1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,270 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,270 --> 00:00:18,220 at ocw.mit.edu. 8 00:00:32,210 --> 00:00:33,150 HERBERT GROSS: Hi. 9 00:00:33,150 --> 00:00:35,790 Today, we're going to conclude our survey 10 00:00:35,790 --> 00:00:37,740 of complex variables. 11 00:00:37,740 --> 00:00:41,430 And a rather fitting topic, I imagine, 12 00:00:41,430 --> 00:00:44,970 would be what it means to integrate a complex valued 13 00:00:44,970 --> 00:00:47,730 function of a complex variable. 14 00:00:47,730 --> 00:00:51,000 And I thought that that's what we would talk about today. 15 00:00:51,000 --> 00:00:54,690 And in fact, let's just call today's lecture "Integrating 16 00:00:54,690 --> 00:00:55,770 Complex Functions." 17 00:00:55,770 --> 00:00:57,840 But let me emphasize, I really do 18 00:00:57,840 --> 00:01:00,210 mean by a complex function the thing 19 00:01:00,210 --> 00:01:02,370 that we've been stressing throughout the course, 20 00:01:02,370 --> 00:01:06,810 a complex valued function of a complex variable. 21 00:01:06,810 --> 00:01:10,020 The idea being that if you have a complex function 22 00:01:10,020 --> 00:01:12,910 of a real variable, essentially, this breaks down, 23 00:01:12,910 --> 00:01:15,240 as we've seen in the previous assignment, 24 00:01:15,240 --> 00:01:21,010 into studying regular functions of a single real variable. 25 00:01:21,010 --> 00:01:25,180 In other words, if you have u of t plus iv of t, 26 00:01:25,180 --> 00:01:28,410 where t is a real variable, this really gives us no problem. 27 00:01:28,410 --> 00:01:32,040 This is sort of like the vector treatment, a vector function, 28 00:01:32,040 --> 00:01:34,240 of a scalar variable, et cetera. 29 00:01:34,240 --> 00:01:35,880 So what I'm saying here, though, is 30 00:01:35,880 --> 00:01:38,340 let's make sure that it's clear that we're still 31 00:01:38,340 --> 00:01:41,490 talking about mappings that map complex numbers 32 00:01:41,490 --> 00:01:43,120 into complex numbers. 33 00:01:43,120 --> 00:01:47,130 Now by way of review, how did we define the definite integral 34 00:01:47,130 --> 00:01:49,380 in the case of a single real variable? 35 00:01:49,380 --> 00:01:51,830 We define the integral from x0 to x1 36 00:01:51,830 --> 00:01:55,230 f of x dx to be a certain limit. 37 00:01:55,230 --> 00:01:59,430 Namely, we partitioned the interval from x0 to x1 38 00:01:59,430 --> 00:02:04,320 into n pieces, calling the pieces delta x1 up to delta xn. 39 00:02:04,320 --> 00:02:08,789 We took a number, say, c sub k star in the k partition, 40 00:02:08,789 --> 00:02:14,370 formed the sum f of c sub k star delta xk-- 41 00:02:14,370 --> 00:02:16,620 xk went from 1 to n-- 42 00:02:16,620 --> 00:02:19,830 and took that limit as the maximum-sized interval. 43 00:02:19,830 --> 00:02:22,530 The maximum delta x sub k went to 0. 44 00:02:22,530 --> 00:02:25,080 And if that limit existed, that was defined 45 00:02:25,080 --> 00:02:27,160 to be the definite integral. 46 00:02:27,160 --> 00:02:30,240 Now, we did see a few other things along the way. 47 00:02:30,240 --> 00:02:32,308 But this is the main definition. 48 00:02:32,308 --> 00:02:34,350 You see, this is the definition that we're using. 49 00:02:34,350 --> 00:02:36,330 Some of the consequences of this definition 50 00:02:36,330 --> 00:02:39,660 were one, if we could find the function capital 51 00:02:39,660 --> 00:02:42,420 F, whose derivative was little f, 52 00:02:42,420 --> 00:02:44,820 then this integral was simply capital F 53 00:02:44,820 --> 00:02:48,220 of x1 minus capital F of x0. 54 00:02:48,220 --> 00:02:51,030 And we also saw that if we wished, and we didn't 55 00:02:51,030 --> 00:02:53,970 have to, but if we wished, we could view this thing 56 00:02:53,970 --> 00:02:58,590 geometrically by observing that the x-axis was the domain of f. 57 00:02:58,590 --> 00:03:00,960 The y-axis was the range of f. 58 00:03:00,960 --> 00:03:02,880 We could then plot the curve y equals 59 00:03:02,880 --> 00:03:09,390 f of x, look at the area of the region R between x0 and x1, 60 00:03:09,390 --> 00:03:12,570 and the area of the region R was the value 61 00:03:12,570 --> 00:03:14,370 of this definite integral. 62 00:03:14,370 --> 00:03:16,050 But the important point is what? 63 00:03:16,050 --> 00:03:17,880 That the definition of the integral 64 00:03:17,880 --> 00:03:21,410 is as a limit of an infinite sum. 65 00:03:21,410 --> 00:03:23,640 And now the question is, how shall we 66 00:03:23,640 --> 00:03:27,990 define the analogous thing for a complex valued 67 00:03:27,990 --> 00:03:29,940 function of a complex variable? 68 00:03:29,940 --> 00:03:32,700 In other words, what shall we mean by integral 69 00:03:32,700 --> 00:03:37,020 from z0 to z1, f of z dz? 70 00:03:37,020 --> 00:03:39,300 And because we've been so successful 71 00:03:39,300 --> 00:03:41,190 with this technique in the past, it 72 00:03:41,190 --> 00:03:43,800 would seem that the simplest thing to do 73 00:03:43,800 --> 00:03:47,730 would be to simply replace x by z 74 00:03:47,730 --> 00:03:50,130 every place in this definition. 75 00:03:50,130 --> 00:03:53,910 In other words, let's simply do this. 76 00:03:53,910 --> 00:03:59,640 Let's define the integral z0 to z1 f of z dz 77 00:03:59,640 --> 00:04:05,070 to be the limit as the maximum delta z sub k approaches 0. 78 00:04:05,070 --> 00:04:11,130 Summation-- k goes from 1 to n, f of c sub k star times 79 00:04:11,130 --> 00:04:14,490 delta z sub k, where all we have done 80 00:04:14,490 --> 00:04:19,750 is replaced xs every place by zs wherever they appeared. 81 00:04:19,750 --> 00:04:22,560 Now, the reason I've put question marks here 82 00:04:22,560 --> 00:04:24,420 is that all of a sudden, a problem 83 00:04:24,420 --> 00:04:27,690 occurs that never occurred in the real case. 84 00:04:27,690 --> 00:04:29,730 You see, the key reason that these problems are 85 00:04:29,730 --> 00:04:32,970 going to occur, as I'll explain in a moment, hinges on the fact 86 00:04:32,970 --> 00:04:35,190 that in the real case, the domain of f 87 00:04:35,190 --> 00:04:36,210 was one-dimensional. 88 00:04:36,210 --> 00:04:38,340 It was an axis, the x-axis. 89 00:04:38,340 --> 00:04:41,490 In the complex variable case, the domain of f 90 00:04:41,490 --> 00:04:42,520 is two-dimensional. 91 00:04:42,520 --> 00:04:45,840 It's the entire xy plane, the Argand diagram. 