1 00:00:00,000 --> 00:00:00,499 2 00:00:00,499 --> 00:00:03,144 The following content is provided under a Creative 3 00:00:03,144 --> 00:00:03,810 Commons license. 4 00:00:03,810 --> 00:00:05,518 Your support will help MIT OpenCourseWare 5 00:00:05,518 --> 00:00:10,405 continue to offer high-quality educational resources for free. 6 00:00:10,405 --> 00:00:12,530 To make a donation, or to view additional materials 7 00:00:12,530 --> 00:00:15,840 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15,840 --> 00:00:20,570 at ocw.mit.edu. 9 00:00:20,570 --> 00:00:25,370 PROFESSOR STRANG: OK, so review session on the first part 10 00:00:25,370 --> 00:00:30,900 of the Fourier chapter, these two topics that we've done 11 00:00:30,900 --> 00:00:34,400 and that homework is now coming on. 12 00:00:34,400 --> 00:00:37,960 Fourier series, the classical facts. 13 00:00:37,960 --> 00:00:41,760 Plus, paying attention to the rate 14 00:00:41,760 --> 00:00:45,570 of decay of the Fourier coefficients, 15 00:00:45,570 --> 00:00:48,310 it's an aspect not always mentioned. 16 00:00:48,310 --> 00:00:52,400 And the energy equality is important. 17 00:00:52,400 --> 00:00:56,570 So it's not just here's the function, find the coefficient. 18 00:00:56,570 --> 00:01:02,480 That's part of it but not all of it. 19 00:01:02,480 --> 00:01:07,800 And then the discrete series we were doing today. 20 00:01:07,800 --> 00:01:10,120 OK, so those are the two topics for today, 21 00:01:10,120 --> 00:01:12,580 and then the next review session, 22 00:01:12,580 --> 00:01:14,140 which would be two weeks from now, 23 00:01:14,140 --> 00:01:18,470 would focus especially on convolution and Fourier 24 00:01:18,470 --> 00:01:20,680 integrals. 25 00:01:20,680 --> 00:01:25,140 OK, so I'm open to questions on the homework, 26 00:01:25,140 --> 00:01:27,260 or off the homework. 27 00:01:27,260 --> 00:01:32,150 Always fine. 28 00:01:32,150 --> 00:01:35,460 I didn't know how many questions to ask you on the homework. 29 00:01:35,460 --> 00:01:41,410 I wanted you to have enough practice doing this stuff 30 00:01:41,410 --> 00:01:46,120 because the time for this Fourier part of the course 31 00:01:46,120 --> 00:01:47,710 is a little shorter. 32 00:01:47,710 --> 00:01:54,740 Thanksgiving comes into it, so needed to do some exercise. 33 00:01:54,740 --> 00:01:56,210 And you've got a good question. 34 00:01:56,210 --> 00:01:56,710 Thanks. 35 00:01:56,710 --> 00:02:00,660 AUDIENCE: [INAUDIBLE] 36 00:02:00,660 --> 00:02:05,510 PROFESSOR STRANG: Number 18 on the homework, OK. 37 00:02:05,510 --> 00:02:08,860 Ah yes, OK. 38 00:02:08,860 --> 00:02:10,300 Right, alright. 39 00:02:10,300 --> 00:02:15,490 So the idea of that problem, I'm really 40 00:02:15,490 --> 00:02:20,860 asking you to read the two pages, the last two 41 00:02:20,860 --> 00:02:22,490 pages of the section. 42 00:02:22,490 --> 00:02:26,480 That use Fourier series to solve the heat equation. 43 00:02:26,480 --> 00:02:29,840 So we've used, briefly, Fourier series 44 00:02:29,840 --> 00:02:32,110 to solve Laplace's equation. 45 00:02:32,110 --> 00:02:34,380 So that was, just to recall. 46 00:02:34,380 --> 00:02:37,210 So Fourier series to solve Laplace's equation 47 00:02:37,210 --> 00:02:40,370 was when the region was a circle. 48 00:02:40,370 --> 00:02:44,330 The function was given, the boundary values were given. 49 00:02:44,330 --> 00:02:47,860 It's 2pi periodic because it is a circle. 50 00:02:47,860 --> 00:02:53,140 And we solved Laplace inside. 