WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high quality educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.270 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.270 --> 00:00:18.210 at ocw.mit.edu. 00:00:31.630 --> 00:00:32.380 HERBERT GROSS: Hi. 00:00:32.380 --> 00:00:34.960 I hope enough time has elapsed since our last lecture 00:00:34.960 --> 00:00:38.830 so that you feel at home doing arithmetic of complex numbers. 00:00:38.830 --> 00:00:42.100 Because you see now, what we'd like to do today 00:00:42.100 --> 00:00:46.090 is the next phase that leads into our calculus study 00:00:46.090 --> 00:00:47.470 of complex numbers. 00:00:47.470 --> 00:00:50.530 Namely, what we would like to do is study functions 00:00:50.530 --> 00:00:52.330 of a complex variable. 00:00:52.330 --> 00:00:54.460 And again, in terms of a function machine, 00:00:54.460 --> 00:00:58.960 I'm visualizing the input of my machine being a complex number 00:00:58.960 --> 00:01:01.570 and the output being a complex number. 00:01:01.570 --> 00:01:04.390 By the way, I should mention that there are situations, 00:01:04.390 --> 00:01:07.570 and we'll cover these in the homework exercises, where 00:01:07.570 --> 00:01:09.700 one sometimes likes the input to be 00:01:09.700 --> 00:01:12.670 a real number and the output a complex number 00:01:12.670 --> 00:01:17.200 or the input a complex number and the output a real number. 00:01:17.200 --> 00:01:20.320 But for reasons that I hope will become apparent in a moment, 00:01:20.320 --> 00:01:22.810 we're going to focus our attention on the case 00:01:22.810 --> 00:01:26.030 where both the input and the output are complex numbers. 00:01:26.030 --> 00:01:28.120 Noticing that if we do that, especially 00:01:28.120 --> 00:01:29.770 from a geometrical point of view, 00:01:29.770 --> 00:01:33.370 a vector point of view, our input is a two tuple, 00:01:33.370 --> 00:01:38.290 say the complex number x comma y or x plus i-y as represented 00:01:38.290 --> 00:01:40.450 by the two tuple x comma y. 00:01:40.450 --> 00:01:46.090 The output, w equals f of z, is u plus i-v, say, 00:01:46.090 --> 00:01:49.720 or as a two tuple, u comma v. The idea being 00:01:49.720 --> 00:01:52.900 that graphically we can view a function that 00:01:52.900 --> 00:01:56.770 maps the complex numbers into the complex numbers 00:01:56.770 --> 00:02:01.590 as a mapping from the xy plane into the uv plane 00:02:01.590 --> 00:02:05.110 where the complex number z is identified with the pair x 00:02:05.110 --> 00:02:10.370 comma y, the complex number w is identified 00:02:10.370 --> 00:02:13.880 with the ordered pair u comma v. The xy plane 00:02:13.880 --> 00:02:15.620 is the domain of our function. 00:02:15.620 --> 00:02:18.830 The uv plane is the range of our function. 00:02:18.830 --> 00:02:22.040 By way of example, suppose I take the function f of z 00:02:22.040 --> 00:02:23.570 equals z squared. 00:02:23.570 --> 00:02:27.680 Recalling that z is x plus i-y, if I square this I 00:02:27.680 --> 00:02:32.090 get as the real part x squared minus y squared. 00:02:32.090 --> 00:02:35.240 The imaginary part is 2xy. 00:02:35.240 --> 00:02:39.830 So as a two tuple type mapping, f maps x 00:02:39.830 --> 00:02:44.420 comma y into x squared minus y squared comma 2xy. 00:02:44.420 --> 00:02:49.130 In other words, u is x squared minus y squared. v is 2xy. 00:02:49.130 --> 00:02:55.100 So in summary, notice that the complex value function 00:02:55.100 --> 00:02:57.210 of a complex variable-- in other words, 00:02:57.210 --> 00:03:00.680 we're mapping complex inputs into complex outputs. 00:03:00.680 --> 00:03:02.500 With that particular mapping, f of z 00:03:02.500 --> 00:03:06.020 equals z squared is equivalent to the-- and this is 00:03:06.020 --> 00:03:09.710 important-- to the real system of equations 00:03:09.710 --> 00:03:13.850 u equals x squared minus y squared, v equals 2xy. 00:03:13.850 --> 00:03:16.760 You see, this is a system of two real equations and two 00:03:16.760 --> 00:03:17.880 unknowns. 00:03:17.880 --> 00:03:21.200 And what I'm saying is that I cannot distinguish between this 00:03:21.200 --> 00:03:23.420 problem and this problem. 00:03:23.420 --> 00:03:25.970 In other words, somehow or other, studying complex 00:03:25.970 --> 00:03:29.690 valued functions is going to help me replace 00:03:29.690 --> 00:03:32.300 or to view from a different vantage point 00:03:32.300 --> 00:03:35.390 the study of systems of real functions. 00:03:35.390 --> 00:03:39.090 But again, more about that as we proceed. 00:03:39.090 --> 00:03:42.140 Anyway, leaving further drill-- 00:03:42.140 --> 00:03:42.770 further drill? 00:03:42.770 --> 00:03:44.150 We haven't done any drill. 00:03:44.150 --> 00:03:47.810 Leaving drill to the exercises because I think conceptually 00:03:47.810 --> 00:03:49.840 this is all old hat. 00:03:49.840 --> 00:03:51.680 And why do I say it's old hat? 00:03:51.680 --> 00:03:54.620 Well, heck, we've spent a whole block of material working 00:03:54.620 --> 00:03:56.510 on these kinds of mappings. 00:03:56.510 --> 00:04:00.080 Also, geometrically, we're viewing the complex numbers 00:04:00.080 --> 00:04:03.560 as being vectors so that in another manner of speaking, 00:04:03.560 --> 00:04:06.890 complex valued functions of a complex variable 00:04:06.890 --> 00:04:08.630 are equivalent to what? 00:04:08.630 --> 00:04:11.240 Functions that map two dimensional space into two 00:04:11.240 --> 00:04:12.380 dimensional space. 00:04:12.380 --> 00:04:15.440 And we talked about that in our treatment of vector functions. 00:04:15.440 --> 00:04:17.540 So whichever way we look at this thing, 00:04:17.540 --> 00:04:20.300 I think that the drill is fairly straightforward. 00:04:20.300 --> 00:04:21.200 Let's go on. 00:04:21.200 --> 00:04:23.650 The next thing after we have functions. 00:04:23.650 --> 00:04:26.780 See, ultimately, we want to get to derivatives and integrals. 00:04:26.780 --> 00:04:29.360 And the one concept that we need for a derivative, 00:04:29.360 --> 00:04:31.730 after we know the theory of functions, 00:04:31.730 --> 00:04:35.150 is the concept of a limit. 00:04:35.150 --> 00:04:36.420 So we want limits next. 00:04:36.420 --> 00:04:38.000 And what are we going to do letting 00:04:38.000 --> 00:04:41.540 C denote the complex number just for purposes of identification. 00:04:41.540 --> 00:04:46.330 Suppose we have a mapping from C into C. Suppose a is in C. 00:04:46.330 --> 00:04:48.170 In other words, a is some complex number. 