92 00:04:45,840 --> 00:04:47,920 You see, the problem is simply this. 93 00:04:47,920 --> 00:04:50,490 What do you mean by delta z sub k? 94 00:04:50,490 --> 00:04:53,790 Intuitively, it certainly should seem to mean what? 95 00:04:53,790 --> 00:04:58,710 The line that joins the point in the Argand diagram z 96 00:04:58,710 --> 00:05:01,828 sub k minus 1 to the point z sub k. 97 00:05:01,828 --> 00:05:03,870 In other words, remember, we view complex numbers 98 00:05:03,870 --> 00:05:04,710 as being vectors. 99 00:05:04,710 --> 00:05:06,420 They have a magnitude and a direction. 100 00:05:06,420 --> 00:05:10,200 So delta z sub k can be viewed as the vector which 101 00:05:10,200 --> 00:05:12,240 joins these two points. 102 00:05:12,240 --> 00:05:13,560 The trouble is this. 103 00:05:13,560 --> 00:05:16,110 Let's take a look in the Argand diagram 104 00:05:16,110 --> 00:05:18,830 and locate the points z0 and z1. 105 00:05:18,830 --> 00:05:20,070 Here's z0. 106 00:05:20,070 --> 00:05:20,820 Here's z1. 107 00:05:20,820 --> 00:05:22,410 That's all that's given. 108 00:05:22,410 --> 00:05:26,910 Where shall these intermediary points z sub-- 109 00:05:26,910 --> 00:05:28,530 where should these intermediary points 110 00:05:28,530 --> 00:05:31,380 that are going to give rise to delta z sub k be? 111 00:05:31,380 --> 00:05:33,660 In other words, there seem to be many points that we 112 00:05:33,660 --> 00:05:35,550 can pick for a partition. 113 00:05:35,550 --> 00:05:38,040 In particular, it would seem that the most 114 00:05:38,040 --> 00:05:43,410 natural definition would be to specify a particular curve C 115 00:05:43,410 --> 00:05:47,880 and then look at delta z sub k in terms of the curve 116 00:05:47,880 --> 00:05:52,140 C. Because you see once, the curve C is specified, 117 00:05:52,140 --> 00:05:55,620 if I partition things now, notice that the delta z sub 118 00:05:55,620 --> 00:05:59,160 ks do not connect random points, but rather points on the curve 119 00:05:59,160 --> 00:05:59,880 C. 120 00:05:59,880 --> 00:06:03,060 The difficult point is that there are many different curves 121 00:06:03,060 --> 00:06:05,970 that join z0 to z1. 122 00:06:05,970 --> 00:06:08,610 So perhaps the first thing that we should do 123 00:06:08,610 --> 00:06:11,310 is, keeping in mind this ambiguity, maybe 124 00:06:11,310 --> 00:06:14,880 we should come back to this definition and put in a C 125 00:06:14,880 --> 00:06:18,030 to indicate that we're going to talk about the integral of f 126 00:06:18,030 --> 00:06:23,760 of z dz from the point z0 to the point z1 along the curve 127 00:06:23,760 --> 00:06:28,290 C, keeping in mind, you see, that the reason that we have 128 00:06:28,290 --> 00:06:32,700 to specify the curve here is that z0 and z1 are 129 00:06:32,700 --> 00:06:34,180 in the plane. 130 00:06:34,180 --> 00:06:34,680 You see? 131 00:06:34,680 --> 00:06:38,880 Going back to this example here, notice that here, 132 00:06:38,880 --> 00:06:41,700 when we said let's go from x0 to x1, 133 00:06:41,700 --> 00:06:44,940 there was only one direction in which 134 00:06:44,940 --> 00:06:47,010 you could go from x0 to x1. 135 00:06:47,010 --> 00:06:50,880 Namely, since the domain of definition of f was the x-axis, 136 00:06:50,880 --> 00:06:54,930 the direction had to be in the direction of the x-axis. 137 00:06:54,930 --> 00:06:56,850 You see, in the Argand diagram case, 138 00:06:56,850 --> 00:06:59,250 I can go along any curve in the plane, 139 00:06:59,250 --> 00:07:04,420 provided that the curve connects the two points in question. 140 00:07:04,420 --> 00:07:07,080 Now, the other question that I think I should point out here, 141 00:07:07,080 --> 00:07:09,780 even though it may be bringing up a question that you might 142 00:07:09,780 --> 00:07:11,850 not have anticipated-- notice that when 143 00:07:11,850 --> 00:07:15,240 I interpret the definite integral as an area here, 144 00:07:15,240 --> 00:07:20,820 notice that the lower bound is the domain of f itself. 145 00:07:20,820 --> 00:07:24,450 And the upper bound is already from the curve y 146 00:07:24,450 --> 00:07:28,300 equals f of x-- that the upper curve brings out the function 147 00:07:28,300 --> 00:07:28,800 here. 148 00:07:28,800 --> 00:07:31,140 You see, the x-axis is the domain. 149 00:07:31,140 --> 00:07:33,000 The y-axis is the range. 150 00:07:33,000 --> 00:07:35,160 I would like you to observe that there 151 00:07:35,160 --> 00:07:39,230 is no similar interpretation along the curve 152 00:07:39,230 --> 00:07:42,000 C. In other words, if we were to drop perpendiculars 153 00:07:42,000 --> 00:07:46,860 from z0 to z1 down to the x-axis and talk about that area, 154 00:07:46,860 --> 00:07:49,770 that would be the wrong area to talk about. 155 00:07:49,770 --> 00:07:52,740 Because you see, the point is that that region is 156 00:07:52,740 --> 00:07:55,140 determined just by our domain. 157 00:07:55,140 --> 00:07:56,400 Notice that this curve-- 158 00:07:56,400 --> 00:07:57,900 and this is very, very crucial-- 159 00:07:57,900 --> 00:08:00,630 that the curve that we're talking about here, C, 160 00:08:00,630 --> 00:08:05,760 has absolutely nothing in the world to do with this f of z. 161 00:08:05,760 --> 00:08:09,840 Just like when we computed mass using density, 162 00:08:09,840 --> 00:08:12,450 the density had nothing to do with the region 163 00:08:12,450 --> 00:08:14,550 that you were integrating with respect to. 164 00:08:14,550 --> 00:08:19,230 The point to observe here is that we're assuming that f of z 165 00:08:19,230 --> 00:08:21,180 is defined every place in the plane. 166 00:08:21,180 --> 00:08:23,190 f of z is defined here. 167 00:08:23,190 --> 00:08:24,060 It's defined here. 168 00:08:24,060 --> 00:08:25,230 It's defined here. 169 00:08:25,230 --> 00:08:28,080 It's also defined on the curve C. 170 00:08:28,080 --> 00:08:31,780 Notice that f, in this case, is a complex number, 171 00:08:31,780 --> 00:08:34,500 which means that it has a real and imaginary part. 172 00:08:34,500 --> 00:08:36,690 f is no place in this diagram. 173 00:08:36,690 --> 00:08:39,240 And to plot what f does, I would have 174 00:08:39,240 --> 00:08:42,850 to use the uv plane because f itself is two-dimensional. 175 00:08:42,850 --> 00:08:44,940 In other words, I would need two dimensions 176 00:08:44,940 --> 00:08:48,780 for the domain and two dimensions for the range. 