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 --> 00:00:18,210 at ocw.mit.edu. 8 00:00:31,630 --> 00:00:32,380 HERBERT GROSS: Hi. 9 00:00:32,380 --> 00:00:34,960 I hope enough time has elapsed since our last lecture 10 00:00:34,960 --> 00:00:38,830 so that you feel at home doing arithmetic of complex numbers. 11 00:00:38,830 --> 00:00:42,100 Because you see now, what we'd like to do today 12 00:00:42,100 --> 00:00:46,090 is the next phase that leads into our calculus study 13 00:00:46,090 --> 00:00:47,470 of complex numbers. 14 00:00:47,470 --> 00:00:50,530 Namely, what we would like to do is study functions 15 00:00:50,530 --> 00:00:52,330 of a complex variable. 16 00:00:52,330 --> 00:00:54,460 And again, in terms of a function machine, 17 00:00:54,460 --> 00:00:58,960 I'm visualizing the input of my machine being a complex number 18 00:00:58,960 --> 00:01:01,570 and the output being a complex number. 19 00:01:01,570 --> 00:01:04,390 By the way, I should mention that there are situations, 20 00:01:04,390 --> 00:01:07,570 and we'll cover these in the homework exercises, where 21 00:01:07,570 --> 00:01:09,700 one sometimes likes the input to be 22 00:01:09,700 --> 00:01:12,670 a real number and the output a complex number 23 00:01:12,670 --> 00:01:17,200 or the input a complex number and the output a real number. 24 00:01:17,200 --> 00:01:20,320 But for reasons that I hope will become apparent in a moment, 25 00:01:20,320 --> 00:01:22,810 we're going to focus our attention on the case 26 00:01:22,810 --> 00:01:26,030 where both the input and the output are complex numbers. 27 00:01:26,030 --> 00:01:28,120 Noticing that if we do that, especially 28 00:01:28,120 --> 00:01:29,770 from a geometrical point of view, 29 00:01:29,770 --> 00:01:33,370 a vector point of view, our input is a two tuple, 30 00:01:33,370 --> 00:01:38,290 say the complex number x comma y or x plus i-y as represented 31 00:01:38,290 --> 00:01:40,450 by the two tuple x comma y. 32 00:01:40,450 --> 00:01:46,090 The output, w equals f of z, is u plus i-v, say, 33 00:01:46,090 --> 00:01:49,720 or as a two tuple, u comma v. The idea being 34 00:01:49,720 --> 00:01:52,900 that graphically we can view a function that 35 00:01:52,900 --> 00:01:56,770 maps the complex numbers into the complex numbers 36 00:01:56,770 --> 00:02:01,590 as a mapping from the xy plane into the uv plane 37 00:02:01,590 --> 00:02:05,110 where the complex number z is identified with the pair x 38 00:02:05,110 --> 00:02:10,370 comma y, the complex number w is identified 39 00:02:10,370 --> 00:02:13,880 with the ordered pair u comma v. The xy plane 40 00:02:13,880 --> 00:02:15,620 is the domain of our function. 41 00:02:15,620 --> 00:02:18,830 The uv plane is the range of our function. 42 00:02:18,830 --> 00:02:22,040 By way of example, suppose I take the function f of z 43 00:02:22,040 --> 00:02:23,570 equals z squared. 44 00:02:23,570 --> 00:02:27,680 Recalling that z is x plus i-y, if I square this I 45 00:02:27,680 --> 00:02:32,090 get as the real part x squared minus y squared. 46 00:02:32,090 --> 00:02:35,240 The imaginary part is 2xy. 47 00:02:35,240 --> 00:02:39,830 So as a two tuple type mapping, f maps x 48 00:02:39,830 --> 00:02:44,420 comma y into x squared minus y squared comma 2xy. 49 00:02:44,420 --> 00:02:49,130 In other words, u is x squared minus y squared. v is 2xy. 50 00:02:49,130 --> 00:02:55,100 So in summary, notice that the complex value function 51 00:02:55,100 --> 00:02:57,210 of a complex variable-- in other words, 52 00:02:57,210 --> 00:03:00,680 we're mapping complex inputs into complex outputs. 53 00:03:00,680 --> 00:03:02,500 With that particular mapping, f of z 54 00:03:02,500 --> 00:03:06,020 equals z squared is equivalent to the-- and this is 55 00:03:06,020 --> 00:03:09,710 important-- to the real system of equations 56 00:03:09,710 --> 00:03:13,850 u equals x squared minus y squared, v equals 2xy. 57 00:03:13,850 --> 00:03:16,760 You see, this is a system of two real equations and two 58 00:03:16,760 --> 00:03:17,880 unknowns. 59 00:03:17,880 --> 00:03:21,200 And what I'm saying is that I cannot distinguish between this 60 00:03:21,200 --> 00:03:23,420 problem and this problem. 61 00:03:23,420 --> 00:03:25,970 In other words, somehow or other, studying complex 62 00:03:25,970 --> 00:03:29,690 valued functions is going to help me replace 63 00:03:29,690 --> 00:03:32,300 or to view from a different vantage point 64 00:03:32,300 --> 00:03:35,390 the study of systems of real functions. 65 00:03:35,390 --> 00:03:39,090 But again, more about that as we proceed. 66 00:03:39,090 --> 00:03:42,140 Anyway, leaving further drill-- 67 00:03:42,140 --> 00:03:42,770 further drill? 68 00:03:42,770 --> 00:03:44,150 We haven't done any drill. 69 00:03:44,150 --> 00:03:47,810 Leaving drill to the exercises because I think conceptually 70 00:03:47,810 --> 00:03:49,840 this is all old hat. 71 00:03:49,840 --> 00:03:51,680 And why do I say it's old hat? 72 00:03:51,680 --> 00:03:54,620 Well, heck, we've spent a whole block of material working 73 00:03:54,620 --> 00:03:56,510 on these kinds of mappings. 74 00:03:56,510 --> 00:04:00,080 Also, geometrically, we're viewing the complex numbers 75 00:04:00,080 --> 00:04:03,560 as being vectors so that in another manner of speaking, 76 00:04:03,560 --> 00:04:06,890 complex valued functions of a complex variable 77 00:04:06,890 --> 00:04:08,630 are equivalent to what? 78 00:04:08,630 --> 00:04:11,240 Functions that map two dimensional space into two 79 00:04:11,240 --> 00:04:12,380 dimensional space. 80 00:04:12,380 --> 00:04:15,440 And we talked about that in our treatment of vector functions. 81 00:04:15,440 --> 00:04:17,540 So whichever way we look at this thing, 82 00:04:17,540 --> 00:04:20,300 I think that the drill is fairly straightforward. 83 00:04:20,300 --> 00:04:21,200 Let's go on. 84 00:04:21,200 --> 00:04:23,650 The next thing after we have functions. 85 00:04:23,650 --> 00:04:26,780 See, ultimately, we want to get to derivatives and integrals. 86 00:04:26,780 --> 00:04:29,360 And the one concept that we need for a derivative, 87 00:04:29,360 --> 00:04:31,730 after we know the theory of functions, 88 00:04:31,730 --> 00:04:35,150 is the concept of a limit. 89 00:04:35,150 --> 00:04:36,420 So we want limits next. 90 00:04:36,420 --> 00:04:38,000 And what are we going to do letting 91 00:04:38,000 --> 00:04:41,540 C denote the complex number just for purposes of identification. 92 00:04:41,540 --> 00:04:46,330 Suppose we have a mapping from C into C. Suppose a is in C. 93 00:04:46,330 --> 00:04:48,170 In other words, a is some complex number. 94 00:04:48,170 --> 00:04:51,905 We define the limit of f of z as z approaches a to [INAUDIBLE].. 