00:04:48.170 --> 00:04:51.905 We define the limit of f of z as z approaches a to [INAUDIBLE].. 00:04:51.905 --> 00:04:53.840 And I'm hoping that by this time you're 00:04:53.840 --> 00:04:56.180 way ahead of me because we've done this many times. 00:04:56.180 --> 00:04:57.650 What am I going to do here? 00:04:57.650 --> 00:05:01.220 I am going to, word for word, mimic the real definition 00:05:01.220 --> 00:05:04.340 that was the case for functions of a real variable, 00:05:04.340 --> 00:05:06.500 or if you wish for vector functions. 00:05:06.500 --> 00:05:08.100 See this is complex. 00:05:08.100 --> 00:05:08.990 This is complex. 00:05:08.990 --> 00:05:11.420 I can think of this as a vector, this as a vector. 00:05:11.420 --> 00:05:13.130 Same definition that we had before, 00:05:13.130 --> 00:05:16.490 namely given epsilon greater than 0 00:05:16.490 --> 00:05:19.010 there exists a delta greater than 0 such 00:05:19.010 --> 00:05:22.160 that when the absolute value of z minus a is greater than 0 00:05:22.160 --> 00:05:24.650 but less than delta, the absolute value of f 00:05:24.650 --> 00:05:27.620 of z minus L is less than epsilon, noticing 00:05:27.620 --> 00:05:31.970 that even though z, a, f of z, and L are complex, 00:05:31.970 --> 00:05:34.640 magnitudes are real. 00:05:34.640 --> 00:05:37.070 So all of these symbols make sense. 00:05:37.070 --> 00:05:40.310 And do they capture intuitively what we'd like them to mean? 00:05:40.310 --> 00:05:43.370 Again, by way of a quick pictorial review, 00:05:43.370 --> 00:05:45.740 the answer is exactly yes. 00:05:45.740 --> 00:05:52.790 Namely if we think of wanting to be with an epsilon of L, 00:05:52.790 --> 00:05:54.620 we simply now in two dimensions take 00:05:54.620 --> 00:05:58.040 a circle of radius epsilon centered et L. 00:05:58.040 --> 00:06:01.070 And what we're saying is to be with an epsilon of L, 00:06:01.070 --> 00:06:05.300 we can find a circle of radius delta centered at a such 00:06:05.300 --> 00:06:08.690 that whenever z is in this circle, f of z 00:06:08.690 --> 00:06:12.770 will be in this circle, which is exactly the intuitive feeling 00:06:12.770 --> 00:06:15.560 that limit is supposed to convey. 00:06:15.560 --> 00:06:18.650 Because structurally the definition of limit 00:06:18.650 --> 00:06:22.670 is the same now as it was in the real case and in the real case 00:06:22.670 --> 00:06:24.620 we only used properties of a definition 00:06:24.620 --> 00:06:28.010 to prove our theorems, it means that the usual limit theorems 00:06:28.010 --> 00:06:29.190 will hold. 00:06:29.190 --> 00:06:33.260 And in particular, by the way, if we decide to write f of z 00:06:33.260 --> 00:06:36.605 as a real plus an imaginary part, in other words f of z 00:06:36.605 --> 00:06:37.250 is what? 00:06:37.250 --> 00:06:41.060 It's u plus i-v where u and v are functions of x and y. 00:06:41.060 --> 00:06:45.260 The complex number L has a real part and an imaginary part. 00:06:45.260 --> 00:06:49.880 L is L1 plus iL2 where these are real, you see. 00:06:49.880 --> 00:06:53.100 So this is complex. 00:06:53.100 --> 00:06:54.290 These are real. 00:06:58.530 --> 00:07:00.510 And the complex number a can be written 00:07:00.510 --> 00:07:05.015 in the form a-1 plus ia-2 where a1 and a2 are real. 00:07:05.015 --> 00:07:06.390 And essentially what we're saying 00:07:06.390 --> 00:07:10.710 is that if the limit as z approaches a, f of z is L, 00:07:10.710 --> 00:07:13.530 it must be that the real part of f of z 00:07:13.530 --> 00:07:16.740 approaches the real part of L and the imaginary part of f 00:07:16.740 --> 00:07:19.810 of z approaches the imaginary part of L. In other words, 00:07:19.810 --> 00:07:23.490 this condition here is equivalent to a system 00:07:23.490 --> 00:07:29.340 of two limits in real variables, namely the limit of u of xy 00:07:29.340 --> 00:07:33.390 as xy approaches a1 and a2 must be the real number L1. 00:07:33.390 --> 00:07:37.370 And the limit of v of xy as xy approaches the real ordered 00:07:37.370 --> 00:07:39.870 pair a1, a2 must be L2. 00:07:39.870 --> 00:07:42.750 And again what this means is that we can often 00:07:42.750 --> 00:07:46.530 view systems of limit problems as being the limit 00:07:46.530 --> 00:07:49.920 of a complex variable function. 00:07:49.920 --> 00:07:52.540 Again, that's all we're saying from that point of view. 00:07:52.540 --> 00:07:55.800 I simply want to rush through this because this is old hat. 00:07:55.800 --> 00:07:58.180 There is really nothing new theoretically here. 00:07:58.180 --> 00:08:02.190 All I want you to see is that the same structure that 00:08:02.190 --> 00:08:05.160 held in our study for real valued functions 00:08:05.160 --> 00:08:07.500 holds for complex valued functions. 00:08:07.500 --> 00:08:10.230 But in particular, when we map complex numbers 00:08:10.230 --> 00:08:13.740 into complex numbers, there are many real interpretations 00:08:13.740 --> 00:08:16.650 identified with the pair of real mappings 00:08:16.650 --> 00:08:20.340 u of some function of x and y and v of some function of x 00:08:20.340 --> 00:08:21.090 and y. 00:08:21.090 --> 00:08:23.100 At any rate, with that as background, 00:08:23.100 --> 00:08:27.540 we can immediately launch into the definition of a derivative. 00:08:27.540 --> 00:08:30.660 Now we may not be able to immediately launch into what 00:08:30.660 --> 00:08:32.220 does a derivative mean. 00:08:32.220 --> 00:08:35.400 This is one of the beauties, you see, of mathematical structure. 00:08:35.400 --> 00:08:38.039 One of the beauties of mathematical structure is 00:08:38.039 --> 00:08:41.280 is that when I'm mimicking a structure, all I have to do 00:08:41.280 --> 00:08:42.030 is copy it. 00:08:42.030 --> 00:08:43.890 That's what mimic means, in fact. 00:08:43.890 --> 00:08:45.960 The beauty isn't that you can copy it. 00:08:45.960 --> 00:08:49.260 The beauty is is that since all of the powerful results 00:08:49.260 --> 00:08:52.200 of the thing that you're copying came logically-- 00:08:52.200 --> 00:08:54.660 in other words, they were valid conclusions based 00:08:54.660 --> 00:08:56.550 on the structure you started with-- 00:08:56.550 --> 00:09:00.330 the new structure that you got by mimicking the old one will 00:09:00.330 --> 00:09:03.690 inherit the same structure , provided, of course, 00:09:03.690 --> 00:09:08.090 that in mimicking the resulting expression still makes sense. 