177 00:08:48,780 --> 00:08:51,780 And I hope that that part is fairly clear to you now. 178 00:08:51,780 --> 00:08:55,950 Don't confuse the path of integration with the integrand 179 00:08:55,950 --> 00:08:56,850 f of z. 180 00:08:56,850 --> 00:09:01,170 They are entirely different concepts. 181 00:09:01,170 --> 00:09:02,640 At any rate, I hope that this does 182 00:09:02,640 --> 00:09:06,480 suggest the concept of a line integral in some sense. 183 00:09:06,480 --> 00:09:10,110 What I'm saying now is, OK, I want to integrate this 184 00:09:10,110 --> 00:09:11,850 from z0 to z1. 185 00:09:11,850 --> 00:09:14,160 That means I want to compute a particular limit. 186 00:09:14,160 --> 00:09:17,190 I picked the curve that joins z0 to z1. 187 00:09:17,190 --> 00:09:20,040 Let's assume that in Cartesian coordinates, 188 00:09:20,040 --> 00:09:22,080 that curve parametrically has the form 189 00:09:22,080 --> 00:09:23,790 x is some function of t. 190 00:09:23,790 --> 00:09:26,910 y is some function of t, where t is a real variable. 191 00:09:26,910 --> 00:09:29,370 That's important. t is a real variable. 192 00:09:29,370 --> 00:09:32,460 By the way, as we mentioned when we were studying vectors, 193 00:09:32,460 --> 00:09:36,690 if I want to introduce vector notation, the radius vector, 194 00:09:36,690 --> 00:09:39,030 and write the curve C in vector form, 195 00:09:39,030 --> 00:09:41,460 notice that the curve C also has the form 196 00:09:41,460 --> 00:09:45,300 that r is x of ti plus y of tj. 197 00:09:45,300 --> 00:09:49,140 And finally, if I now want to write the curve in terms 198 00:09:49,140 --> 00:09:52,200 of complex numbers-- after all, the Argand diagram 199 00:09:52,200 --> 00:09:54,940 now is the domain of the complex numbers. 200 00:09:54,940 --> 00:09:57,900 What is the connection between the xy plane and the Argand 201 00:09:57,900 --> 00:09:58,800 diagram? 202 00:09:58,800 --> 00:10:03,430 The x-axis names the real part, the y-axis, the imaginary part. 203 00:10:03,430 --> 00:10:06,780 In other words, the unit-- 204 00:10:06,780 --> 00:10:09,810 we think of the i vector as representing 205 00:10:09,810 --> 00:10:13,210 the real part, the j vector as the imaginary part. 206 00:10:13,210 --> 00:10:16,380 Notice that changing this notation 207 00:10:16,380 --> 00:10:19,590 into the language of the Argand diagram results 208 00:10:19,590 --> 00:10:22,590 in this equation being replaced by z 209 00:10:22,590 --> 00:10:26,250 equals x of t plus iy of t. 210 00:10:26,250 --> 00:10:30,930 Where notice now that z is a complex valued function 211 00:10:30,930 --> 00:10:34,280 of the real variable t. 212 00:10:34,280 --> 00:10:36,920 And t, of course, goes from some fixed value 213 00:10:36,920 --> 00:10:39,830 t0 to some fixed value t1. 214 00:10:39,830 --> 00:10:41,540 Because after all, what we're saying 215 00:10:41,540 --> 00:10:46,190 is that the curve Z equals x of t plus iy of t 216 00:10:46,190 --> 00:10:48,380 is being traced out in terms of t. 217 00:10:48,380 --> 00:10:51,980 When t0 corresponds to being at z0, 218 00:10:51,980 --> 00:10:54,770 t1 corresponds to being at z1. 219 00:10:54,770 --> 00:10:57,800 The idea now is this. 220 00:10:57,800 --> 00:11:02,000 We go back to our definition of what the definite integral 221 00:11:02,000 --> 00:11:03,590 meant in terms of a limit. 222 00:11:03,590 --> 00:11:06,140 We observe that we'll assume that we'll 223 00:11:06,140 --> 00:11:09,650 make the further restriction that our curves are smooth, 224 00:11:09,650 --> 00:11:12,440 so that dz, dt will exist. 225 00:11:12,440 --> 00:11:15,620 What we now do is go back to our infinite sum 226 00:11:15,620 --> 00:11:20,560 and multiply and divide by delta t sub k, the idea being-- 227 00:11:20,560 --> 00:11:22,460 notice that this is what? 228 00:11:22,460 --> 00:11:26,510 A complex valued function of a real variable. 229 00:11:26,510 --> 00:11:29,810 Because I can handle complex valued functions of a real 230 00:11:29,810 --> 00:11:32,360 variable-- namely, I just break them down into real 231 00:11:32,360 --> 00:11:35,570 and imaginary parts and use the same calculus that we used 232 00:11:35,570 --> 00:11:37,290 in part one of this course-- 233 00:11:37,290 --> 00:11:40,730 the idea is that with this suggestive notation, 234 00:11:40,730 --> 00:11:46,670 and remembering how the infinite sum gave rise 235 00:11:46,670 --> 00:11:50,300 to the integral notation, looking at this, 236 00:11:50,300 --> 00:11:55,310 you see we arrive at the fact that the integral from z0 to z1 237 00:11:55,310 --> 00:11:58,190 along the curve C, f of z dz. 238 00:11:58,190 --> 00:12:01,220 If we want to write that in terms of an integral 239 00:12:01,220 --> 00:12:05,180 involving a real variable t, it's simply what? 240 00:12:05,180 --> 00:12:08,558 The integral from t0 to t1 f of z of t. 241 00:12:08,558 --> 00:12:10,100 In other words, we're assuming that z 242 00:12:10,100 --> 00:12:13,220 is a function of t along the curve C-- 243 00:12:13,220 --> 00:12:17,060 we know what z looks like along the curve C in terms of t-- 244 00:12:17,060 --> 00:12:20,960 times the derivative of z with respect to t, z prime of t, 245 00:12:20,960 --> 00:12:22,430 times dt. 246 00:12:22,430 --> 00:12:24,800 And we'll have plenty of exercises, including 247 00:12:24,800 --> 00:12:26,930 one in this lecture, to show you how 248 00:12:26,930 --> 00:12:30,230 we use this particular result. On the other hand, 249 00:12:30,230 --> 00:12:33,170 keep in mind that this is just what? 250 00:12:33,170 --> 00:12:36,350 One way of visualizing what the infinite sum is. 251 00:12:36,350 --> 00:12:39,530 A second way to visualize the same problem 252 00:12:39,530 --> 00:12:41,950 is to recall that we very frequently like 253 00:12:41,950 --> 00:12:45,140 things in terms of u and v components, coordinates. 254 00:12:45,140 --> 00:12:47,660 The idea is that if we now want to introduce 255 00:12:47,660 --> 00:12:52,520 u and v in the usual manner, f of z is then u plus iv. 256 00:12:52,520 --> 00:12:55,730 Delta z is delta x plus i delta y. 257 00:12:55,730 --> 00:13:00,590 And coming back to our basic definition over here, you see, 258 00:13:00,590 --> 00:13:05,570 what we can now do is rewrite f as u plus iv, 259 00:13:05,570 --> 00:13:11,870 rewrite delta z, sub k as delta x sub k plus i delta y sub k, 260 00:13:11,870 --> 00:13:14,120 multiply this thing out, pick off 261 00:13:14,120 --> 00:13:16,370 the real and the imaginary parts, 262 00:13:16,370 --> 00:13:18,800 write this as two separate sums, et cetera, 263 00:13:18,800 --> 00:13:20,630 and take the limit separately. 