95 00:04:51,905 --> 00:04:53,840 And I'm hoping that by this time you're 96 00:04:53,840 --> 00:04:56,180 way ahead of me because we've done this many times. 97 00:04:56,180 --> 00:04:57,650 What am I going to do here? 98 00:04:57,650 --> 00:05:01,220 I am going to, word for word, mimic the real definition 99 00:05:01,220 --> 00:05:04,340 that was the case for functions of a real variable, 100 00:05:04,340 --> 00:05:06,500 or if you wish for vector functions. 101 00:05:06,500 --> 00:05:08,100 See this is complex. 102 00:05:08,100 --> 00:05:08,990 This is complex. 103 00:05:08,990 --> 00:05:11,420 I can think of this as a vector, this as a vector. 104 00:05:11,420 --> 00:05:13,130 Same definition that we had before, 105 00:05:13,130 --> 00:05:16,490 namely given epsilon greater than 0 106 00:05:16,490 --> 00:05:19,010 there exists a delta greater than 0 such 107 00:05:19,010 --> 00:05:22,160 that when the absolute value of z minus a is greater than 0 108 00:05:22,160 --> 00:05:24,650 but less than delta, the absolute value of f 109 00:05:24,650 --> 00:05:27,620 of z minus L is less than epsilon, noticing 110 00:05:27,620 --> 00:05:31,970 that even though z, a, f of z, and L are complex, 111 00:05:31,970 --> 00:05:34,640 magnitudes are real. 112 00:05:34,640 --> 00:05:37,070 So all of these symbols make sense. 113 00:05:37,070 --> 00:05:40,310 And do they capture intuitively what we'd like them to mean? 114 00:05:40,310 --> 00:05:43,370 Again, by way of a quick pictorial review, 115 00:05:43,370 --> 00:05:45,740 the answer is exactly yes. 116 00:05:45,740 --> 00:05:52,790 Namely if we think of wanting to be with an epsilon of L, 117 00:05:52,790 --> 00:05:54,620 we simply now in two dimensions take 118 00:05:54,620 --> 00:05:58,040 a circle of radius epsilon centered et L. 119 00:05:58,040 --> 00:06:01,070 And what we're saying is to be with an epsilon of L, 120 00:06:01,070 --> 00:06:05,300 we can find a circle of radius delta centered at a such 121 00:06:05,300 --> 00:06:08,690 that whenever z is in this circle, f of z 122 00:06:08,690 --> 00:06:12,770 will be in this circle, which is exactly the intuitive feeling 123 00:06:12,770 --> 00:06:15,560 that limit is supposed to convey. 124 00:06:15,560 --> 00:06:18,650 Because structurally the definition of limit 125 00:06:18,650 --> 00:06:22,670 is the same now as it was in the real case and in the real case 126 00:06:22,670 --> 00:06:24,620 we only used properties of a definition 127 00:06:24,620 --> 00:06:28,010 to prove our theorems, it means that the usual limit theorems 128 00:06:28,010 --> 00:06:29,190 will hold. 129 00:06:29,190 --> 00:06:33,260 And in particular, by the way, if we decide to write f of z 130 00:06:33,260 --> 00:06:36,605 as a real plus an imaginary part, in other words f of z 131 00:06:36,605 --> 00:06:37,250 is what? 132 00:06:37,250 --> 00:06:41,060 It's u plus i-v where u and v are functions of x and y. 133 00:06:41,060 --> 00:06:45,260 The complex number L has a real part and an imaginary part. 134 00:06:45,260 --> 00:06:49,880 L is L1 plus iL2 where these are real, you see. 135 00:06:49,880 --> 00:06:53,100 So this is complex. 136 00:06:53,100 --> 00:06:54,290 These are real. 137 00:06:58,530 --> 00:07:00,510 And the complex number a can be written 138 00:07:00,510 --> 00:07:05,015 in the form a-1 plus ia-2 where a1 and a2 are real. 139 00:07:05,015 --> 00:07:06,390 And essentially what we're saying 140 00:07:06,390 --> 00:07:10,710 is that if the limit as z approaches a, f of z is L, 141 00:07:10,710 --> 00:07:13,530 it must be that the real part of f of z 142 00:07:13,530 --> 00:07:16,740 approaches the real part of L and the imaginary part of f 143 00:07:16,740 --> 00:07:19,810 of z approaches the imaginary part of L. In other words, 144 00:07:19,810 --> 00:07:23,490 this condition here is equivalent to a system 145 00:07:23,490 --> 00:07:29,340 of two limits in real variables, namely the limit of u of xy 146 00:07:29,340 --> 00:07:33,390 as xy approaches a1 and a2 must be the real number L1. 147 00:07:33,390 --> 00:07:37,370 And the limit of v of xy as xy approaches the real ordered 148 00:07:37,370 --> 00:07:39,870 pair a1, a2 must be L2. 149 00:07:39,870 --> 00:07:42,750 And again what this means is that we can often 150 00:07:42,750 --> 00:07:46,530 view systems of limit problems as being the limit 151 00:07:46,530 --> 00:07:49,920 of a complex variable function. 152 00:07:49,920 --> 00:07:52,540 Again, that's all we're saying from that point of view. 153 00:07:52,540 --> 00:07:55,800 I simply want to rush through this because this is old hat. 154 00:07:55,800 --> 00:07:58,180 There is really nothing new theoretically here. 155 00:07:58,180 --> 00:08:02,190 All I want you to see is that the same structure that 156 00:08:02,190 --> 00:08:05,160 held in our study for real valued functions 157 00:08:05,160 --> 00:08:07,500 holds for complex valued functions. 158 00:08:07,500 --> 00:08:10,230 But in particular, when we map complex numbers 159 00:08:10,230 --> 00:08:13,740 into complex numbers, there are many real interpretations 160 00:08:13,740 --> 00:08:16,650 identified with the pair of real mappings 161 00:08:16,650 --> 00:08:20,340 u of some function of x and y and v of some function of x 162 00:08:20,340 --> 00:08:21,090 and y. 163 00:08:21,090 --> 00:08:23,100 At any rate, with that as background, 164 00:08:23,100 --> 00:08:27,540 we can immediately launch into the definition of a derivative. 165 00:08:27,540 --> 00:08:30,660 Now we may not be able to immediately launch into what 166 00:08:30,660 --> 00:08:32,220 does a derivative mean. 167 00:08:32,220 --> 00:08:35,400 This is one of the beauties, you see, of mathematical structure. 168 00:08:35,400 --> 00:08:38,039 One of the beauties of mathematical structure is 169 00:08:38,039 --> 00:08:41,280 is that when I'm mimicking a structure, all I have to do 170 00:08:41,280 --> 00:08:42,030 is copy it. 171 00:08:42,030 --> 00:08:43,890 That's what mimic means, in fact. 172 00:08:43,890 --> 00:08:45,960 The beauty isn't that you can copy it. 173 00:08:45,960 --> 00:08:49,260 The beauty is is that since all of the powerful results 174 00:08:49,260 --> 00:08:52,200 of the thing that you're copying came logically-- 175 00:08:52,200 --> 00:08:54,660 in other words, they were valid conclusions based 176 00:08:54,660 --> 00:08:56,550 on the structure you started with-- 177 00:08:56,550 --> 00:09:00,330 the new structure that you got by mimicking the old one will 178 00:09:00,330 --> 00:09:03,690 inherit the same structure , provided, of course, 179 00:09:03,690 --> 00:09:08,090 that in mimicking the resulting expression still makes sense. 