00:09:08.090 --> 00:09:10.790 Without beating around the bush, all I'm saying is why don't we 00:09:10.790 --> 00:09:15.340 copy as our definition of f prime of z sub 0 where z sub 0 00:09:15.340 --> 00:09:18.530 is some complex number, why don't we mimic what we would 00:09:18.530 --> 00:09:21.020 have done for f prime as x sub 0, 00:09:21.020 --> 00:09:23.720 only replacing the x's by the z's. 00:09:23.720 --> 00:09:25.490 And if we do that, remembering what 00:09:25.490 --> 00:09:27.500 our definition for derivative was 00:09:27.500 --> 00:09:29.720 in the case of a real variable, we simply 00:09:29.720 --> 00:09:35.540 define f prime of z0 to be the limit as delta z approaches 0, 00:09:35.540 --> 00:09:40.820 f of z0 plus delta z minus f of z0, all over delta z. 00:09:40.820 --> 00:09:43.530 The questions, of course, that we raise are, first of all, 00:09:43.530 --> 00:09:44.940 does this make sense. 00:09:44.940 --> 00:09:49.410 Well, remember f is defined for complex numbers. 00:09:49.410 --> 00:09:52.950 And an f of a complex number is a complex number. 00:09:52.950 --> 00:09:55.490 So this is the difference of two complex numbers. 00:09:55.490 --> 00:09:59.450 The difference of two complex numbers is a complex number. 00:09:59.450 --> 00:10:02.790 We're dividing that difference by another complex number, 00:10:02.790 --> 00:10:04.250 which by the way can't be 0. 00:10:04.250 --> 00:10:06.860 Remember, delta z is not 0. 00:10:06.860 --> 00:10:07.820 It approaches 0. 00:10:07.820 --> 00:10:10.250 It means that delta z is greater than 0 00:10:10.250 --> 00:10:13.720 but less than delta, et cetera, little delta, the same as 00:10:13.720 --> 00:10:15.830 in the epsilon-delta definition. 00:10:15.830 --> 00:10:16.760 This says what? 00:10:16.760 --> 00:10:18.860 The delta z gets arbitrarily close to 0 00:10:18.860 --> 00:10:20.330 but it's never equal to 0. 00:10:20.330 --> 00:10:21.800 Consequently, this is the quotient 00:10:21.800 --> 00:10:23.390 of two complex numbers. 00:10:23.390 --> 00:10:25.820 The denominator is not 0. 00:10:25.820 --> 00:10:28.010 Consequently, as we saw in the last lecture, 00:10:28.010 --> 00:10:29.780 this is again a complex number. 00:10:29.780 --> 00:10:32.000 And this makes sense to compute this limit. 00:10:32.000 --> 00:10:33.770 And we'll actually do this in a problem 00:10:33.770 --> 00:10:35.090 before we're through today. 00:10:35.090 --> 00:10:37.530 And we'll also have a lot of homework problems on this. 00:10:37.530 --> 00:10:39.530 But to try to give you some feeling as to what 00:10:39.530 --> 00:10:42.080 this thing means geometrically, think of this 00:10:42.080 --> 00:10:44.010 in terms of u and v again. 00:10:44.010 --> 00:10:47.990 In other words, think of mapping the xy plane into the uv plane. 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 00:10:53.690 --> 00:10:59.570 prime of z0 is just the symbol dw dz whereby the dw dz 00:10:59.570 --> 00:11:02.630 we mean the limit as delta z approaches 0, 00:11:02.630 --> 00:11:04.342 delta w over delta z. 00:11:04.342 --> 00:11:06.800 And by the way, to help you see that hopefully a little bit 00:11:06.800 --> 00:11:11.930 more clearly, if this is the w, delta w involves a change in u 00:11:11.930 --> 00:11:14.570 and a change in v. So delta w would 00:11:14.570 --> 00:11:18.350 be delta u plus i-delta v. Delta z, by definition, 00:11:18.350 --> 00:11:20.510 is delta x plus i-delta y. 00:11:20.510 --> 00:11:24.208 So computing the derivative involves computing this limit. 00:11:24.208 --> 00:11:26.000 And notice that with the exception of the i 00:11:26.000 --> 00:11:29.470 appearing here, everything else involves real numbers. 00:11:29.470 --> 00:11:33.410 The very, very crucial thing to notice is that not only must 00:11:33.410 --> 00:11:34.970 this limit exist. 00:11:34.970 --> 00:11:38.540 But it must exist independently of the direction 00:11:38.540 --> 00:11:40.790 in which delta z approaches 0. 00:11:40.790 --> 00:11:44.150 This is much stronger than the corresponding notion 00:11:44.150 --> 00:11:46.130 of a directional derivative when we 00:11:46.130 --> 00:11:48.950 were talking about real valued functions 00:11:48.950 --> 00:11:50.720 of several real variables. 00:11:50.720 --> 00:11:52.310 You see, for a directional derivative 00:11:52.310 --> 00:11:54.380 to exist in every direction all you 00:11:54.380 --> 00:11:58.740 require is that in each direction the limit exists, 00:11:58.740 --> 00:12:00.710 but that the limit could be different in two 00:12:00.710 --> 00:12:02.000 different directions. 00:12:02.000 --> 00:12:05.150 Notice that in our definition, this says what? 00:12:05.150 --> 00:12:09.170 That this particular limit must exist no matter 00:12:09.170 --> 00:12:12.030 what direction delta z approaches 0 in. 00:12:12.030 --> 00:12:13.890 Now to help you see this geometrically, 00:12:13.890 --> 00:12:16.490 and for heaven's sakes I'm not the greatest guy 00:12:16.490 --> 00:12:18.230 at visualizing things geometrically 00:12:18.230 --> 00:12:20.810 myself, so if I either confuse you 00:12:20.810 --> 00:12:24.570 or else you're not turned on by this, ignore it. 00:12:24.570 --> 00:12:26.750 Because we don't need this geometric interpretation 00:12:26.750 --> 00:12:28.670 until next lecture. 00:12:28.670 --> 00:12:31.310 But for those of you who might be interested in pursuing this, 00:12:31.310 --> 00:12:32.500 here's what we're saying. 00:12:32.500 --> 00:12:36.030 Notice that f prime of z0 is a complex number. 00:12:36.030 --> 00:12:37.580 It's in the uv plane. 00:12:37.580 --> 00:12:40.130 Complex numbers are like vectors. 00:12:40.130 --> 00:12:43.100 Consequently, I can visualize the complex number 00:12:43.100 --> 00:12:45.810 as a directed length, an arrow. 00:12:45.810 --> 00:12:48.410 If I take f prime of z0 let me assume 00:12:48.410 --> 00:12:54.440 it originates at w0 where w0 is the image of z0 under f. 00:12:54.440 --> 00:12:57.770 So as a vector, f prime of z0 is this. 00:12:57.770 --> 00:12:59.120 Now here's what we're saying. 00:12:59.120 --> 00:13:03.050 Pick any point z1 near z0. 00:13:03.050 --> 00:13:05.150 Any point at all, z1. 00:13:05.150 --> 00:13:08.400 It maps into some point w1. 