264 00:13:20,630 --> 00:13:22,370 What this leads to, if you want to go 265 00:13:22,370 --> 00:13:25,220 through the whole rigorous process quickly, 266 00:13:25,220 --> 00:13:27,110 just coming up with the right answer, 267 00:13:27,110 --> 00:13:28,760 is that essentially, what we do is, 268 00:13:28,760 --> 00:13:33,500 given this particular integral, we write f as u plus iv. 269 00:13:33,500 --> 00:13:36,230 We write dz as dx plus idy. 270 00:13:36,230 --> 00:13:37,400 Y 271 00:13:37,400 --> 00:13:40,280 We then multiply the way we ordinarily 272 00:13:40,280 --> 00:13:41,600 would with complex numbers. 273 00:13:41,600 --> 00:13:44,300 See, udx minus vdy. 274 00:13:44,300 --> 00:13:48,260 And the imaginary part is vdx plus udy. 275 00:13:48,260 --> 00:13:51,020 Break this up as two separate integrals. 276 00:13:51,020 --> 00:13:54,020 And the key point is notice that both of these 277 00:13:54,020 --> 00:13:57,440 integrals-- forgetting about the coefficient of here being i, 278 00:13:57,440 --> 00:13:59,530 this is the only place where i appears. 279 00:13:59,530 --> 00:14:04,520 Notice that both integrals are now ordinary line integrals. 280 00:14:04,520 --> 00:14:08,220 You see u as a real valued function of x and y. 281 00:14:08,220 --> 00:14:11,150 v is a real valued function of x and y. 282 00:14:11,150 --> 00:14:15,290 Along the curve C, x and y are functions of t. 283 00:14:15,290 --> 00:14:17,720 Remember, the equation of the curve C 284 00:14:17,720 --> 00:14:20,010 was x was some function of t. 285 00:14:20,010 --> 00:14:22,700 y was some function of t. 286 00:14:22,700 --> 00:14:25,700 In other words, again, this is not only a line integral. 287 00:14:25,700 --> 00:14:28,070 But if you want to use this parametric form, 288 00:14:28,070 --> 00:14:31,700 both of these integrals involve real integrals 289 00:14:31,700 --> 00:14:32,600 of real variables. 290 00:14:32,600 --> 00:14:36,680 In other words, both u, v, dx, and dy 291 00:14:36,680 --> 00:14:39,470 are all expressible in terms of t and dt. 292 00:14:39,470 --> 00:14:43,110 And we can integrate this expression in the usual way. 293 00:14:43,110 --> 00:14:45,530 But the reason I like the u and v form 294 00:14:45,530 --> 00:14:48,080 is that we've stressed what it means for a function 295 00:14:48,080 --> 00:14:51,980 to be analytic in terms of the real and imaginary parts, 296 00:14:51,980 --> 00:14:52,760 et cetera. 297 00:14:52,760 --> 00:14:55,160 Remember, keep an eye on this. 298 00:14:55,160 --> 00:14:57,440 Remember this particular expression here. 299 00:14:57,440 --> 00:14:59,070 And here's the key point. 300 00:14:59,070 --> 00:15:03,440 If it turns out that u and v are the real and imaginary parts 301 00:15:03,440 --> 00:15:05,430 of an analytic function-- in other words, 302 00:15:05,430 --> 00:15:08,120 if u plus iv is some analytic function f-- 303 00:15:08,120 --> 00:15:11,390 remember that the Cauchy-Riemann conditions told us 304 00:15:11,390 --> 00:15:13,550 that the partial of u with respect to x 305 00:15:13,550 --> 00:15:15,980 is equal to the partial of v with respect to y. 306 00:15:15,980 --> 00:15:17,630 The partial of u with respect to y 307 00:15:17,630 --> 00:15:20,390 is minus the partial of v with respect to x. 308 00:15:20,390 --> 00:15:23,120 And if you look at these two conditions coupled 309 00:15:23,120 --> 00:15:24,950 with our definition of what it meant 310 00:15:24,950 --> 00:15:28,670 for a real differential in two variables to be exact, 311 00:15:28,670 --> 00:15:33,790 notice that this condition here tells us that this is exact. 312 00:15:33,790 --> 00:15:37,240 And this condition here tells us that this is exact. 313 00:15:37,240 --> 00:15:41,350 In other words, both of these ordinary line integrals 314 00:15:41,350 --> 00:15:44,350 are exact if u plus iv is analytic. 315 00:15:44,350 --> 00:15:46,840 And again, if you want to leave the complex number part out 316 00:15:46,840 --> 00:15:49,120 of this, if all we know is that the partial 317 00:15:49,120 --> 00:15:51,430 of the real function u with respect to x 318 00:15:51,430 --> 00:15:53,350 equals the partial of the real valued function 319 00:15:53,350 --> 00:15:56,380 v with respect to y, et cetera. 320 00:15:56,380 --> 00:15:58,900 And in particular then, because this is exact, 321 00:15:58,900 --> 00:16:01,600 what happened when a differential was exact? 322 00:16:01,600 --> 00:16:03,940 Remember, if the differential was exact, 323 00:16:03,940 --> 00:16:08,140 the line integral was what? 324 00:16:08,140 --> 00:16:11,080 Dependent only on the endpoints, but not 325 00:16:11,080 --> 00:16:13,600 on the path that connected the end points. 326 00:16:13,600 --> 00:16:17,830 Or stated in still other words, if the integrand was exact, 327 00:16:17,830 --> 00:16:21,820 the line integral around the closed curve was 0. 328 00:16:21,820 --> 00:16:24,130 So what we're saying in summary is 329 00:16:24,130 --> 00:16:28,450 that notice that if f equals u plus iv is analytic, 330 00:16:28,450 --> 00:16:35,050 then the integral from z0 to z1 f of z dz is independent of C. 331 00:16:35,050 --> 00:16:37,810 It depends only on z0 and z1. 332 00:16:37,810 --> 00:16:40,960 And in that case, I do not have to specify what 333 00:16:40,960 --> 00:16:44,440 curve C is joining z0 and z1. 334 00:16:44,440 --> 00:16:49,600 And in particular, if f of z happens to be analytic, 335 00:16:49,600 --> 00:16:54,220 so that this is exact, then the integral around a closed curve 336 00:16:54,220 --> 00:16:58,030 is 0 for all closed curves C. And that means again 337 00:16:58,030 --> 00:17:01,570 in particular, if I know that f of z is analytic, 338 00:17:01,570 --> 00:17:06,220 I do not have to indicate the curve that I'm forming 339 00:17:06,220 --> 00:17:08,190 the contour or integral around. 340 00:17:08,190 --> 00:17:08,690 You see? 341 00:17:08,690 --> 00:17:10,750 But I do have to put this in here 342 00:17:10,750 --> 00:17:13,750 if the function is not analytic. 343 00:17:13,750 --> 00:17:16,869 And by the way, let me just make one more remark in passing. 344 00:17:16,869 --> 00:17:21,400 And we'll emphasize this more in the learning exercises. 345 00:17:21,400 --> 00:17:26,440 Not only is it true that integral z0 to z1 f of z dz 346 00:17:26,440 --> 00:17:29,500 depends only on the end points if f is analytic. 