that a necessary and sufficient condition for a continuous function to be analytic in a simply connected domain is that the integral be independent of the path of integration. 2 Winding Number or Index of a Curve Now we introduce the concept of winding number (or index) of a closed curve with respect to a point. Suppose that γ is a simple closed curve in the complex plane C and a is a given point in C n γ: Then there is a useful formula that measures how often γ winds around a. For example, if γ : = fz : z − a = reiθ; 0 ≤ θ ≤ 2kπg; then γ encircles the point a k times in positive sense. Also, Z dz Z 2kπ ireiθ = iθ dθ = 2kπi γ z − a 0 re 1 Z dz i:e: = k: 2πi γ z − a From this we also see that if γ encircles the point a k times in negative sense, then 1 Z dz = −k: 2πi γ z − a 1 R dz In both the cases, either in positive sense or in negative sense, 2πi γ z−a is an integer. Now we give the analytic definition of winding number. Definition 1. Let γ be a closed rectifiable curve and a be a point not on the curve γ: Then, the index or winding number n(γ; a) of γ with respect to the point a is given by the integral 1 Z 1 n(γ; a) = dz: 2πi γ z − a Geometrically speaking, the winding number counts the number of rounds of a curve around a point. It is a positive integer for positive oriented curves and a negative integer for curves with negative orientation. If the point a is enclosed by a simple closed rectifiable 6 curve, then n(γ; a) = 1 and, if a is outside the curve, then n(γ; a) = 0 (see Fig. 3). Figure 3 shows how the index number changes with respect to the orientation and nature Fig. 3: of the curves. Here we see that, for first one n(γ; a) = 1, n(γ; b) = 0; for second one n(γ; a) = −1, n(γ; b) = 0 and for last n(γ; a) = 1, n(γ; b) = 0, n(γ; c) = 2. Properties of the Winding Number (i) For every closed rectifiable curve γ in C and every a 2 C n γ, n(γ; a) is an integer. (ii) If γ is a closed rectifiable curve in C, then the mapping a ! n(γ; a) is a continuous function of a at any a 62 γ. (iii) If γ is the sum of two closed rectifiable curves γ1 and γ2 in a complex plane, then for every a 62 γ, we have n(γ; a) = n(γ1; a) + n(γ2; a): Example 2.