00:13:08.400 --> 00:13:11.660 The directed length from z0 to z1 00:13:11.660 --> 00:13:14.540 is the vector of the complex number delta z. 00:13:14.540 --> 00:13:18.200 The directed distance from w0 to w1 00:13:18.200 --> 00:13:21.920 is the vector, the complex number, delta w. 00:13:21.920 --> 00:13:23.750 In general, one would expect that when 00:13:23.750 --> 00:13:27.830 you divide delta w by delta z, what you get 00:13:27.830 --> 00:13:30.920 should depend very strongly on what z1 is. 00:13:30.920 --> 00:13:32.690 See, different values of z1 would have 00:13:32.690 --> 00:13:34.820 led to different values if w1. 00:13:34.820 --> 00:13:40.130 The fact that the limit of delta w over delta z is f prime of z0 00:13:40.130 --> 00:13:43.220 means that if z1 is very close to z0, 00:13:43.220 --> 00:13:47.870 no matter where you pick it, the ratio delta w by delta z 00:13:47.870 --> 00:13:49.230 is essentially a constant. 00:13:49.230 --> 00:13:50.750 In fact, what constant is it? 00:13:50.750 --> 00:13:55.850 It's essentially the vector f prime of z0, the complex number 00:13:55.850 --> 00:14:00.750 f prime of z0, contrary to what you might suspect 00:14:00.750 --> 00:14:04.230 that in general different z's will lead to such drastically 00:14:04.230 --> 00:14:08.100 different w's that the ratio delta w by delta z 00:14:08.100 --> 00:14:09.420 is bound to change. 00:14:09.420 --> 00:14:12.240 What we're saying is if a function is differentiable, 00:14:12.240 --> 00:14:13.970 that ratio doesn't change. 00:14:13.970 --> 00:14:17.090 And it's always equal to f prime of z0. 00:14:17.090 --> 00:14:20.910 Now the question comes up is it's very, very strong 00:14:20.910 --> 00:14:23.400 condition to insist not only that this limit exists 00:14:23.400 --> 00:14:26.500 in every direction but be the same in all directions. 00:14:26.500 --> 00:14:30.000 It's possible that this imposes some very strong conditions 00:14:30.000 --> 00:14:32.550 on delta u and delta v. And what we're 00:14:32.550 --> 00:14:34.770 going to do is to say lookit, if this 00:14:34.770 --> 00:14:37.110 must be the same for all directions, 00:14:37.110 --> 00:14:39.930 let's pick two particularly attractive directions 00:14:39.930 --> 00:14:40.830 to look at. 00:14:40.830 --> 00:14:42.390 What directions do we like when we 00:14:42.390 --> 00:14:46.310 deal with real functions of two real variables? 00:14:46.310 --> 00:14:48.560 What we like is, in that case we're dealing with what? 00:14:48.560 --> 00:14:50.730 A function of x and y, in which case 00:14:50.730 --> 00:14:53.730 the usual derivatives involve the partials with respect 00:14:53.730 --> 00:14:56.220 to x and the partials with respect to y. 00:14:56.220 --> 00:15:01.380 That means, in one case, holding delta x equal to 0. 00:15:01.380 --> 00:15:03.683 You're holding x constant so delta x is 0. 00:15:03.683 --> 00:15:05.350 The The other case, you hold y constant. 00:15:05.350 --> 00:15:06.750 So delta y is 0. 00:15:06.750 --> 00:15:09.420 Let's look at these two cases specially. 00:15:09.420 --> 00:15:11.700 In other words, let's look at this first 00:15:11.700 --> 00:15:15.390 of all in the special case that delta y is identically 0. 00:15:15.390 --> 00:15:18.210 You see, if delta y is identically 0, 00:15:18.210 --> 00:15:21.160 delta z is just delta x in that case. 00:15:21.160 --> 00:15:22.620 And so this drops out. 00:15:22.620 --> 00:15:24.810 And the problem would be to compute the limit 00:15:24.810 --> 00:15:30.000 as delta x approaches 0 delta u plus i-delta v over delta x. 00:15:30.000 --> 00:15:32.850 And saying that more slowly so that you can see it better, 00:15:32.850 --> 00:15:35.800 if I let delta y be identically 0, 00:15:35.800 --> 00:15:41.100 it means that I'm allowing delta z to approach 0 horizontally. 00:15:41.100 --> 00:15:42.900 In other words, I'm allowing the approach 00:15:42.900 --> 00:15:44.550 to be parallel to the real axis. 00:15:44.550 --> 00:15:46.770 That's how we're approaching z0 now. 00:15:46.770 --> 00:15:51.690 In that case, since delta y is 0, delta z is just delta x. 00:15:51.690 --> 00:15:55.230 This is the limit as delta x approaches 0 delta 00:15:55.230 --> 00:15:57.525 u plus i-delta v over delta x. 00:15:57.525 --> 00:15:59.400 That means I can break this down and write it 00:15:59.400 --> 00:16:03.150 as delta u over delta x plus i-delta v over delta x. 00:16:03.150 --> 00:16:06.300 The limit of a sum is the sum of the limits, et cetera. 00:16:06.300 --> 00:16:09.810 I take the limits here recalling that the limit of delta 00:16:09.810 --> 00:16:12.030 u over delta x as delta x approaches 00:16:12.030 --> 00:16:14.480 0 holding y constant-- 00:16:14.480 --> 00:16:15.740 see delta y is 0. 00:16:15.740 --> 00:16:18.660 Here's what I mean by the partial with respect to x. 00:16:18.660 --> 00:16:22.230 Similarly, the limit of delta v over delta x as delta 00:16:22.230 --> 00:16:25.292 x approaches 0 and delta y is 0-- 00:16:25.292 --> 00:16:27.000 in other words y is constant-- is what we 00:16:27.000 --> 00:16:31.320 mean by delta v over delta x. 00:16:31.320 --> 00:16:35.040 And consequently, what we're saying is if f prime of z0 00:16:35.040 --> 00:16:37.860 exists, it must be the partial of u with respect 00:16:37.860 --> 00:16:41.400 to x plus i times the partial of v with respect to x. 00:16:41.400 --> 00:16:44.220 And I'm computing this at z0, which 00:16:44.220 --> 00:16:49.120 corresponds to the point in the xy plane, say x0, y0. 00:16:49.120 --> 00:16:52.140 On the other hand, I know that f prime of z0 00:16:52.140 --> 00:16:54.990 has to exist and be the same if delta 00:16:54.990 --> 00:16:57.390 x is held to be identically 0. 00:16:57.390 --> 00:17:01.170 Holding delta x0-- in other words, holding x constant-- 00:17:01.170 --> 00:17:04.470 means that you're now allowing z to approach z0. 00:17:04.470 --> 00:17:07.109 Delta z is approaching 0 parallel 00:17:07.109 --> 00:17:10.270 to the imaginary axis, the y-axis. 00:17:10.270 --> 00:17:11.910 Now in this case, what happens? 00:17:11.910 --> 00:17:16.710 If delta x is identically 0, this term drops out. 00:17:16.710 --> 00:17:19.079 Delta z is just i-delta y. 00:17:19.079 --> 00:17:22.020 Since i is a constant different from 0, 00:17:22.020 --> 00:17:24.599 the only way that i-delta y can approach 0 00:17:24.599 --> 00:17:26.609 is if delta y approaches 0. 00:17:26.609 --> 00:17:32.490 Consequently, with delta x0, this reads the limit as delta y 00:17:32.490 --> 00:17:37.840 approaches 0 delta u plus i-delta v over. 00:17:37.840 --> 00:17:39.630 i-delta y. 00:17:39.630 --> 00:17:43.648 And if I now divide each term in here by i-delta y, 00:17:43.648 --> 00:17:44.940 you see what's going to happen. 