347 00:17:29,500 --> 00:17:33,130 But the parallel structure that is obeyed in the real case 348 00:17:33,130 --> 00:17:34,630 also is true here. 349 00:17:34,630 --> 00:17:39,580 Namely, if f is analytic, not only does the integral from z0 350 00:17:39,580 --> 00:17:43,570 to z1 f of z dz not depend on the path, 351 00:17:43,570 --> 00:17:46,240 but it can be evaluated very simply just 352 00:17:46,240 --> 00:17:50,950 by computing capital F of z1 minus capital F of z0, 353 00:17:50,950 --> 00:17:55,150 where capital F is any function whose derivative is little f-- 354 00:17:55,150 --> 00:17:57,090 the same structure as before. 355 00:17:57,090 --> 00:18:00,490 But-- and this is extremely important to note-- 356 00:18:00,490 --> 00:18:04,750 the integral around the closed curve C, f of z dz, 357 00:18:04,750 --> 00:18:09,520 need not be 0 if f is not analytic. 358 00:18:09,520 --> 00:18:11,050 And I think the best way to show you 359 00:18:11,050 --> 00:18:13,270 that is by means of an example. 360 00:18:13,270 --> 00:18:16,480 Let's compute the integral around the closed curve 361 00:18:16,480 --> 00:18:22,270 C, dz over z, where C is the circle in the Argand 362 00:18:22,270 --> 00:18:27,550 diagram, centered at the origin with radius equal to r. 363 00:18:27,550 --> 00:18:30,130 And again, to emphasize what I said to you earlier 364 00:18:30,130 --> 00:18:35,380 in the lesson, do not confuse the integrand with the path C. 365 00:18:35,380 --> 00:18:37,630 Notice that the integrand here is what? 366 00:18:37,630 --> 00:18:38,800 What is the integrand? 367 00:18:38,800 --> 00:18:40,410 It's 1 over z. 368 00:18:40,410 --> 00:18:45,220 Notice that 1 over z is defined at any point in the Argand 369 00:18:45,220 --> 00:18:47,450 plane, all right? 370 00:18:47,450 --> 00:18:49,690 Notice also from our previous lecture 371 00:18:49,690 --> 00:18:53,710 on derivatives that 1 over z is analytic, except when z 372 00:18:53,710 --> 00:18:57,190 equals 0, in which case 1 over z isn't even defined. 373 00:18:57,190 --> 00:19:01,810 You see, what we're saying is notice that on the curve C, 374 00:19:01,810 --> 00:19:06,100 I can talk about f of 1 over z for each point on the curve. 375 00:19:06,100 --> 00:19:09,760 I can talk about 1 over z for every point inside the curve. 376 00:19:09,760 --> 00:19:13,390 I can talk about 1 over z for every point outside the curve. 377 00:19:13,390 --> 00:19:16,840 What I want to compute here is the integral 378 00:19:16,840 --> 00:19:19,560 as I move along this curve-- 379 00:19:19,560 --> 00:19:22,150 1 over z dz. 380 00:19:22,150 --> 00:19:25,420 Notice, by the way, that inside the interval, 381 00:19:25,420 --> 00:19:29,590 inside the circle, f of z is not analytic. f of z 382 00:19:29,590 --> 00:19:30,520 has a bad point. 383 00:19:30,520 --> 00:19:36,610 Namely, 1 over z is trouble when z equals 0. 384 00:19:36,610 --> 00:19:39,500 And notice that z equals 0 is inside my region. 385 00:19:39,500 --> 00:19:43,480 In other words, notice that my function f of z 386 00:19:43,480 --> 00:19:46,930 is analytic on the circle, but it's not analytic 387 00:19:46,930 --> 00:19:48,460 every place inside the circle. 388 00:19:48,460 --> 00:19:52,300 The one bad spot is when z is 0. 389 00:19:52,300 --> 00:19:54,580 So I do have a case where what? 390 00:19:54,580 --> 00:19:56,830 The integrand, which is 1 over z, 391 00:19:56,830 --> 00:19:59,230 is not analytic in the entire region, 392 00:19:59,230 --> 00:20:02,135 even though it does happen to be analytic on the boundary. 393 00:20:02,135 --> 00:20:04,510 You see, the boundary is made up of points, none of which 394 00:20:04,510 --> 00:20:07,600 is the origin because this is a circle of positive radius 395 00:20:07,600 --> 00:20:09,680 r centered at the origin. 396 00:20:09,680 --> 00:20:13,690 And again, keep in mind that f does not appear in this diagram 397 00:20:13,690 --> 00:20:14,380 anywhere. 398 00:20:14,380 --> 00:20:18,910 If I wanted to plot 1 over z, I have to use the uv plane. 399 00:20:18,910 --> 00:20:22,570 If I want to think of this in terms of a real interpretation, 400 00:20:22,570 --> 00:20:26,650 what I'm saying is visualize a force, which 401 00:20:26,650 --> 00:20:30,905 when written in the language of complex variables, is 1 over z. 402 00:20:30,905 --> 00:20:32,780 We don't have to worry about what that means. 403 00:20:32,780 --> 00:20:34,363 What it really means is write the real 404 00:20:34,363 --> 00:20:35,780 and the imaginary parts of this. 405 00:20:35,780 --> 00:20:38,490 And u is the real part, and v is the imaginary part. 406 00:20:38,490 --> 00:20:41,600 It's like computing the work as you go along this circle 407 00:20:41,600 --> 00:20:44,000 under the influence of that particular force. 408 00:20:44,000 --> 00:20:46,640 And the force has two components, you see, 409 00:20:46,640 --> 00:20:48,770 an i and a j component-- in terms 410 00:20:48,770 --> 00:20:52,220 of the Argand diagram, a real and an imaginary component. 411 00:20:52,220 --> 00:20:54,590 So I can't stress that point too much. 412 00:20:54,590 --> 00:20:57,530 Separate the integrand from the path 413 00:20:57,530 --> 00:20:59,220 that you're integrating with respect to. 414 00:20:59,220 --> 00:21:03,050 Now, what I claim is, is that if I integrate 1 over z 415 00:21:03,050 --> 00:21:07,250 along the curve C from beginning to end, let's 416 00:21:07,250 --> 00:21:09,010 say for the sake of definiteness, 417 00:21:09,010 --> 00:21:12,070 I start at the point 1 over here and go around-- 418 00:21:12,070 --> 00:21:14,870 at the point r and go around like this. 419 00:21:14,870 --> 00:21:17,570 I claim that this integral will not be 0. 420 00:21:17,570 --> 00:21:19,810 And I'm going to do it two ways for you, 421 00:21:19,810 --> 00:21:24,650 one which emphasizes the uv definition of integral, 422 00:21:24,650 --> 00:21:30,110 and one that emphasizes the f of z f prime dt definition. 423 00:21:30,110 --> 00:21:32,180 Method one is what? 424 00:21:32,180 --> 00:21:34,370 To compute this particular integral, 425 00:21:34,370 --> 00:21:37,550 you simply write the integrand in terms of its real 426 00:21:37,550 --> 00:21:41,690 and imaginary parts and write dz as dx plus i dy. 427 00:21:41,690 --> 00:21:46,760 If I do that, I mechanically get from here to here, remembering 428 00:21:46,760 --> 00:21:50,450 that to get into the standard form of a real plus 429 00:21:50,450 --> 00:21:53,390 i times a real, I must multiply the denominator 430 00:21:53,390 --> 00:21:55,010 by its complex conjugate. 