180 00:09:08,090 --> 00:09:10,790 Without beating around the bush, all I'm saying is why don't we 181 00:09:10,790 --> 00:09:15,340 copy as our definition of f prime of z sub 0 where z sub 0 182 00:09:15,340 --> 00:09:18,530 is some complex number, why don't we mimic what we would 183 00:09:18,530 --> 00:09:21,020 have done for f prime as x sub 0, 184 00:09:21,020 --> 00:09:23,720 only replacing the x's by the z's. 185 00:09:23,720 --> 00:09:25,490 And if we do that, remembering what 186 00:09:25,490 --> 00:09:27,500 our definition for derivative was 187 00:09:27,500 --> 00:09:29,720 in the case of a real variable, we simply 188 00:09:29,720 --> 00:09:35,540 define f prime of z0 to be the limit as delta z approaches 0, 189 00:09:35,540 --> 00:09:40,820 f of z0 plus delta z minus f of z0, all over delta z. 190 00:09:40,820 --> 00:09:43,530 The questions, of course, that we raise are, first of all, 191 00:09:43,530 --> 00:09:44,940 does this make sense. 192 00:09:44,940 --> 00:09:49,410 Well, remember f is defined for complex numbers. 193 00:09:49,410 --> 00:09:52,950 And an f of a complex number is a complex number. 194 00:09:52,950 --> 00:09:55,490 So this is the difference of two complex numbers. 195 00:09:55,490 --> 00:09:59,450 The difference of two complex numbers is a complex number. 196 00:09:59,450 --> 00:10:02,790 We're dividing that difference by another complex number, 197 00:10:02,790 --> 00:10:04,250 which by the way can't be 0. 198 00:10:04,250 --> 00:10:06,860 Remember, delta z is not 0. 199 00:10:06,860 --> 00:10:07,820 It approaches 0. 200 00:10:07,820 --> 00:10:10,250 It means that delta z is greater than 0 201 00:10:10,250 --> 00:10:13,720 but less than delta, et cetera, little delta, the same as 202 00:10:13,720 --> 00:10:15,830 in the epsilon-delta definition. 203 00:10:15,830 --> 00:10:16,760 This says what? 204 00:10:16,760 --> 00:10:18,860 The delta z gets arbitrarily close to 0 205 00:10:18,860 --> 00:10:20,330 but it's never equal to 0. 206 00:10:20,330 --> 00:10:21,800 Consequently, this is the quotient 207 00:10:21,800 --> 00:10:23,390 of two complex numbers. 208 00:10:23,390 --> 00:10:25,820 The denominator is not 0. 209 00:10:25,820 --> 00:10:28,010 Consequently, as we saw in the last lecture, 210 00:10:28,010 --> 00:10:29,780 this is again a complex number. 211 00:10:29,780 --> 00:10:32,000 And this makes sense to compute this limit. 212 00:10:32,000 --> 00:10:33,770 And we'll actually do this in a problem 213 00:10:33,770 --> 00:10:35,090 before we're through today. 214 00:10:35,090 --> 00:10:37,530 And we'll also have a lot of homework problems on this. 215 00:10:37,530 --> 00:10:39,530 But to try to give you some feeling as to what 216 00:10:39,530 --> 00:10:42,080 this thing means geometrically, think of this 217 00:10:42,080 --> 00:10:44,010 in terms of u and v again. 218 00:10:44,010 --> 00:10:47,990 In other words, think of mapping the xy plane into the uv plane. 219 00:10:47,990 --> 00:10:53,690 If w is f of z where f of z is what, u plus i-v, then f 220 00:10:53,690 --> 00:10:59,570 prime of z0 is just the symbol dw dz whereby the dw dz 221 00:10:59,570 --> 00:11:02,630 we mean the limit as delta z approaches 0, 222 00:11:02,630 --> 00:11:04,342 delta w over delta z. 223 00:11:04,342 --> 00:11:06,800 And by the way, to help you see that hopefully a little bit 224 00:11:06,800 --> 00:11:11,930 more clearly, if this is the w, delta w involves a change in u 225 00:11:11,930 --> 00:11:14,570 and a change in v. So delta w would 226 00:11:14,570 --> 00:11:18,350 be delta u plus i-delta v. Delta z, by definition, 227 00:11:18,350 --> 00:11:20,510 is delta x plus i-delta y. 228 00:11:20,510 --> 00:11:24,208 So computing the derivative involves computing this limit. 229 00:11:24,208 --> 00:11:26,000 And notice that with the exception of the i 230 00:11:26,000 --> 00:11:29,470 appearing here, everything else involves real numbers. 231 00:11:29,470 --> 00:11:33,410 The very, very crucial thing to notice is that not only must 232 00:11:33,410 --> 00:11:34,970 this limit exist. 233 00:11:34,970 --> 00:11:38,540 But it must exist independently of the direction 234 00:11:38,540 --> 00:11:40,790 in which delta z approaches 0. 235 00:11:40,790 --> 00:11:44,150 This is much stronger than the corresponding notion 236 00:11:44,150 --> 00:11:46,130 of a directional derivative when we 237 00:11:46,130 --> 00:11:48,950 were talking about real valued functions 238 00:11:48,950 --> 00:11:50,720 of several real variables. 239 00:11:50,720 --> 00:11:52,310 You see, for a directional derivative 240 00:11:52,310 --> 00:11:54,380 to exist in every direction all you 241 00:11:54,380 --> 00:11:58,740 require is that in each direction the limit exists, 242 00:11:58,740 --> 00:12:00,710 but that the limit could be different in two 243 00:12:00,710 --> 00:12:02,000 different directions. 244 00:12:02,000 --> 00:12:05,150 Notice that in our definition, this says what? 245 00:12:05,150 --> 00:12:09,170 That this particular limit must exist no matter 246 00:12:09,170 --> 00:12:12,030 what direction delta z approaches 0 in. 247 00:12:12,030 --> 00:12:13,890 Now to help you see this geometrically, 248 00:12:13,890 --> 00:12:16,490 and for heaven's sakes I'm not the greatest guy 249 00:12:16,490 --> 00:12:18,230 at visualizing things geometrically 250 00:12:18,230 --> 00:12:20,810 myself, so if I either confuse you 251 00:12:20,810 --> 00:12:24,570 or else you're not turned on by this, ignore it. 252 00:12:24,570 --> 00:12:26,750 Because we don't need this geometric interpretation 253 00:12:26,750 --> 00:12:28,670 until next lecture. 254 00:12:28,670 --> 00:12:31,310 But for those of you who might be interested in pursuing this, 255 00:12:31,310 --> 00:12:32,500 here's what we're saying. 256 00:12:32,500 --> 00:12:36,030 Notice that f prime of z0 is a complex number. 257 00:12:36,030 --> 00:12:37,580 It's in the uv plane. 258 00:12:37,580 --> 00:12:40,130 Complex numbers are like vectors. 259 00:12:40,130 --> 00:12:43,100 Consequently, I can visualize the complex number 260 00:12:43,100 --> 00:12:45,810 as a directed length, an arrow. 261 00:12:45,810 --> 00:12:48,410 If I take f prime of z0 let me assume 262 00:12:48,410 --> 00:12:54,440 it originates at w0 where w0 is the image of z0 under f. 263 00:12:54,440 --> 00:12:57,770 So as a vector, f prime of z0 is this. 264 00:12:57,770 --> 00:12:59,120 Now here's what we're saying. 265 00:12:59,120 --> 00:13:03,050 Pick any point z1 near z0. 266 00:13:03,050 --> 00:13:05,150 Any point at all, z1. 267 00:13:05,150 --> 00:13:08,400 It maps into some point w1. 268 00:13:08,400 --> 00:13:11,660 The directed length from z0 to z1 269 00:13:11,660 --> 00:13:14,540 is the vector of the complex number delta z. 270 00:13:14,540 --> 00:13:18,200 The directed distance from w0 to w1 271 00:13:18,200 --> 00:13:21,920 is the vector, the complex number, delta w. 272 00:13:21,920 --> 00:13:23,750 In general, one would expect that when 273 00:13:23,750 --> 00:13:27,830 you divide delta w by delta z, what you get 274 00:13:27,830 --> 00:13:30,920 should depend very strongly on what z1 is. 275 00:13:30,920 --> 00:13:32,690 See, different values of z1 would have 276 00:13:32,690 --> 00:13:34,820 led to different values if w1. 