00:17:44.940 --> 00:17:47.850 I'm going to get a delta u over i-delta y here. 00:17:47.850 --> 00:17:49.710 And the i's will cancel here. 00:17:49.710 --> 00:17:53.820 So that f prime of z0 will turn out to be the limit as delta y 00:17:53.820 --> 00:17:57.210 approaches 0 delta u over i-delta y 00:17:57.210 --> 00:17:59.130 plus delta v over delta y. 00:17:59.130 --> 00:18:02.070 Again taking to the limit as delta y approaches 0, 00:18:02.070 --> 00:18:04.440 remembering that x is being held constant, 00:18:04.440 --> 00:18:06.270 and remembering what the definition 00:18:06.270 --> 00:18:10.050 for the partials of u and v with respect to y are. 00:18:10.050 --> 00:18:12.750 This limit simply turns out to be this. 00:18:12.750 --> 00:18:15.210 I would like to write this in the standard form-- 00:18:15.210 --> 00:18:19.650 something real plus something real times i, not 1 over i. 00:18:19.650 --> 00:18:23.640 And so I use that usual thing of the complex conjugate. 00:18:23.640 --> 00:18:27.960 I multiply numerator and denominator here by minus i. 00:18:27.960 --> 00:18:32.040 That gives me my new numerator is minus i. 00:18:32.040 --> 00:18:34.350 My new denominator is minus i times 00:18:34.350 --> 00:18:38.040 i, which is minus minus 1, or plus 1. 00:18:38.040 --> 00:18:41.610 So that f prime of z0 is the partial of u with respect 00:18:41.610 --> 00:18:45.900 to y minus i partial of u with respect to y. 00:18:45.900 --> 00:18:49.080 Or if you wish, it's plus i times 00:18:49.080 --> 00:18:51.420 minus partial u with respect to y. 00:18:51.420 --> 00:18:53.160 At any rate, without making a big issue 00:18:53.160 --> 00:18:56.700 over this, since these two must be the same 00:18:56.700 --> 00:19:00.290 and since the only way that two complex numbers can be equal 00:19:00.290 --> 00:19:02.640 is if they're equal component by component. 00:19:02.640 --> 00:19:04.940 In other words, the real parts must be equal. 00:19:04.940 --> 00:19:07.100 And the imaginary parts must be equal . 00:19:07.100 --> 00:19:11.330 I compare the real parts here and the imaginary parts 00:19:11.330 --> 00:19:13.400 and conclude that they must be equal. 00:19:13.400 --> 00:19:17.480 So in summary if f is equal to u plus i-v and f 00:19:17.480 --> 00:19:21.170 is differentiable-- and by the way a very common word that's 00:19:21.170 --> 00:19:24.620 used when a complex valued function of a complex variable 00:19:24.620 --> 00:19:27.810 is differentiable is that we say it's analytic. 00:19:27.810 --> 00:19:31.100 If f equals u plus i-v is analytic, 00:19:31.100 --> 00:19:34.933 then the partial of u with respect to x is v sub y. 00:19:34.933 --> 00:19:37.100 In other words, the partial of u with respect the y. 00:19:37.100 --> 00:19:39.140 And the partial of u with respect to y 00:19:39.140 --> 00:19:41.570 must be minus the partial of u with respect to x. 00:19:41.570 --> 00:19:44.810 You see, this equals this. 00:19:44.810 --> 00:19:48.410 And this must line up with this. 00:19:48.410 --> 00:19:50.750 By the way, notice that we arrived 00:19:50.750 --> 00:19:54.410 at this result using only the information of two 00:19:54.410 --> 00:19:55.400 special directions. 00:19:55.400 --> 00:19:58.190 We had delta x0 and delta y0. 00:19:58.190 --> 00:20:02.090 So all this proves is is that if the function is differentiable 00:20:02.090 --> 00:20:03.830 this must be true. 00:20:03.830 --> 00:20:06.750 It happens that the converse is also true. 00:20:06.750 --> 00:20:11.060 And we're going to prove that as an exercise in the unit. 00:20:11.060 --> 00:20:13.290 Intuitively, to respect why this is true, 00:20:13.290 --> 00:20:16.730 it's the same reason as why the directional derivative is 00:20:16.730 --> 00:20:19.700 determined by the partials with respect to x and y. 00:20:19.700 --> 00:20:23.510 In other words, if the partials exist and are continuous, 00:20:23.510 --> 00:20:26.060 knowing what happens in the x and y direction 00:20:26.060 --> 00:20:28.790 is enough to tell you what's happening in every direction. 00:20:28.790 --> 00:20:33.170 And we'll give you the rigorous details as a exercise later on. 00:20:33.170 --> 00:20:35.840 But to put this from a different perspective, 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 00:20:41.390 --> 00:20:45.950 equal to minus v sub x, then u plus i-v cannot be an analytic 00:20:45.950 --> 00:20:46.700 function. 00:20:46.700 --> 00:20:48.290 And the easiest way to talk about this 00:20:48.290 --> 00:20:49.950 is in terms of examples. 00:20:49.950 --> 00:20:51.230 Let's look at this. 00:20:51.230 --> 00:20:54.440 Let's take three examples, but hopefully one at a time. 00:20:54.440 --> 00:20:58.143 The first example is f of z equals z squared. 00:20:58.143 --> 00:20:59.810 And by the way, before I go any further, 00:20:59.810 --> 00:21:02.840 notice that if you had not thought of any subtleties here 00:21:02.840 --> 00:21:05.270 and I said what is f prime of z, the odds 00:21:05.270 --> 00:21:08.130 are overwhelming that you would have said 2z right away. 00:21:08.130 --> 00:21:10.130 I want to show you that that really is the case. 00:21:10.130 --> 00:21:12.090 And I want to do it in two different ways. 00:21:12.090 --> 00:21:15.290 One is, as we've seen before, z squared 00:21:15.290 --> 00:21:17.510 can be written as x squared minus y 00:21:17.510 --> 00:21:20.450 squared plus i times 2xy. 00:21:20.450 --> 00:21:23.750 So that corresponds to u equals x squared minus y squared, 00:21:23.750 --> 00:21:28.490 v equals 2xy from which you see we can compute u sub 00:21:28.490 --> 00:21:32.630 x, u sub y, v sub x, and v sub y. 00:21:32.630 --> 00:21:35.090 And what we see right away is that u 00:21:35.090 --> 00:21:39.500 sub x does equal v sub y, u sub y does equal minus v sub x. 00:21:39.500 --> 00:21:42.673 So in particular, these conditions are obeyed. 00:21:42.673 --> 00:21:44.840 This is written up in the text by the way, the proof 00:21:44.840 --> 00:21:46.040 of what we've just done. 00:21:46.040 --> 00:21:49.070 The conditions are called the Cauchy-Riemann conditions. 00:21:49.070 --> 00:21:50.660 That's also mentioned in the text. 00:21:50.660 --> 00:21:52.610 So I won't take the time to write that now. 00:21:52.610 --> 00:21:54.830 But the idea is that what? 00:21:54.830 --> 00:21:57.950 Because the Cauchy-Riemann conditions are obeyed, 00:21:57.950 --> 00:22:01.160 it means that this function should be differentiable. 