431 00:21:55,010 --> 00:21:59,420 I multiply the numerator and denominator here by x minus iy. 432 00:21:59,420 --> 00:22:01,700 I then get this expression. 433 00:22:01,700 --> 00:22:04,160 You see x plus iy times x minus iy 434 00:22:04,160 --> 00:22:06,020 being x squared plus y squared. 435 00:22:06,020 --> 00:22:08,820 I now pick off the real and the imaginary parts. 436 00:22:08,820 --> 00:22:13,630 The real part here is xdx plus ydy because minus i times i 437 00:22:13,630 --> 00:22:15,020 is plus 1. 438 00:22:15,020 --> 00:22:20,540 The imaginary part is going to be minus ydx plus xdy. 439 00:22:20,540 --> 00:22:23,420 And so in terms of u and v components, 440 00:22:23,420 --> 00:22:26,690 this is the integral that I'm evaluating. 441 00:22:26,690 --> 00:22:29,990 And you see, notice that both of the integrands 442 00:22:29,990 --> 00:22:34,850 are line integrals-- real line integrals. 443 00:22:34,850 --> 00:22:39,200 Now, notice that on C, how do I describe the curve C? 444 00:22:39,200 --> 00:22:42,650 Remember, I'm trying to go so that my area always 445 00:22:42,650 --> 00:22:43,970 stays to the left. 446 00:22:43,970 --> 00:22:45,830 I'm starting at this point. 447 00:22:45,830 --> 00:22:48,200 So why not write this parametrically 448 00:22:48,200 --> 00:22:53,120 as x equals capital R cosine theta and y equals capital R 449 00:22:53,120 --> 00:22:53,870 sine theta? 450 00:22:53,870 --> 00:22:54,370 You see? 451 00:22:54,370 --> 00:22:56,780 That puts me in here when theta is 0, 452 00:22:56,780 --> 00:22:59,660 brings me around in the counterclockwise direction, 453 00:22:59,660 --> 00:23:02,720 back to the same point when theta is 2 pi. 454 00:23:02,720 --> 00:23:06,980 In other words, parametrically, the curve C is given how? 455 00:23:06,980 --> 00:23:08,720 x is our cosine theta. 456 00:23:08,720 --> 00:23:10,450 y is our sine theta. 457 00:23:10,450 --> 00:23:13,580 I could have used t in here if I wanted to, but why bother? 458 00:23:13,580 --> 00:23:16,700 dx is just minus R sine theta d theta. 459 00:23:16,700 --> 00:23:19,490 dy is our cosine theta d theta. 460 00:23:19,490 --> 00:23:23,420 x squared plus y squared on C is just R squared. 461 00:23:23,420 --> 00:23:27,000 And theta goes continuously from 0 to 2pi. 462 00:23:27,000 --> 00:23:30,200 So you see, if I now put all of this information into here, 463 00:23:30,200 --> 00:23:31,940 see, xdx is what? 464 00:23:31,940 --> 00:23:35,870 It's minus R squared sine theta cosine theta d theta. 465 00:23:35,870 --> 00:23:44,450 ydy is plus R squared sine theta cosine theta d theta. 466 00:23:44,450 --> 00:23:46,640 So therefore, the sum is 0. 467 00:23:46,640 --> 00:23:48,590 So therefore, this integral will be 0 468 00:23:48,590 --> 00:23:50,900 because the integrand is 0. 469 00:23:50,900 --> 00:23:56,000 On the other hand, minus ydx is minus R sine theta times 470 00:23:56,000 --> 00:23:58,040 minus R sine theta d theta. 471 00:23:58,040 --> 00:24:01,670 That's R squared sine squared theta d theta. 472 00:24:01,670 --> 00:24:06,270 xdy is R squared cosine squared theta d theta. 473 00:24:06,270 --> 00:24:10,340 So I add those together, and then I divide by R squared. 474 00:24:10,340 --> 00:24:14,100 And I have now converted this in terms of theta. 475 00:24:14,100 --> 00:24:16,940 So the integral goes from 0 to 2pi, the same 476 00:24:16,940 --> 00:24:19,550 as we did, you see, with ordinary line integrals. 477 00:24:19,550 --> 00:24:21,710 And to summarize this for you, because I 478 00:24:21,710 --> 00:24:24,080 may have talked kind of fast, all we're saying 479 00:24:24,080 --> 00:24:28,820 is that the integral around the closed curve C, dz over z, 480 00:24:28,820 --> 00:24:31,190 is the real part is 0. 481 00:24:31,190 --> 00:24:34,160 And the imaginary part, the coefficient of i, 482 00:24:34,160 --> 00:24:37,340 is integral from 0 to 2pi R squared 483 00:24:37,340 --> 00:24:41,590 sine squared theta plus cosine squared theta d theta over R 484 00:24:41,590 --> 00:24:42,480 squared. 485 00:24:42,480 --> 00:24:46,640 Notice that R cannot be 0 because C is the circle 486 00:24:46,640 --> 00:24:48,860 of radius R centered at the origin. 487 00:24:48,860 --> 00:24:50,000 On that circle-- 488 00:24:50,000 --> 00:24:53,220 I mean, R is a positive number, here. 489 00:24:53,220 --> 00:24:54,890 So it's just not 0. 490 00:24:54,890 --> 00:24:59,540 I can cancel the Rs because the radius of a circle can't be 0. 491 00:24:59,540 --> 00:25:01,460 And all I'm left with is sine squared theta 492 00:25:01,460 --> 00:25:03,780 plus cosine squared theta, which is 1. 493 00:25:03,780 --> 00:25:07,740 The integral d theta from 0 to 2pi is just 2pi. 494 00:25:07,740 --> 00:25:11,030 So this integral is just 2pi i. 495 00:25:11,030 --> 00:25:11,630 All right? 496 00:25:11,630 --> 00:25:16,700 And the second method is to use the formula what? 497 00:25:16,700 --> 00:25:21,600 Integral f of z dz is the integral f of z of theta dz 498 00:25:21,600 --> 00:25:23,750 d theta times d theta. 499 00:25:23,750 --> 00:25:26,690 In this case, notice that the circle centered 500 00:25:26,690 --> 00:25:31,640 at the origin with radius R is given in polar complex form 501 00:25:31,640 --> 00:25:34,640 by z equals R e to the i theta. 502 00:25:34,640 --> 00:25:37,670 Remember, in polar form, R is the magnitude. 503 00:25:37,670 --> 00:25:39,500 And theta is the angle. 504 00:25:39,500 --> 00:25:42,110 To say that you're on the circle of radius R 505 00:25:42,110 --> 00:25:46,460 centered origin simply says that the magnitude of the point 506 00:25:46,460 --> 00:25:48,230 must be R, the radius. 507 00:25:48,230 --> 00:25:52,790 And the angle can be any angle whatsoever between 0 and 2pi 508 00:25:52,790 --> 00:25:57,710 if you wanted to reverse the curve C just one time. 509 00:25:57,710 --> 00:26:02,600 At any rate, from this, notice that dz d theta is simply what? 510 00:26:02,600 --> 00:26:04,970 iRe to the i theta. 511 00:26:04,970 --> 00:26:09,440 Therefore, dz over z around C is just the integral 512 00:26:09,440 --> 00:26:11,480 from 0 to 2pi. 513 00:26:11,480 --> 00:26:15,140 f of z of t-- in this case, that's f of z of theta-- 514 00:26:15,140 --> 00:26:17,240 that's 1 over z of theta. 515 00:26:17,240 --> 00:26:21,890 z of theta is Re to the i theta, dz d theta, which is this, 516 00:26:21,890 --> 00:26:23,330 times d theta. 517 00:26:23,330 --> 00:26:26,490 You see, I've now converted this into a complex valued 518 00:26:26,490 --> 00:26:29,150 function of a real variable. 