277 00:13:34,820 --> 00:13:40,130 The fact that the limit of delta w over delta z is f prime of z0 278 00:13:40,130 --> 00:13:43,220 means that if z1 is very close to z0, 279 00:13:43,220 --> 00:13:47,870 no matter where you pick it, the ratio delta w by delta z 280 00:13:47,870 --> 00:13:49,230 is essentially a constant. 281 00:13:49,230 --> 00:13:50,750 In fact, what constant is it? 282 00:13:50,750 --> 00:13:55,850 It's essentially the vector f prime of z0, the complex number 283 00:13:55,850 --> 00:14:00,750 f prime of z0, contrary to what you might suspect 284 00:14:00,750 --> 00:14:04,230 that in general different z's will lead to such drastically 285 00:14:04,230 --> 00:14:08,100 different w's that the ratio delta w by delta z 286 00:14:08,100 --> 00:14:09,420 is bound to change. 287 00:14:09,420 --> 00:14:12,240 What we're saying is if a function is differentiable, 288 00:14:12,240 --> 00:14:13,970 that ratio doesn't change. 289 00:14:13,970 --> 00:14:17,090 And it's always equal to f prime of z0. 290 00:14:17,090 --> 00:14:20,910 Now the question comes up is it's very, very strong 291 00:14:20,910 --> 00:14:23,400 condition to insist not only that this limit exists 292 00:14:23,400 --> 00:14:26,500 in every direction but be the same in all directions. 293 00:14:26,500 --> 00:14:30,000 It's possible that this imposes some very strong conditions 294 00:14:30,000 --> 00:14:32,550 on delta u and delta v. And what we're 295 00:14:32,550 --> 00:14:34,770 going to do is to say lookit, if this 296 00:14:34,770 --> 00:14:37,110 must be the same for all directions, 297 00:14:37,110 --> 00:14:39,930 let's pick two particularly attractive directions 298 00:14:39,930 --> 00:14:40,830 to look at. 299 00:14:40,830 --> 00:14:42,390 What directions do we like when we 300 00:14:42,390 --> 00:14:46,310 deal with real functions of two real variables? 301 00:14:46,310 --> 00:14:48,560 What we like is, in that case we're dealing with what? 302 00:14:48,560 --> 00:14:50,730 A function of x and y, in which case 303 00:14:50,730 --> 00:14:53,730 the usual derivatives involve the partials with respect 304 00:14:53,730 --> 00:14:56,220 to x and the partials with respect to y. 305 00:14:56,220 --> 00:15:01,380 That means, in one case, holding delta x equal to 0. 306 00:15:01,380 --> 00:15:03,683 You're holding x constant so delta x is 0. 307 00:15:03,683 --> 00:15:05,350 The The other case, you hold y constant. 308 00:15:05,350 --> 00:15:06,750 So delta y is 0. 309 00:15:06,750 --> 00:15:09,420 Let's look at these two cases specially. 310 00:15:09,420 --> 00:15:11,700 In other words, let's look at this first 311 00:15:11,700 --> 00:15:15,390 of all in the special case that delta y is identically 0. 312 00:15:15,390 --> 00:15:18,210 You see, if delta y is identically 0, 313 00:15:18,210 --> 00:15:21,160 delta z is just delta x in that case. 314 00:15:21,160 --> 00:15:22,620 And so this drops out. 315 00:15:22,620 --> 00:15:24,810 And the problem would be to compute the limit 316 00:15:24,810 --> 00:15:30,000 as delta x approaches 0 delta u plus i-delta v over delta x. 317 00:15:30,000 --> 00:15:32,850 And saying that more slowly so that you can see it better, 318 00:15:32,850 --> 00:15:35,800 if I let delta y be identically 0, 319 00:15:35,800 --> 00:15:41,100 it means that I'm allowing delta z to approach 0 horizontally. 320 00:15:41,100 --> 00:15:42,900 In other words, I'm allowing the approach 321 00:15:42,900 --> 00:15:44,550 to be parallel to the real axis. 322 00:15:44,550 --> 00:15:46,770 That's how we're approaching z0 now. 323 00:15:46,770 --> 00:15:51,690 In that case, since delta y is 0, delta z is just delta x. 324 00:15:51,690 --> 00:15:55,230 This is the limit as delta x approaches 0 delta 325 00:15:55,230 --> 00:15:57,525 u plus i-delta v over delta x. 326 00:15:57,525 --> 00:15:59,400 That means I can break this down and write it 327 00:15:59,400 --> 00:16:03,150 as delta u over delta x plus i-delta v over delta x. 328 00:16:03,150 --> 00:16:06,300 The limit of a sum is the sum of the limits, et cetera. 329 00:16:06,300 --> 00:16:09,810 I take the limits here recalling that the limit of delta 330 00:16:09,810 --> 00:16:12,030 u over delta x as delta x approaches 331 00:16:12,030 --> 00:16:14,480 0 holding y constant-- 332 00:16:14,480 --> 00:16:15,740 see delta y is 0. 333 00:16:15,740 --> 00:16:18,660 Here's what I mean by the partial with respect to x. 334 00:16:18,660 --> 00:16:22,230 Similarly, the limit of delta v over delta x as delta 335 00:16:22,230 --> 00:16:25,292 x approaches 0 and delta y is 0-- 336 00:16:25,292 --> 00:16:27,000 in other words y is constant-- is what we 337 00:16:27,000 --> 00:16:31,320 mean by delta v over delta x. 338 00:16:31,320 --> 00:16:35,040 And consequently, what we're saying is if f prime of z0 339 00:16:35,040 --> 00:16:37,860 exists, it must be the partial of u with respect 340 00:16:37,860 --> 00:16:41,400 to x plus i times the partial of v with respect to x. 341 00:16:41,400 --> 00:16:44,220 And I'm computing this at z0, which 342 00:16:44,220 --> 00:16:49,120 corresponds to the point in the xy plane, say x0, y0. 343 00:16:49,120 --> 00:16:52,140 On the other hand, I know that f prime of z0 344 00:16:52,140 --> 00:16:54,990 has to exist and be the same if delta 345 00:16:54,990 --> 00:16:57,390 x is held to be identically 0. 346 00:16:57,390 --> 00:17:01,170 Holding delta x0-- in other words, holding x constant-- 347 00:17:01,170 --> 00:17:04,470 means that you're now allowing z to approach z0. 348 00:17:04,470 --> 00:17:07,109 Delta z is approaching 0 parallel 349 00:17:07,109 --> 00:17:10,270 to the imaginary axis, the y-axis. 350 00:17:10,270 --> 00:17:11,910 Now in this case, what happens? 351 00:17:11,910 --> 00:17:16,710 If delta x is identically 0, this term drops out. 352 00:17:16,710 --> 00:17:19,079 Delta z is just i-delta y. 353 00:17:19,079 --> 00:17:22,020 Since i is a constant different from 0, 354 00:17:22,020 --> 00:17:24,599 the only way that i-delta y can approach 0 355 00:17:24,599 --> 00:17:26,609 is if delta y approaches 0. 356 00:17:26,609 --> 00:17:32,490 Consequently, with delta x0, this reads the limit as delta y 357 00:17:32,490 --> 00:17:37,840 approaches 0 delta u plus i-delta v over. 358 00:17:37,840 --> 00:17:39,630 i-delta y. 359 00:17:39,630 --> 00:17:43,648 And if I now divide each term in here by i-delta y, 360 00:17:43,648 --> 00:17:44,940 you see what's going to happen. 