00:22:01.160 --> 00:22:03.327 By the way, knowing that it should be differentiable 00:22:03.327 --> 00:22:04.743 is not the same as saying that you 00:22:04.743 --> 00:22:06.020 know what the derivative is. 00:22:06.020 --> 00:22:09.020 Let's now take the alternative approach 00:22:09.020 --> 00:22:12.140 that somehow or other we don't look at u and v 00:22:12.140 --> 00:22:15.020 but work directly from our basic definition. 00:22:15.020 --> 00:22:17.630 After all, we define what f prime of z 00:22:17.630 --> 00:22:21.230 meant in terms of f of z and f of z plus delta z. 00:22:21.230 --> 00:22:24.860 Consequently, this should have meaning regardless of any 00:22:24.860 --> 00:22:27.200 interpretation in terms of u's and v's. 00:22:27.200 --> 00:22:31.370 Notice that if I write this, by definition of this, 00:22:31.370 --> 00:22:36.380 this quotient f of z0 plus delta z minus f of z0 over delta z 00:22:36.380 --> 00:22:41.120 is just z0 plus delta z squared minus z0 squared over delta z. 00:22:41.120 --> 00:22:44.690 The binomial theorem works for complex numbers the same way 00:22:44.690 --> 00:22:46.130 as it does for real numbers. 00:22:46.130 --> 00:22:49.280 In fact, that was what motivated our definition for multiplying 00:22:49.280 --> 00:22:50.550 complex numbers. 00:22:50.550 --> 00:22:52.220 if you recall, our lecture of last time 00:22:52.220 --> 00:22:55.160 was to mimic what happens in the real case. 00:22:55.160 --> 00:22:57.050 This, the same as before, comes out 00:22:57.050 --> 00:23:02.270 to be what? z0 squared, which kills off this minus z0 squared 00:23:02.270 --> 00:23:07.592 plus 2z0 delta z plus delta z square, all over delta z. 00:23:07.592 --> 00:23:09.800 I'm assuming I'm now going to take the limit as delta 00:23:09.800 --> 00:23:11.150 z approaches 0. 00:23:11.150 --> 00:23:13.880 That means, in particular , delta z is not 0. 00:23:13.880 --> 00:23:17.330 Because delta z is not 0, I can divide through by it. 00:23:17.330 --> 00:23:20.510 That gives me 2z0 plus delta z. 00:23:20.510 --> 00:23:25.410 And now notice if I let delta z approach 0, z0 00:23:25.410 --> 00:23:26.660 is a fixed number. 00:23:26.660 --> 00:23:29.420 That doesn't change as delta z approaches 0. 00:23:29.420 --> 00:23:32.780 On the other hand, delta z approaches 0 00:23:32.780 --> 00:23:35.450 as delta z approaches 0, that's a truism. 00:23:35.450 --> 00:23:38.690 Delta z approaches 0, no matter what the direction of approach 00:23:38.690 --> 00:23:39.280 is. 00:23:39.280 --> 00:23:41.720 This is delta z goes to 0, it goes to 0. 00:23:41.720 --> 00:23:43.340 This term does drop out. 00:23:43.340 --> 00:23:45.830 And you wind up with the anticipated result 00:23:45.830 --> 00:23:49.550 that f prime of z0 is 2z0. 00:23:49.550 --> 00:23:53.090 Notice again, structurally, that in the case where there is 00:23:53.090 --> 00:23:57.080 an analogy between the complex variable and the real variable, 00:23:57.080 --> 00:24:00.930 the steps are verbatim because the definitions were mimicked 00:24:00.930 --> 00:24:04.140 to be verbatim, that you cannot tell the difference between 00:24:04.140 --> 00:24:08.550 the proof that f prime of z is 2z in this case with the proof 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 00:24:13.200 --> 00:24:14.740 in the real case. 00:24:14.740 --> 00:24:17.400 Or as another example, one would expect 00:24:17.400 --> 00:24:21.480 that the derivative of f of z, if f of z is 1 over z, 00:24:21.480 --> 00:24:24.090 should be minus 1 over z squared. 00:24:24.090 --> 00:24:28.620 Except that when z is 0, this shouldn't be differentiable. 00:24:28.620 --> 00:24:31.860 I leave it again as a voluntary exercise for you 00:24:31.860 --> 00:24:34.980 to apply this definition to 1 over z, 00:24:34.980 --> 00:24:38.040 1 over z plus delta z, et cetera, and show that step 00:24:38.040 --> 00:24:41.490 by step you get the same result as in the corresponding problem 00:24:41.490 --> 00:24:43.500 involving 1 over x. 00:24:43.500 --> 00:24:46.140 To show you how this works in terms of u's and v's, just 00:24:46.140 --> 00:24:50.030 for the drill, if z is x plus i-y, 1 over z 00:24:50.030 --> 00:24:52.920 is 1 over x plus i-y. 00:24:52.920 --> 00:24:54.840 Multiplying numerator and denominator 00:24:54.840 --> 00:24:57.990 by the complex conjugate x minus i-y here, 00:24:57.990 --> 00:25:02.250 my denominator becomes x squared plus y squared. 00:25:02.250 --> 00:25:04.530 My numerator is x minus i-y. 00:25:04.530 --> 00:25:07.020 That's the same as x over x squared 00:25:07.020 --> 00:25:12.030 plus y squared plus i times minus y 00:25:12.030 --> 00:25:16.000 over x squared plus y squared. 00:25:16.000 --> 00:25:17.160 This is the u part. 00:25:17.160 --> 00:25:18.360 This is the v part. 00:25:18.360 --> 00:25:21.240 u equals x over x squared plus y squared. 00:25:21.240 --> 00:25:25.530 v equals minus y over x squared plus y squared. 00:25:25.530 --> 00:25:28.530 I again leave it as an exercise for you 00:25:28.530 --> 00:25:32.040 to show that the Cauchy-Riemann conditions are obeyed. 00:25:32.040 --> 00:25:34.470 In other words, that the partial of u with respect to x 00:25:34.470 --> 00:25:36.780 equals the partial of u with respect to y. 00:25:36.780 --> 00:25:38.700 The partial of u with respect to y 00:25:38.700 --> 00:25:41.160 is minus the partial of u with respect to x. 00:25:41.160 --> 00:25:43.910 And the only place that this isn't true 00:25:43.910 --> 00:25:47.000 turns out to be when this denominator is 0. 00:25:47.000 --> 00:25:49.400 The only time this denominator can be 0 00:25:49.400 --> 00:25:51.470 is when x and y are both 0. 00:25:51.470 --> 00:25:53.540 In other words, the Cauchy-Riemann conditions 00:25:53.540 --> 00:25:57.200 here are obeyed except when x equals y equals 0. 00:25:57.200 --> 00:25:59.930 And that, of course, implies what? 00:25:59.930 --> 00:26:03.737 That z is 0, the real and the imaginary parts of 0. 00:26:03.737 --> 00:26:05.320 And what we're saying is that 1 over z 00:26:05.320 --> 00:26:07.910 is analytic except when z is 0. 00:26:07.910 --> 00:26:11.300 And in fact, the derivative is minus 1 over z squared. 00:26:11.300 --> 00:26:14.120 For a third example, I would like to pick one that we 00:26:14.120 --> 00:26:14.690 can't-- 00:26:14.690 --> 00:26:17.160 that has no analog in the real case. 00:26:17.160 --> 00:26:19.460 I want to pick a genuine complex valued 00:26:19.460 --> 00:26:21.320 function of a complex variable. 