519 00:26:29,150 --> 00:26:31,700 At any rate, making all of these substitutions 520 00:26:31,700 --> 00:26:36,600 and simplifying, dz d theta being iRe to the i theta, 521 00:26:36,600 --> 00:26:39,650 and z being Re to the i theta, these cancel, 522 00:26:39,650 --> 00:26:41,540 leaving only a factor of i. 523 00:26:41,540 --> 00:26:43,040 The i comes outside. 524 00:26:43,040 --> 00:26:48,020 I have i integral 0 to 2pi d theta, which is just 2pi i. 525 00:26:48,020 --> 00:26:51,020 Again, notice the answers are the same 526 00:26:51,020 --> 00:26:52,940 because the techniques are equivalent. 527 00:26:52,940 --> 00:26:54,590 Which of the two ways is better? 528 00:26:54,590 --> 00:26:57,080 It depends on yourself, as to which way 529 00:26:57,080 --> 00:26:58,520 you feel more comfortable with. 530 00:26:58,520 --> 00:27:00,800 It depends on the particular problem that you're 531 00:27:00,800 --> 00:27:01,910 dealing with and the like. 532 00:27:01,910 --> 00:27:04,460 We'll emphasize these things in the exercises. 533 00:27:04,460 --> 00:27:06,740 But for the time being, I just hope 534 00:27:06,740 --> 00:27:08,630 that you have a feeling as to what 535 00:27:08,630 --> 00:27:13,160 we mean by integrating a complex valued function 536 00:27:13,160 --> 00:27:15,230 around a particular contour. 537 00:27:15,230 --> 00:27:17,000 And by the way, the subject called 538 00:27:17,000 --> 00:27:22,670 topology finds a natural inroad to the study 539 00:27:22,670 --> 00:27:25,070 of complex variables from this point of view. 540 00:27:25,070 --> 00:27:28,760 Namely, I call this "rubber sheet" geometry simply 541 00:27:28,760 --> 00:27:30,680 because of a properly that I'll mention 542 00:27:30,680 --> 00:27:31,850 for you in a few moments. 543 00:27:31,850 --> 00:27:33,410 But the idea is this. 544 00:27:33,410 --> 00:27:35,420 One of the very interesting factors 545 00:27:35,420 --> 00:27:38,450 about what we've just done, one of the interesting byproducts 546 00:27:38,450 --> 00:27:39,330 is this. 547 00:27:39,330 --> 00:27:44,230 Let's suppose I'm integrating f of z dz along a simple curve 548 00:27:44,230 --> 00:27:45,520 C1. 549 00:27:45,520 --> 00:27:50,480 And let's suppose that C2 is another curve that encloses C1. 550 00:27:50,480 --> 00:27:53,000 And let's suppose that f is analytic 551 00:27:53,000 --> 00:27:56,270 on the boundaries C1 and C2 and also 552 00:27:56,270 --> 00:27:58,610 in the region between them. 553 00:27:58,610 --> 00:28:03,770 In other words, I want to compute f of z dz along C1. 554 00:28:03,770 --> 00:28:06,980 I want to compute f of z dz along C2. 555 00:28:06,980 --> 00:28:10,790 And all I know is that f of z is analytic 556 00:28:10,790 --> 00:28:13,880 on and between C1 and C2. 557 00:28:13,880 --> 00:28:18,830 Then the amazing result is that the integral around C1, 558 00:28:18,830 --> 00:28:24,800 f of z dz, is equal to the integral around C2, f of z dz. 559 00:28:24,800 --> 00:28:27,920 That's where the word "rubber sheet" geometry comes in. 560 00:28:27,920 --> 00:28:29,540 What you're saying is that if there 561 00:28:29,540 --> 00:28:32,090 are no bad spots between the two curves, 562 00:28:32,090 --> 00:28:35,510 if I were to visualize this curve as being a rubber band, 563 00:28:35,510 --> 00:28:37,850 by stretching that rubber band, I 564 00:28:37,850 --> 00:28:41,120 could make it take the form of the curve C2. 565 00:28:41,120 --> 00:28:43,460 And what you're saying is given any curve that you're 566 00:28:43,460 --> 00:28:46,460 integrating around, if I can stretch that curve out 567 00:28:46,460 --> 00:28:48,620 into any position I want without breaking it, 568 00:28:48,620 --> 00:28:51,170 I stretch that curve out in such a way 569 00:28:51,170 --> 00:28:53,930 that as that curve is being stretched, 570 00:28:53,930 --> 00:28:57,200 it never goes through any points at which f of z 571 00:28:57,200 --> 00:28:58,940 fails to be analytic. 572 00:28:58,940 --> 00:29:02,660 Then it turns out that the integral around the new curve 573 00:29:02,660 --> 00:29:05,600 is the same as the integral around the old curve. 574 00:29:05,600 --> 00:29:07,580 And the easiest way to prove this 575 00:29:07,580 --> 00:29:09,920 is by the method of making cuts here. 576 00:29:09,920 --> 00:29:12,410 Remember we showed in our lecture on Green's theorem 577 00:29:12,410 --> 00:29:13,667 that I could make a cut. 578 00:29:13,667 --> 00:29:15,500 Actually, in the lecture on Green's theorem, 579 00:29:15,500 --> 00:29:17,870 I made two cuts so that I could vividly show 580 00:29:17,870 --> 00:29:19,370 you the two separate pieces. 581 00:29:19,370 --> 00:29:21,440 All you need is one cut. 582 00:29:21,440 --> 00:29:23,480 See, let me cut the region this way. 583 00:29:23,480 --> 00:29:25,470 And to emphasize what I've done here, 584 00:29:25,470 --> 00:29:27,380 I'll pull this apart just so that we 585 00:29:27,380 --> 00:29:28,700 can see what's happened here. 586 00:29:28,700 --> 00:29:30,380 You see, I've made the cut. 587 00:29:30,380 --> 00:29:32,270 And now, here's what I'm saying. 588 00:29:32,270 --> 00:29:34,282 This S and F stands for start and finish. 589 00:29:34,282 --> 00:29:36,240 You see, I'm going to start at this point here. 590 00:29:38,870 --> 00:29:44,390 I'm going to go along the simply connected region that I 591 00:29:44,390 --> 00:29:45,530 have here. 592 00:29:45,530 --> 00:29:48,410 Once I make the cut, this is a simply connected region. 593 00:29:48,410 --> 00:29:52,160 Notice that if I go along here, I am doing what? 594 00:29:52,160 --> 00:29:55,850 I am integrating a complex valued function 595 00:29:55,850 --> 00:30:00,440 over a simply connected region, where the function is analytic. 596 00:30:00,440 --> 00:30:02,877 Consequently, whatever this integral turns out 597 00:30:02,877 --> 00:30:04,460 to be-- in other words, the integral f 598 00:30:04,460 --> 00:30:08,030 of z dz along this particular contour 599 00:30:08,030 --> 00:30:10,970 must be 0 by our previous result, 600 00:30:10,970 --> 00:30:13,130 that if a function is analytic, the integral 601 00:30:13,130 --> 00:30:15,470 around the closed curve is 0. 602 00:30:15,470 --> 00:30:16,550 Now, here's the point. 603 00:30:16,550 --> 00:30:18,020 Notice that with this cut in here, 604 00:30:18,020 --> 00:30:19,580 when I look at the line integral, 605 00:30:19,580 --> 00:30:21,380 I traverse to cut twice-- 606 00:30:21,380 --> 00:30:24,440 once in one sense, once in the opposite sense-- 607 00:30:24,440 --> 00:30:25,890 so that cancels. 