361 00:17:44,940 --> 00:17:47,850 I'm going to get a delta u over i-delta y here. 362 00:17:47,850 --> 00:17:49,710 And the i's will cancel here. 363 00:17:49,710 --> 00:17:53,820 So that f prime of z0 will turn out to be the limit as delta y 364 00:17:53,820 --> 00:17:57,210 approaches 0 delta u over i-delta y 365 00:17:57,210 --> 00:17:59,130 plus delta v over delta y. 366 00:17:59,130 --> 00:18:02,070 Again taking to the limit as delta y approaches 0, 367 00:18:02,070 --> 00:18:04,440 remembering that x is being held constant, 368 00:18:04,440 --> 00:18:06,270 and remembering what the definition 369 00:18:06,270 --> 00:18:10,050 for the partials of u and v with respect to y are. 370 00:18:10,050 --> 00:18:12,750 This limit simply turns out to be this. 371 00:18:12,750 --> 00:18:15,210 I would like to write this in the standard form-- 372 00:18:15,210 --> 00:18:19,650 something real plus something real times i, not 1 over i. 373 00:18:19,650 --> 00:18:23,640 And so I use that usual thing of the complex conjugate. 374 00:18:23,640 --> 00:18:27,960 I multiply numerator and denominator here by minus i. 375 00:18:27,960 --> 00:18:32,040 That gives me my new numerator is minus i. 376 00:18:32,040 --> 00:18:34,350 My new denominator is minus i times 377 00:18:34,350 --> 00:18:38,040 i, which is minus minus 1, or plus 1. 378 00:18:38,040 --> 00:18:41,610 So that f prime of z0 is the partial of u with respect 379 00:18:41,610 --> 00:18:45,900 to y minus i partial of u with respect to y. 380 00:18:45,900 --> 00:18:49,080 Or if you wish, it's plus i times 381 00:18:49,080 --> 00:18:51,420 minus partial u with respect to y. 382 00:18:51,420 --> 00:18:53,160 At any rate, without making a big issue 383 00:18:53,160 --> 00:18:56,700 over this, since these two must be the same 384 00:18:56,700 --> 00:19:00,290 and since the only way that two complex numbers can be equal 385 00:19:00,290 --> 00:19:02,640 is if they're equal component by component. 386 00:19:02,640 --> 00:19:04,940 In other words, the real parts must be equal. 387 00:19:04,940 --> 00:19:07,100 And the imaginary parts must be equal . 388 00:19:07,100 --> 00:19:11,330 I compare the real parts here and the imaginary parts 389 00:19:11,330 --> 00:19:13,400 and conclude that they must be equal. 390 00:19:13,400 --> 00:19:17,480 So in summary if f is equal to u plus i-v and f 391 00:19:17,480 --> 00:19:21,170 is differentiable-- and by the way a very common word that's 392 00:19:21,170 --> 00:19:24,620 used when a complex valued function of a complex variable 393 00:19:24,620 --> 00:19:27,810 is differentiable is that we say it's analytic. 394 00:19:27,810 --> 00:19:31,100 If f equals u plus i-v is analytic, 395 00:19:31,100 --> 00:19:34,933 then the partial of u with respect to x is v sub y. 396 00:19:34,933 --> 00:19:37,100 In other words, the partial of u with respect the y. 397 00:19:37,100 --> 00:19:39,140 And the partial of u with respect to y 398 00:19:39,140 --> 00:19:41,570 must be minus the partial of u with respect to x. 399 00:19:41,570 --> 00:19:44,810 You see, this equals this. 400 00:19:44,810 --> 00:19:48,410 And this must line up with this. 401 00:19:48,410 --> 00:19:50,750 By the way, notice that we arrived 402 00:19:50,750 --> 00:19:54,410 at this result using only the information of two 403 00:19:54,410 --> 00:19:55,400 special directions. 404 00:19:55,400 --> 00:19:58,190 We had delta x0 and delta y0. 405 00:19:58,190 --> 00:20:02,090 So all this proves is is that if the function is differentiable 406 00:20:02,090 --> 00:20:03,830 this must be true. 407 00:20:03,830 --> 00:20:06,750 It happens that the converse is also true. 408 00:20:06,750 --> 00:20:11,060 And we're going to prove that as an exercise in the unit. 409 00:20:11,060 --> 00:20:13,290 Intuitively, to respect why this is true, 410 00:20:13,290 --> 00:20:16,730 it's the same reason as why the directional derivative is 411 00:20:16,730 --> 00:20:19,700 determined by the partials with respect to x and y. 412 00:20:19,700 --> 00:20:23,510 In other words, if the partials exist and are continuous, 413 00:20:23,510 --> 00:20:26,060 knowing what happens in the x and y direction 414 00:20:26,060 --> 00:20:28,790 is enough to tell you what's happening in every direction. 415 00:20:28,790 --> 00:20:33,170 And we'll give you the rigorous details as a exercise later on. 416 00:20:33,170 --> 00:20:35,840 But to put this from a different perspective, 417 00:20:35,840 --> 00:20:41,390 if either u sub x is not equal to v sub y or u sub y is not 418 00:20:41,390 --> 00:20:45,950 equal to minus v sub x, then u plus i-v cannot be an analytic 419 00:20:45,950 --> 00:20:46,700 function. 420 00:20:46,700 --> 00:20:48,290 And the easiest way to talk about this 421 00:20:48,290 --> 00:20:49,950 is in terms of examples. 422 00:20:49,950 --> 00:20:51,230 Let's look at this. 423 00:20:51,230 --> 00:20:54,440 Let's take three examples, but hopefully one at a time. 424 00:20:54,440 --> 00:20:58,143 The first example is f of z equals z squared. 425 00:20:58,143 --> 00:20:59,810 And by the way, before I go any further, 426 00:20:59,810 --> 00:21:02,840 notice that if you had not thought of any subtleties here 427 00:21:02,840 --> 00:21:05,270 and I said what is f prime of z, the odds 428 00:21:05,270 --> 00:21:08,130 are overwhelming that you would have said 2z right away. 429 00:21:08,130 --> 00:21:10,130 I want to show you that that really is the case. 430 00:21:10,130 --> 00:21:12,090 And I want to do it in two different ways. 431 00:21:12,090 --> 00:21:15,290 One is, as we've seen before, z squared 432 00:21:15,290 --> 00:21:17,510 can be written as x squared minus y 433 00:21:17,510 --> 00:21:20,450 squared plus i times 2xy. 434 00:21:20,450 --> 00:21:23,750 So that corresponds to u equals x squared minus y squared, 435 00:21:23,750 --> 00:21:28,490 v equals 2xy from which you see we can compute u sub 436 00:21:28,490 --> 00:21:32,630 x, u sub y, v sub x, and v sub y. 437 00:21:32,630 --> 00:21:35,090 And what we see right away is that u 438 00:21:35,090 --> 00:21:39,500 sub x does equal v sub y, u sub y does equal minus v sub x. 439 00:21:39,500 --> 00:21:42,673 So in particular, these conditions are obeyed. 440 00:21:42,673 --> 00:21:44,840 This is written up in the text by the way, the proof 441 00:21:44,840 --> 00:21:46,040 of what we've just done. 442 00:21:46,040 --> 00:21:49,070 The conditions are called the Cauchy-Riemann conditions. 443 00:21:49,070 --> 00:21:50,660 That's also mentioned in the text. 444 00:21:50,660 --> 00:21:52,610 So I won't take the time to write that now. 445 00:21:52,610 --> 00:21:54,830 But the idea is that what? 446 00:21:54,830 --> 00:21:57,950 Because the Cauchy-Riemann conditions are obeyed, 447 00:21:57,950 --> 00:22:01,160 it means that this function should be differentiable. 