00:26:21.320 --> 00:26:23.710 And I'm going to pick a very simple one. 00:26:23.710 --> 00:26:25.710 And the reason I picked a simple one is twofold. 00:26:25.710 --> 00:26:28.730 One is the computations are easy to do. 00:26:28.730 --> 00:26:32.420 The result is easy to interpret geometrically. 00:26:32.420 --> 00:26:34.940 And thirdly and most importantly, I 00:26:34.940 --> 00:26:39.530 mentioned before that it is not easy for a complex valued 00:26:39.530 --> 00:26:41.260 function to have a derivative. 00:26:41.260 --> 00:26:43.940 A directional derivative must exist in every direction 00:26:43.940 --> 00:26:44.930 and be the same. 00:26:44.930 --> 00:26:46.640 That's a very stringent condition. 00:26:46.640 --> 00:26:48.530 So I picked this third example to show you 00:26:48.530 --> 00:26:51.440 how a relatively simple complex valued 00:26:51.440 --> 00:26:55.220 function of a complex variable can have no derivative. 00:26:55.220 --> 00:26:57.140 And the simple function that I have chosen 00:26:57.140 --> 00:27:01.220 is the one which maps the complex conjugate, the number 00:27:01.220 --> 00:27:02.990 into its complex conjugate. 00:27:02.990 --> 00:27:06.680 The complex number z is mapped by f into f bar. 00:27:06.680 --> 00:27:09.920 In terms of two tuples, if the input of the f machine 00:27:09.920 --> 00:27:14.330 is x comma y, the output will be x comma minus y. 00:27:14.330 --> 00:27:16.820 A very simple thing, very straightforward, 00:27:16.820 --> 00:27:19.690 easy function machine to work with here. 00:27:19.690 --> 00:27:23.570 Notice in this case that u is x. v is minus y. 00:27:23.570 --> 00:27:26.150 The partial of u with respect to x is 1. 00:27:26.150 --> 00:27:29.555 The partial of u with respect to y is minus 1. 00:27:29.555 --> 00:27:31.070 1 is not equal to minus 1. 00:27:31.070 --> 00:27:34.790 Consequently, u sub x is not equal to v sub y. 00:27:34.790 --> 00:27:36.830 That means that the Cauchy-Riemann conditions 00:27:36.830 --> 00:27:37.940 are not obeyed. 00:27:37.940 --> 00:27:40.910 By what we just saw a few minutes ago, that in turn 00:27:40.910 --> 00:27:45.150 means that this function does not have a derivative. 00:27:45.150 --> 00:27:47.550 Well, that's nice to know the function doesn't 00:27:47.550 --> 00:27:48.300 have a derivative. 00:27:48.300 --> 00:27:51.740 It would be even nicer if we could see why it didn't. 00:27:51.740 --> 00:27:53.570 And the best way to do this is, again, 00:27:53.570 --> 00:27:55.730 go back to the definition. 00:27:55.730 --> 00:27:57.980 What we're going to ultimately do to take a derivative 00:27:57.980 --> 00:28:02.960 is we're going to take delta w where w is z bar. 00:28:02.960 --> 00:28:05.840 Delta w over delta z and take the limit 00:28:05.840 --> 00:28:08.950 as delta z approaches 0 along thee various directions. 00:28:08.950 --> 00:28:13.350 Well, delta w is just delta x minus i-delta y. 00:28:13.350 --> 00:28:16.850 Delta z is delta x plus i-delta y. 00:28:16.850 --> 00:28:18.800 To keep in mind that I'm going to move 00:28:18.800 --> 00:28:22.790 in a particular direction in the xy plane to approach z0, 00:28:22.790 --> 00:28:24.990 I'm going to it up the slope over here. 00:28:24.990 --> 00:28:26.870 Namely in this expression, I'm going 00:28:26.870 --> 00:28:29.480 to assume that delta x is not 0. 00:28:29.480 --> 00:28:32.350 And what I'm going to do is divide through a numerator 00:28:32.350 --> 00:28:33.920 and denominator by delta x. 00:28:33.920 --> 00:28:39.020 That gives me 1 minus i times delta y over delta x over 1 00:28:39.020 --> 00:28:41.840 plus i-delta y over delta x. 00:28:41.840 --> 00:28:49.850 And as I approach the point z0 in the xy plane, 00:28:49.850 --> 00:28:55.280 delta y over delta x approaches the slope of the curve at z0, 00:28:55.280 --> 00:28:56.960 which we'll call dy dx. 00:28:56.960 --> 00:28:58.340 And I wind up with this. 00:28:58.340 --> 00:29:01.990 Now to simplify this, let me assume that I approach 00:29:01.990 --> 00:29:04.580 z0 along some straight line. 00:29:04.580 --> 00:29:06.920 Let me pick a fixed straight line. 00:29:06.920 --> 00:29:09.620 I'll pick a straight line which goes through z0 00:29:09.620 --> 00:29:11.150 and has slope m. 00:29:11.150 --> 00:29:14.720 So in that case, dy dx is the same as delta 00:29:14.720 --> 00:29:19.790 x is m for all points on this particular line. 00:29:19.790 --> 00:29:21.690 So what is dy dx in this case? 00:29:21.690 --> 00:29:22.850 It's m. 00:29:22.850 --> 00:29:26.057 Consequently, what is delta w over delta z in this case? 00:29:26.057 --> 00:29:27.890 And by the way, notice what I'm saying here. 00:29:27.890 --> 00:29:30.510 What I'm saying is this is this line segment. 00:29:30.510 --> 00:29:33.610 z0 maps into w0. 00:29:33.610 --> 00:29:37.730 z1 maps into w1, sufficiently close to z0 00:29:37.730 --> 00:29:40.220 I can assume that the image of this straight line 00:29:40.220 --> 00:29:44.120 is a straight line that joins w0 to w1. 00:29:44.120 --> 00:29:47.330 And the ratio that I want is this length 00:29:47.330 --> 00:29:50.270 divided by this length, where by dividing lengths 00:29:50.270 --> 00:29:53.360 I mean the usual complex variable interpretation, 00:29:53.360 --> 00:29:56.030 dividing magnitudes and subtracting arguments. 00:29:56.030 --> 00:29:57.860 At any rate, what we're saying is 00:29:57.860 --> 00:30:03.620 that delta w over delta z in this case is just 1 minus i-m 00:30:03.620 --> 00:30:05.630 over 1 plus i-m. 00:30:05.630 --> 00:30:09.320 And notice that as delta z approaches 0 in this direction, 00:30:09.320 --> 00:30:10.370 this limit exists. 00:30:10.370 --> 00:30:12.800 In fact, there's no variable in here. 00:30:12.800 --> 00:30:15.680 It's 1 minus i-m over 1 plus i-m. 00:30:15.680 --> 00:30:20.900 But notice that even though that limit exists, 00:30:20.900 --> 00:30:23.210 the value depends on m. 00:30:23.210 --> 00:30:25.630 If I change m, I change this ratio. 00:30:25.630 --> 00:30:27.500 And because that limit depends on m, 00:30:27.500 --> 00:30:30.800 it means that even though the directional derivative exists 00:30:30.800 --> 00:30:35.790 different directions will give me-- 00:30:35.790 --> 00:30:37.350 see, if I come in along this line 00:30:37.350 --> 00:30:40.440 I can compute this ratio of delta w over delta z. 00:30:40.440 --> 00:30:41.750 Will I get in that case? 00:30:41.750 --> 00:30:46.440 1 minus i-m 1 over 1 plus i-m 1. 