608 00:30:25,890 --> 00:30:26,390 See? 609 00:30:26,390 --> 00:30:27,860 That part cancels out. 610 00:30:27,860 --> 00:30:32,700 Notice that this integral around here, this part, 611 00:30:32,700 --> 00:30:35,840 is just the integral around the closed curve of C2 f of z 612 00:30:35,840 --> 00:30:39,187 dz because after all, we're assuming that this is together. 613 00:30:39,187 --> 00:30:40,770 I've just separated it here so that we 614 00:30:40,770 --> 00:30:42,060 can see what's happening. 615 00:30:42,060 --> 00:30:46,560 Notice that the inner curve is C1, 616 00:30:46,560 --> 00:30:48,390 but in the opposite orientation. 617 00:30:48,390 --> 00:30:51,600 You see, C1 is traversed in this sense. 618 00:30:51,600 --> 00:30:53,370 But the inner curve here is traversed 619 00:30:53,370 --> 00:30:54,660 in the opposite sense. 620 00:30:54,660 --> 00:30:59,790 So this integrand is just minus the integral C1 f of z dz. 621 00:30:59,790 --> 00:31:02,550 So if we put this whole thing together, what we have 622 00:31:02,550 --> 00:31:07,290 is that this integral with the cut along C 623 00:31:07,290 --> 00:31:09,120 is, on the one hand, 0. 624 00:31:09,120 --> 00:31:12,810 On the other hand, it's the integral around C2 f of z dz 625 00:31:12,810 --> 00:31:16,650 minus the integral around C1 f of z dz. 626 00:31:16,650 --> 00:31:20,970 Because this expression is 0, it means that this equals this. 627 00:31:20,970 --> 00:31:23,190 And that proves the result that we wanted. 628 00:31:23,190 --> 00:31:24,960 In other words, this must equal this 629 00:31:24,960 --> 00:31:26,880 because the difference is 0. 630 00:31:26,880 --> 00:31:30,300 How is this useful in the theory of complex integration? 631 00:31:30,300 --> 00:31:31,930 Well, let me give you an example. 632 00:31:31,930 --> 00:31:36,260 Let's go back to our old friend, integrating dz over z 633 00:31:36,260 --> 00:31:40,230 along the closed curve C. But now, the closed curve C 634 00:31:40,230 --> 00:31:43,410 is going to be much messier than before. 635 00:31:43,410 --> 00:31:45,120 It's not going to be a nice circle. 636 00:31:45,120 --> 00:31:48,160 The closed curve C is going to be something like this. 637 00:31:48,160 --> 00:31:51,810 And what I'd like to do is to find the value of this integral 638 00:31:51,810 --> 00:31:54,210 along the curve C. 639 00:31:54,210 --> 00:31:57,450 Now, the point is I could get into a mess trying 640 00:31:57,450 --> 00:31:59,850 to express C parametrically. 641 00:31:59,850 --> 00:32:01,140 But the beauty is what? 642 00:32:01,140 --> 00:32:05,010 First of all, notice that if C doesn't enclose the origin, 643 00:32:05,010 --> 00:32:07,590 the integral is just 0 because the only place 644 00:32:07,590 --> 00:32:10,680 that the integrand 1 over z fails to be analytic 645 00:32:10,680 --> 00:32:12,060 is when z is 0. 646 00:32:12,060 --> 00:32:14,940 Consequently, if I were to take a region like this-- 647 00:32:14,940 --> 00:32:18,190 see, call this my C. If I were to take a region like this, 648 00:32:18,190 --> 00:32:20,070 notice that the integral around here 649 00:32:20,070 --> 00:32:23,580 would just be 0 because in and on this region, 650 00:32:23,580 --> 00:32:26,510 the function 1 over z is analytic. 651 00:32:26,510 --> 00:32:29,220 At any rate, I know that I'm in a bad situation 652 00:32:29,220 --> 00:32:32,010 here because my function fails to be analytic over here. 653 00:32:32,010 --> 00:32:34,770 The integral around the closed curve doesn't have to be 0. 654 00:32:34,770 --> 00:32:38,520 However, what I do know is how to find the integral 655 00:32:38,520 --> 00:32:41,060 around this very nice curve C1, where 656 00:32:41,060 --> 00:32:44,610 C1 happens to be a circle centered at the origin. 657 00:32:44,610 --> 00:32:47,700 You see, the integral around C1 was just 2pi i. 658 00:32:47,700 --> 00:32:49,750 We did that in our earlier example. 659 00:32:49,750 --> 00:32:52,500 The important point is to observe 660 00:32:52,500 --> 00:32:55,800 that between the circle C1 and the given curve 661 00:32:55,800 --> 00:32:59,910 C, in that particular region, including the two curves C 662 00:32:59,910 --> 00:33:03,780 and C1, the function 1 over z is analytic. 663 00:33:03,780 --> 00:33:08,670 Consequently, the integral dz over z along the curve C 664 00:33:08,670 --> 00:33:13,710 is the same as the integral dz over z along the curve C1. 665 00:33:13,710 --> 00:33:18,580 And therefore, its value will also be 2 pi i. 666 00:33:18,580 --> 00:33:21,900 Now, at any rate, all I've tried to show you 667 00:33:21,900 --> 00:33:25,140 in this particular lecture, and I 668 00:33:25,140 --> 00:33:27,840 hope this part has come through fairly clearly, 669 00:33:27,840 --> 00:33:33,300 is the fact that we can handle complex valued functions 670 00:33:33,300 --> 00:33:35,220 in terms of integration and that this 671 00:33:35,220 --> 00:33:37,710 leads to a treatment of line integrals. 672 00:33:37,710 --> 00:33:44,460 It leads to examining certain types of line integrals 673 00:33:44,460 --> 00:33:47,490 along certain regions in the xy plane, which 674 00:33:47,490 --> 00:33:49,680 we can call the Argand diagram. 675 00:33:49,680 --> 00:33:52,920 We will save applications or identifications 676 00:33:52,920 --> 00:33:55,800 with the real world for the exercises. 677 00:33:55,800 --> 00:33:58,770 But all I hope by now is that we have 678 00:33:58,770 --> 00:34:02,340 had a little bit of insight to the various aspects 679 00:34:02,340 --> 00:34:06,390 of mathematical analysis as it applies to complex variables. 680 00:34:06,390 --> 00:34:09,449 This winds up our treatment of complex variables 681 00:34:09,449 --> 00:34:11,310 as far as this course is concerned. 682 00:34:11,310 --> 00:34:15,330 And next time, we will begin a new block of material 683 00:34:15,330 --> 00:34:19,110 and begin, again, a survey-type investigation 684 00:34:19,110 --> 00:34:23,010 of a subject known as ordinary differential equations. 685 00:34:23,010 --> 00:34:25,530 At any rate then, until next time, goodbye. 686 00:34:30,610 --> 00:34:33,010 Funding for the publication of this video 687 00:34:33,010 --> 00:34:37,870 was provided by the Gabriella and Paul Rosenbaum Foundation. 688 00:34:37,870 --> 00:34:42,040 Help OCW continue to provide free and open access to MIT 689 00:34:42,040 --> 00:34:47,474 courses by making a donation at ocw.mit.edu/donate.