448 00:22:01,160 --> 00:22:03,327 By the way, knowing that it should be differentiable 449 00:22:03,327 --> 00:22:04,743 is not the same as saying that you 450 00:22:04,743 --> 00:22:06,020 know what the derivative is. 451 00:22:06,020 --> 00:22:09,020 Let's now take the alternative approach 452 00:22:09,020 --> 00:22:12,140 that somehow or other we don't look at u and v 453 00:22:12,140 --> 00:22:15,020 but work directly from our basic definition. 454 00:22:15,020 --> 00:22:17,630 After all, we define what f prime of z 455 00:22:17,630 --> 00:22:21,230 meant in terms of f of z and f of z plus delta z. 456 00:22:21,230 --> 00:22:24,860 Consequently, this should have meaning regardless of any 457 00:22:24,860 --> 00:22:27,200 interpretation in terms of u's and v's. 458 00:22:27,200 --> 00:22:31,370 Notice that if I write this, by definition of this, 459 00:22:31,370 --> 00:22:36,380 this quotient f of z0 plus delta z minus f of z0 over delta z 460 00:22:36,380 --> 00:22:41,120 is just z0 plus delta z squared minus z0 squared over delta z. 461 00:22:41,120 --> 00:22:44,690 The binomial theorem works for complex numbers the same way 462 00:22:44,690 --> 00:22:46,130 as it does for real numbers. 463 00:22:46,130 --> 00:22:49,280 In fact, that was what motivated our definition for multiplying 464 00:22:49,280 --> 00:22:50,550 complex numbers. 465 00:22:50,550 --> 00:22:52,220 if you recall, our lecture of last time 466 00:22:52,220 --> 00:22:55,160 was to mimic what happens in the real case. 467 00:22:55,160 --> 00:22:57,050 This, the same as before, comes out 468 00:22:57,050 --> 00:23:02,270 to be what? z0 squared, which kills off this minus z0 squared 469 00:23:02,270 --> 00:23:07,592 plus 2z0 delta z plus delta z square, all over delta z. 470 00:23:07,592 --> 00:23:09,800 I'm assuming I'm now going to take the limit as delta 471 00:23:09,800 --> 00:23:11,150 z approaches 0. 472 00:23:11,150 --> 00:23:13,880 That means, in particular , delta z is not 0. 473 00:23:13,880 --> 00:23:17,330 Because delta z is not 0, I can divide through by it. 474 00:23:17,330 --> 00:23:20,510 That gives me 2z0 plus delta z. 475 00:23:20,510 --> 00:23:25,410 And now notice if I let delta z approach 0, z0 476 00:23:25,410 --> 00:23:26,660 is a fixed number. 477 00:23:26,660 --> 00:23:29,420 That doesn't change as delta z approaches 0. 478 00:23:29,420 --> 00:23:32,780 On the other hand, delta z approaches 0 479 00:23:32,780 --> 00:23:35,450 as delta z approaches 0, that's a truism. 480 00:23:35,450 --> 00:23:38,690 Delta z approaches 0, no matter what the direction of approach 481 00:23:38,690 --> 00:23:39,280 is. 482 00:23:39,280 --> 00:23:41,720 This is delta z goes to 0, it goes to 0. 483 00:23:41,720 --> 00:23:43,340 This term does drop out. 484 00:23:43,340 --> 00:23:45,830 And you wind up with the anticipated result 485 00:23:45,830 --> 00:23:49,550 that f prime of z0 is 2z0. 486 00:23:49,550 --> 00:23:53,090 Notice again, structurally, that in the case where there is 487 00:23:53,090 --> 00:23:57,080 an analogy between the complex variable and the real variable, 488 00:23:57,080 --> 00:24:00,930 the steps are verbatim because the definitions were mimicked 489 00:24:00,930 --> 00:24:04,140 to be verbatim, that you cannot tell the difference between 490 00:24:04,140 --> 00:24:08,550 the proof that f prime of z is 2z in this case with the proof 491 00:24:08,550 --> 00:24:13,200 that f prime of x was to 2x when f of x was equal to x squared 492 00:24:13,200 --> 00:24:14,740 in the real case. 493 00:24:14,740 --> 00:24:17,400 Or as another example, one would expect 494 00:24:17,400 --> 00:24:21,480 that the derivative of f of z, if f of z is 1 over z, 495 00:24:21,480 --> 00:24:24,090 should be minus 1 over z squared. 496 00:24:24,090 --> 00:24:28,620 Except that when z is 0, this shouldn't be differentiable. 497 00:24:28,620 --> 00:24:31,860 I leave it again as a voluntary exercise for you 498 00:24:31,860 --> 00:24:34,980 to apply this definition to 1 over z, 499 00:24:34,980 --> 00:24:38,040 1 over z plus delta z, et cetera, and show that step 500 00:24:38,040 --> 00:24:41,490 by step you get the same result as in the corresponding problem 501 00:24:41,490 --> 00:24:43,500 involving 1 over x. 502 00:24:43,500 --> 00:24:46,140 To show you how this works in terms of u's and v's, just 503 00:24:46,140 --> 00:24:50,030 for the drill, if z is x plus i-y, 1 over z 504 00:24:50,030 --> 00:24:52,920 is 1 over x plus i-y. 505 00:24:52,920 --> 00:24:54,840 Multiplying numerator and denominator 506 00:24:54,840 --> 00:24:57,990 by the complex conjugate x minus i-y here, 507 00:24:57,990 --> 00:25:02,250 my denominator becomes x squared plus y squared. 508 00:25:02,250 --> 00:25:04,530 My numerator is x minus i-y. 509 00:25:04,530 --> 00:25:07,020 That's the same as x over x squared 510 00:25:07,020 --> 00:25:12,030 plus y squared plus i times minus y 511 00:25:12,030 --> 00:25:16,000 over x squared plus y squared. 512 00:25:16,000 --> 00:25:17,160 This is the u part. 513 00:25:17,160 --> 00:25:18,360 This is the v part. 514 00:25:18,360 --> 00:25:21,240 u equals x over x squared plus y squared. 515 00:25:21,240 --> 00:25:25,530 v equals minus y over x squared plus y squared. 516 00:25:25,530 --> 00:25:28,530 I again leave it as an exercise for you 517 00:25:28,530 --> 00:25:32,040 to show that the Cauchy-Riemann conditions are obeyed. 518 00:25:32,040 --> 00:25:34,470 In other words, that the partial of u with respect to x 519 00:25:34,470 --> 00:25:36,780 equals the partial of u with respect to y. 520 00:25:36,780 --> 00:25:38,700 The partial of u with respect to y 521 00:25:38,700 --> 00:25:41,160 is minus the partial of u with respect to x. 522 00:25:41,160 --> 00:25:43,910 And the only place that this isn't true 523 00:25:43,910 --> 00:25:47,000 turns out to be when this denominator is 0. 524 00:25:47,000 --> 00:25:49,400 The only time this denominator can be 0 525 00:25:49,400 --> 00:25:51,470 is when x and y are both 0. 526 00:25:51,470 --> 00:25:53,540 In other words, the Cauchy-Riemann conditions 527 00:25:53,540 --> 00:25:57,200 here are obeyed except when x equals y equals 0. 528 00:25:57,200 --> 00:25:59,930 And that, of course, implies what? 529 00:25:59,930 --> 00:26:03,737 That z is 0, the real and the imaginary parts of 0. 530 00:26:03,737 --> 00:26:05,320 And what we're saying is that 1 over z 531 00:26:05,320 --> 00:26:07,910 is analytic except when z is 0. 532 00:26:07,910 --> 00:26:11,300 And in fact, the derivative is minus 1 over z squared. 533 00:26:11,300 --> 00:26:14,120 For a third example, I would like to pick one that we 534 00:26:14,120 --> 00:26:14,690 can't-- 535 00:26:14,690 --> 00:26:17,160 that has no analog in the real case. 536 00:26:17,160 --> 00:26:19,460 I want to pick a genuine complex valued 537 00:26:19,460 --> 00:26:21,320 function of a complex variable. 