00:30:46.440 --> 00:30:47.980 That limit will exist. 00:30:47.980 --> 00:30:50.250 But it will be unequal to this one. 00:30:50.250 --> 00:30:53.530 You see, the derivative exists in each direction. 00:30:53.530 --> 00:30:56.460 But it's not the same in each direction. 00:30:56.460 --> 00:30:57.780 That's how subtle this is. 00:30:57.780 --> 00:31:01.230 I will leave the remaining exercises-- 00:31:01.230 --> 00:31:05.280 not the remaining exercises but exercises to hit home 00:31:05.280 --> 00:31:07.860 at the remaining fine points. 00:31:07.860 --> 00:31:10.830 What I wanted to do in closing today's lesson 00:31:10.830 --> 00:31:14.190 was to show you one more tremendous connection 00:31:14.190 --> 00:31:19.050 with complex variables, or between complex variables 00:31:19.050 --> 00:31:22.290 and real problems in the physical world. 00:31:22.290 --> 00:31:26.100 You may recall from some of our previous exercises 00:31:26.100 --> 00:31:29.640 in an earlier block that we say that u of xy 00:31:29.640 --> 00:31:33.380 satisfies Laplace's equation if the second partial of u 00:31:33.380 --> 00:31:35.700 with respect to x plus the second partial of u 00:31:35.700 --> 00:31:37.560 with respect to y is 0. 00:31:37.560 --> 00:31:40.320 In fact, if u denotes temperature distribution 00:31:40.320 --> 00:31:41.280 at the point-- 00:31:41.280 --> 00:31:44.010 the temperature at the point x comma y, this equation 00:31:44.010 --> 00:31:47.035 is what's known as the steady state equation. 00:31:47.035 --> 00:31:49.410 We're not going to worry about the physical ramifications 00:31:49.410 --> 00:31:49.910 here. 00:31:49.910 --> 00:31:54.000 What is important is that to solve this partial differential 00:31:54.000 --> 00:31:56.730 equation, which comes up in the physical world, 00:31:56.730 --> 00:31:59.460 this is a real partial differential equation 00:31:59.460 --> 00:32:00.960 meaning real in two senses. 00:32:00.960 --> 00:32:02.670 It comes up in the real world. 00:32:02.670 --> 00:32:07.620 And also in the sense that u is a real valued function of x 00:32:07.620 --> 00:32:08.820 and y. 00:32:08.820 --> 00:32:10.700 Now the interesting point is this. 00:32:10.700 --> 00:32:13.290 That if u plus i-v is analytic, what 00:32:13.290 --> 00:32:16.230 do we know about u plus i-v being analytic? 00:32:16.230 --> 00:32:19.500 It means that the partial of u with respect to x is v sub y. 00:32:19.500 --> 00:32:21.060 The partial of u with respect to y 00:32:21.060 --> 00:32:24.510 is minus the partial of v with respect to x. 00:32:24.510 --> 00:32:26.970 If I differentiate both sides here with respect 00:32:26.970 --> 00:32:30.780 to x and both sides here with respect to y, 00:32:30.780 --> 00:32:34.110 I get u sub xx equals v sub yz. 00:32:34.110 --> 00:32:37.260 u sub yy is minus v sub xy. 00:32:37.260 --> 00:32:40.950 Assuming continuity so that these are equal, 00:32:40.950 --> 00:32:44.170 this says by addition that if I add these two, 00:32:44.170 --> 00:32:45.970 I get the second partial of u with respect 00:32:45.970 --> 00:32:49.920 to x plus the second partial of u with respect to y is 0. 00:32:49.920 --> 00:32:51.870 A similar argument would have shown 00:32:51.870 --> 00:32:54.482 that the same is true for the imaginary part, 00:32:54.482 --> 00:32:56.190 that the second partial of u with respect 00:32:56.190 --> 00:32:59.250 to x plus the second partial of u with respect to y also 00:32:59.250 --> 00:33:00.450 will be 0. 00:33:00.450 --> 00:33:03.570 In other words, the real and the imaginary parts 00:33:03.570 --> 00:33:06.030 which by themselves are real valued 00:33:06.030 --> 00:33:09.300 functions of two real variables, the real and imaginary parts 00:33:09.300 --> 00:33:11.730 each satisfy Laplace's equation. 00:33:11.730 --> 00:33:13.950 The converse, by the way, is also true. 00:33:13.950 --> 00:33:15.460 I'll mention that in a moment. 00:33:15.460 --> 00:33:17.220 But for the time being, let me close 00:33:17.220 --> 00:33:19.150 today's lesson with an example. 00:33:19.150 --> 00:33:23.000 We already know that f of z equals z squared is analytic. 00:33:23.000 --> 00:33:24.210 It's differentiable. 00:33:24.210 --> 00:33:28.320 The real part of z squared u is x squared minus y squared. 00:33:28.320 --> 00:33:31.590 The imaginary part v is 2xy. 00:33:31.590 --> 00:33:33.810 What that tells me without even checking 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 00:33:39.450 --> 00:33:44.580 xx plus v sub yy, those would be 0. 00:33:44.580 --> 00:33:46.260 Because of the fact that they satisfy 00:33:46.260 --> 00:33:48.210 the Cauchy-Riemann conditions, which 00:33:48.210 --> 00:33:53.020 is what we'll talk about in more detail during the exercises. 00:33:53.020 --> 00:33:55.260 And we'll bring this up more in the next lecture. 00:33:55.260 --> 00:33:57.030 But for the time being, what I wanted 00:33:57.030 --> 00:34:00.330 to close on aside from the calculus was the application. 00:34:00.330 --> 00:34:03.900 I want you to see that solving LaPlace's equation 00:34:03.900 --> 00:34:06.610 is a very important real world problem. 00:34:06.610 --> 00:34:09.870 The interesting point is that the solutions of the LaPlace's 00:34:09.870 --> 00:34:13.380 equations have to be the real and imaginary parts 00:34:13.380 --> 00:34:14.820 of analytic functions. 00:34:14.820 --> 00:34:17.190 And there aren't that many analytic functions. 00:34:17.190 --> 00:34:19.960 And consequently by knowing analytic functions, 00:34:19.960 --> 00:34:24.690 we have a good hold on what solutions of Laplace's equation 00:34:24.690 --> 00:34:26.250 in the real world look like. 00:34:26.250 --> 00:34:29.010 Well, I think that's enough for one session. 00:34:29.010 --> 00:34:30.780 Work on the exercises. 00:34:30.780 --> 00:34:33.929 And next time we're going to pound out 00:34:33.929 --> 00:34:36.900 some other interesting ramifications of what 00:34:36.900 --> 00:34:40.469 it means for a complex valued function to be differentiable. 00:34:40.469 --> 00:34:42.929 At any rate, then, until next time, goodbye. 00:34:46.820 --> 00:34:49.219 Funding for the publication of this video 00:34:49.219 --> 00:34:54.110 was provided by the Gabriella and Paul Rosenbaum Foundation. 00:34:54.110 --> 00:34:58.250 Help OCW continue to provide free and open access to MIT 00:34:58.250 --> 00:35:03.685 courses by making a donation at ocw.mit.edu/donate.