538 00:26:21,320 --> 00:26:23,710 And I'm going to pick a very simple one. 539 00:26:23,710 --> 00:26:25,710 And the reason I picked a simple one is twofold. 540 00:26:25,710 --> 00:26:28,730 One is the computations are easy to do. 541 00:26:28,730 --> 00:26:32,420 The result is easy to interpret geometrically. 542 00:26:32,420 --> 00:26:34,940 And thirdly and most importantly, I 543 00:26:34,940 --> 00:26:39,530 mentioned before that it is not easy for a complex valued 544 00:26:39,530 --> 00:26:41,260 function to have a derivative. 545 00:26:41,260 --> 00:26:43,940 A directional derivative must exist in every direction 546 00:26:43,940 --> 00:26:44,930 and be the same. 547 00:26:44,930 --> 00:26:46,640 That's a very stringent condition. 548 00:26:46,640 --> 00:26:48,530 So I picked this third example to show you 549 00:26:48,530 --> 00:26:51,440 how a relatively simple complex valued 550 00:26:51,440 --> 00:26:55,220 function of a complex variable can have no derivative. 551 00:26:55,220 --> 00:26:57,140 And the simple function that I have chosen 552 00:26:57,140 --> 00:27:01,220 is the one which maps the complex conjugate, the number 553 00:27:01,220 --> 00:27:02,990 into its complex conjugate. 554 00:27:02,990 --> 00:27:06,680 The complex number z is mapped by f into f bar. 555 00:27:06,680 --> 00:27:09,920 In terms of two tuples, if the input of the f machine 556 00:27:09,920 --> 00:27:14,330 is x comma y, the output will be x comma minus y. 557 00:27:14,330 --> 00:27:16,820 A very simple thing, very straightforward, 558 00:27:16,820 --> 00:27:19,690 easy function machine to work with here. 559 00:27:19,690 --> 00:27:23,570 Notice in this case that u is x. v is minus y. 560 00:27:23,570 --> 00:27:26,150 The partial of u with respect to x is 1. 561 00:27:26,150 --> 00:27:29,555 The partial of u with respect to y is minus 1. 562 00:27:29,555 --> 00:27:31,070 1 is not equal to minus 1. 563 00:27:31,070 --> 00:27:34,790 Consequently, u sub x is not equal to v sub y. 564 00:27:34,790 --> 00:27:36,830 That means that the Cauchy-Riemann conditions 565 00:27:36,830 --> 00:27:37,940 are not obeyed. 566 00:27:37,940 --> 00:27:40,910 By what we just saw a few minutes ago, that in turn 567 00:27:40,910 --> 00:27:45,150 means that this function does not have a derivative. 568 00:27:45,150 --> 00:27:47,550 Well, that's nice to know the function doesn't 569 00:27:47,550 --> 00:27:48,300 have a derivative. 570 00:27:48,300 --> 00:27:51,740 It would be even nicer if we could see why it didn't. 571 00:27:51,740 --> 00:27:53,570 And the best way to do this is, again, 572 00:27:53,570 --> 00:27:55,730 go back to the definition. 573 00:27:55,730 --> 00:27:57,980 What we're going to ultimately do to take a derivative 574 00:27:57,980 --> 00:28:02,960 is we're going to take delta w where w is z bar. 575 00:28:02,960 --> 00:28:05,840 Delta w over delta z and take the limit 576 00:28:05,840 --> 00:28:08,950 as delta z approaches 0 along thee various directions. 577 00:28:08,950 --> 00:28:13,350 Well, delta w is just delta x minus i-delta y. 578 00:28:13,350 --> 00:28:16,850 Delta z is delta x plus i-delta y. 579 00:28:16,850 --> 00:28:18,800 To keep in mind that I'm going to move 580 00:28:18,800 --> 00:28:22,790 in a particular direction in the xy plane to approach z0, 581 00:28:22,790 --> 00:28:24,990 I'm going to it up the slope over here. 582 00:28:24,990 --> 00:28:26,870 Namely in this expression, I'm going 583 00:28:26,870 --> 00:28:29,480 to assume that delta x is not 0. 584 00:28:29,480 --> 00:28:32,350 And what I'm going to do is divide through a numerator 585 00:28:32,350 --> 00:28:33,920 and denominator by delta x. 586 00:28:33,920 --> 00:28:39,020 That gives me 1 minus i times delta y over delta x over 1 587 00:28:39,020 --> 00:28:41,840 plus i-delta y over delta x. 588 00:28:41,840 --> 00:28:49,850 And as I approach the point z0 in the xy plane, 589 00:28:49,850 --> 00:28:55,280 delta y over delta x approaches the slope of the curve at z0, 590 00:28:55,280 --> 00:28:56,960 which we'll call dy dx. 591 00:28:56,960 --> 00:28:58,340 And I wind up with this. 592 00:28:58,340 --> 00:29:01,990 Now to simplify this, let me assume that I approach 593 00:29:01,990 --> 00:29:04,580 z0 along some straight line. 594 00:29:04,580 --> 00:29:06,920 Let me pick a fixed straight line. 595 00:29:06,920 --> 00:29:09,620 I'll pick a straight line which goes through z0 596 00:29:09,620 --> 00:29:11,150 and has slope m. 597 00:29:11,150 --> 00:29:14,720 So in that case, dy dx is the same as delta 598 00:29:14,720 --> 00:29:19,790 x is m for all points on this particular line. 599 00:29:19,790 --> 00:29:21,690 So what is dy dx in this case? 600 00:29:21,690 --> 00:29:22,850 It's m. 601 00:29:22,850 --> 00:29:26,057 Consequently, what is delta w over delta z in this case? 602 00:29:26,057 --> 00:29:27,890 And by the way, notice what I'm saying here. 603 00:29:27,890 --> 00:29:30,510 What I'm saying is this is this line segment. 604 00:29:30,510 --> 00:29:33,610 z0 maps into w0. 605 00:29:33,610 --> 00:29:37,730 z1 maps into w1, sufficiently close to z0 606 00:29:37,730 --> 00:29:40,220 I can assume that the image of this straight line 607 00:29:40,220 --> 00:29:44,120 is a straight line that joins w0 to w1. 608 00:29:44,120 --> 00:29:47,330 And the ratio that I want is this length 609 00:29:47,330 --> 00:29:50,270 divided by this length, where by dividing lengths 610 00:29:50,270 --> 00:29:53,360 I mean the usual complex variable interpretation, 611 00:29:53,360 --> 00:29:56,030 dividing magnitudes and subtracting arguments. 612 00:29:56,030 --> 00:29:57,860 At any rate, what we're saying is 613 00:29:57,860 --> 00:30:03,620 that delta w over delta z in this case is just 1 minus i-m 614 00:30:03,620 --> 00:30:05,630 over 1 plus i-m. 615 00:30:05,630 --> 00:30:09,320 And notice that as delta z approaches 0 in this direction, 616 00:30:09,320 --> 00:30:10,370 this limit exists. 617 00:30:10,370 --> 00:30:12,800 In fact, there's no variable in here. 618 00:30:12,800 --> 00:30:15,680 It's 1 minus i-m over 1 plus i-m. 619 00:30:15,680 --> 00:30:20,900 But notice that even though that limit exists, 620 00:30:20,900 --> 00:30:23,210 the value depends on m. 621 00:30:23,210 --> 00:30:25,630 If I change m, I change this ratio. 622 00:30:25,630 --> 00:30:27,500 And because that limit depends on m, 623 00:30:27,500 --> 00:30:30,800 it means that even though the directional derivative exists 624 00:30:30,800 --> 00:30:35,790 different directions will give me-- 625 00:30:35,790 --> 00:30:37,350 see, if I come in along this line 626 00:30:37,350 --> 00:30:40,440 I can compute this ratio of delta w over delta z. 627 00:30:40,440 --> 00:30:41,750 Will I get in that case? 628 00:30:41,750 --> 00:30:46,440 1 minus i-m 1 over 1 plus i-m 1. 629 00:30:46,440 --> 00:30:47,980 That limit will exist. 630 00:30:47,980 --> 00:30:50,250 But it will be unequal to this one. 631 00:30:50,250 --> 00:30:53,530 You see, the derivative exists in each direction. 632 00:30:53,530 --> 00:30:56,460 But it's not the same in each direction. 633 00:30:56,460 --> 00:30:57,780 That's how subtle this is. 634 00:30:57,780 --> 00:31:01,230 I will leave the remaining exercises-- 635 00:31:01,230 --> 00:31:05,280 not the remaining exercises but exercises to hit home 636 00:31:05,280 --> 00:31:07,860 at the remaining fine points. 637 00:31:07,860 --> 00:31:10,830 What I wanted to do in closing today's lesson 638 00:31:10,830 --> 00:31:14,190 was to show you one more tremendous connection 639 00:31:14,190 --> 00:31:19,050 with complex variables, or between complex variables 640 00:31:19,050 --> 00:31:22,290 and real problems in the physical world. 641 00:31:22,290 --> 00:31:26,100 You may recall from some of our previous exercises 642 00:31:26,100 --> 00:31:29,640 in an earlier block that we say that u of xy 643 00:31:29,640 --> 00:31:33,380 satisfies Laplace's equation if the second partial of u 644 00:31:33,380 --> 00:31:35,700 with respect to x plus the second partial of u 645 00:31:35,700 --> 00:31:37,560 with respect to y is 0. 646 00:31:37,560 --> 00:31:40,320 In fact, if u denotes temperature distribution 647 00:31:40,320 --> 00:31:41,280 at the point-- 648 00:31:41,280 --> 00:31:44,010 the temperature at the point x comma y, this equation 649 00:31:44,010 --> 00:31:47,035 is what's known as the steady state equation. 650 00:31:47,035 --> 00:31:49,410 We're not going to worry about the physical ramifications 651 00:31:49,410 --> 00:31:49,910 here. 652 00:31:49,910 --> 00:31:54,000 What is important is that to solve this partial differential 653 00:31:54,000 --> 00:31:56,730 equation, which comes up in the physical world, 654 00:31:56,730 --> 00:31:59,460 this is a real partial differential equation 655 00:31:59,460 --> 00:32:00,960 meaning real in two senses. 656 00:32:00,960 --> 00:32:02,670 It comes up in the real world. 657 00:32:02,670 --> 00:32:07,620 And also in the sense that u is a real valued function of x 658 00:32:07,620 --> 00:32:08,820 and y. 659 00:32:08,820 --> 00:32:10,700 Now the interesting point is this. 660 00:32:10,700 --> 00:32:13,290 That if u plus i-v is analytic, what 661 00:32:13,290 --> 00:32:16,230 do we know about u plus i-v being analytic? 662 00:32:16,230 --> 00:32:19,500 It means that the partial of u with respect to x is v sub y. 663 00:32:19,500 --> 00:32:21,060 The partial of u with respect to y 664 00:32:21,060 --> 00:32:24,510 is minus the partial of v with respect to x. 665 00:32:24,510 --> 00:32:26,970 If I differentiate both sides here with respect 666 00:32:26,970 --> 00:32:30,780 to x and both sides here with respect to y, 667 00:32:30,780 --> 00:32:34,110 I get u sub xx equals v sub yz. 668 00:32:34,110 --> 00:32:37,260 u sub yy is minus v sub xy. 669 00:32:37,260 --> 00:32:40,950 Assuming continuity so that these are equal, 670 00:32:40,950 --> 00:32:44,170 this says by addition that if I add these two, 671 00:32:44,170 --> 00:32:45,970 I get the second partial of u with respect 672 00:32:45,970 --> 00:32:49,920 to x plus the second partial of u with respect to y is 0. 673 00:32:49,920 --> 00:32:51,870 A similar argument would have shown 674 00:32:51,870 --> 00:32:54,482 that the same is true for the imaginary part, 675 00:32:54,482 --> 00:32:56,190 that the second partial of u with respect 676 00:32:56,190 --> 00:32:59,250 to x plus the second partial of u with respect to y also 677 00:32:59,250 --> 00:33:00,450 will be 0. 678 00:33:00,450 --> 00:33:03,570 In other words, the real and the imaginary parts 679 00:33:03,570 --> 00:33:06,030 which by themselves are real valued 680 00:33:06,030 --> 00:33:09,300 functions of two real variables, the real and imaginary parts 681 00:33:09,300 --> 00:33:11,730 each satisfy Laplace's equation. 682 00:33:11,730 --> 00:33:13,950 The converse, by the way, is also true. 683 00:33:13,950 --> 00:33:15,460 I'll mention that in a moment. 684 00:33:15,460 --> 00:33:17,220 But for the time being, let me close 685 00:33:17,220 --> 00:33:19,150 today's lesson with an example. 686 00:33:19,150 --> 00:33:23,000 We already know that f of z equals z squared is analytic. 687 00:33:23,000 --> 00:33:24,210 It's differentiable. 688 00:33:24,210 --> 00:33:28,320 The real part of z squared u is x squared minus y squared. 689 00:33:28,320 --> 00:33:31,590 The imaginary part v is 2xy. 690 00:33:31,590 --> 00:33:33,810 What that tells me without even checking 691 00:33:33,810 --> 00:33:39,450 is that if I were to form u sub xx plus u sub yy or v sub 692 00:33:39,450 --> 00:33:44,580 xx plus v sub yy, those would be 0. 693 00:33:44,580 --> 00:33:46,260 Because of the fact that they satisfy 694 00:33:46,260 --> 00:33:48,210 the Cauchy-Riemann conditions, which 695 00:33:48,210 --> 00:33:53,020 is what we'll talk about in more detail during the exercises. 696 00:33:53,020 --> 00:33:55,260 And we'll bring this up more in the next lecture. 697 00:33:55,260 --> 00:33:57,030 But for the time being, what I wanted 698 00:33:57,030 --> 00:34:00,330 to close on aside from the calculus was the application. 699 00:34:00,330 --> 00:34:03,900 I want you to see that solving LaPlace's equation 700 00:34:03,900 --> 00:34:06,610 is a very important real world problem. 701 00:34:06,610 --> 00:34:09,870 The interesting point is that the solutions of the LaPlace's 702 00:34:09,870 --> 00:34:13,380 equations have to be the real and imaginary parts 703 00:34:13,380 --> 00:34:14,820 of analytic functions. 704 00:34:14,820 --> 00:34:17,190 And there aren't that many analytic functions. 705 00:34:17,190 --> 00:34:19,960 And consequently by knowing analytic functions, 706 00:34:19,960 --> 00:34:24,690 we have a good hold on what solutions of Laplace's equation 707 00:34:24,690 --> 00:34:26,250 in the real world look like. 708 00:34:26,250 --> 00:34:29,010 Well, I think that's enough for one session. 709 00:34:29,010 --> 00:34:30,780 Work on the exercises. 710 00:34:30,780 --> 00:34:33,929 And next time we're going to pound out 711 00:34:33,929 --> 00:34:36,900 some other interesting ramifications of what 712 00:34:36,900 --> 00:34:40,469 it means for a complex valued function to be differentiable. 713 00:34:40,469 --> 00:34:42,929 At any rate, then, until next time, goodbye. 714 00:34:46,820 --> 00:34:49,219 Funding for the publication of this video 715 00:34:49,219 --> 00:34:54,110 was provided by the Gabriella and Paul Rosenbaum Foundation. 716 00:34:54,110 --> 00:34:58,250 Help OCW continue to provide free and open access to MIT 717 00:34:58,250 --> 00:35:03,685 courses by making a donation at ocw.mit.edu/donate.