51 00:02:53,140 --> 00:02:58,510 Because on the boundary, the perfect thing we needed 52 00:02:58,510 --> 00:03:00,980 was the Fourier series to match the boundary. 53 00:03:00,980 --> 00:03:04,630 Now, I'm taking another, classical, classical 54 00:03:04,630 --> 00:03:06,230 application too. 55 00:03:06,230 --> 00:03:07,590 The heat equation. 56 00:03:07,590 --> 00:03:10,100 So I made it heat equation. 57 00:03:10,100 --> 00:03:15,210 So this direction is time, this direction is space. 58 00:03:15,210 --> 00:03:23,230 The heat equation is u_t=u_xx, if the coefficient-- 59 00:03:23,230 --> 00:03:26,220 Everybody here knows there would be a c in there, 60 00:03:26,220 --> 00:03:29,080 but let's take it to be one. 61 00:03:29,080 --> 00:03:34,560 Then what are the solutions, and how does a Fourier series 62 00:03:34,560 --> 00:03:37,150 help you to match the initial functions? 63 00:03:37,150 --> 00:03:43,450 So I'm matching, I'm given u(x,0) here. 64 00:03:43,450 --> 00:03:44,630 OK. 65 00:03:44,630 --> 00:03:47,070 Along this is at time zero. 66 00:03:47,070 --> 00:03:50,210 So that says at a time zero, so I have a bar. 67 00:03:50,210 --> 00:03:53,440 I have a conducting bar. 68 00:03:53,440 --> 00:03:57,790 And this is such a classical example 69 00:03:57,790 --> 00:04:02,410 that I didn't feel you could miss it completely. 70 00:04:02,410 --> 00:04:06,810 Even though we look beyond formulas. 71 00:04:06,810 --> 00:04:10,320 But here's one where the formula shows us something important. 72 00:04:10,320 --> 00:04:15,520 OK, so what are solutions to this equation? 73 00:04:15,520 --> 00:04:18,700 You look for solutions, so the classical idea 74 00:04:18,700 --> 00:04:22,880 of separate variables. 75 00:04:22,880 --> 00:04:29,670 Look for solutions that are of the form some function of t, 76 00:04:29,670 --> 00:04:32,760 maybe I'll try to use the same letters as the text. 77 00:04:32,760 --> 00:04:38,090 Some function of t, times some function of x, OK? 78 00:04:38,090 --> 00:04:43,600 And the text uses, look for solutions u(x,t) that are 79 00:04:43,600 --> 00:04:44,340 of this form. 80 00:04:44,340 --> 00:04:50,460 Some function of t, and I didn't remember. 81 00:04:50,460 --> 00:04:56,450 Ah yes, it's A(x)B(t). 82 00:04:56,450 --> 00:04:59,450 OK, that's what I mean by separating. 83 00:04:59,450 --> 00:05:02,470 So those will be especially simple solutions. 84 00:05:02,470 --> 00:05:05,770 And when we go to match the initial condition, 85 00:05:05,770 --> 00:05:09,840 I'll just plug in t=0 and I'll see the A(x)'s. 86 00:05:09,840 --> 00:05:12,280 Well, what are they? 87 00:05:12,280 --> 00:05:15,260 In this case, so their eigenfunction-- oh, 88 00:05:15,260 --> 00:05:18,020 I have to tell you about the remaining boundary conditions, 89 00:05:18,020 --> 00:05:18,885 don't I? 90 00:05:18,885 --> 00:05:24,590 Because that will decide what the A(x) has to satisfy, 91 00:05:24,590 --> 00:05:28,540 and will decide what those eigenfunctions are. 92 00:05:28,540 --> 00:05:29,770 So, let's see. 93 00:05:29,770 --> 00:05:35,970 In this problem I think I picked free conditions. 94 00:05:35,970 --> 00:05:40,210 So I made the interval minus pi to pi, that's a change. 95 00:05:40,210 --> 00:05:43,690 Minus pi to pi just so we have nice Fourier series. 96 00:05:43,690 --> 00:05:54,140 And here I have this boundary is free, du/dx, u', at x=-pi, 97 00:05:54,140 --> 00:05:59,160 for all time, so up this line, is zero. 98 00:05:59,160 --> 00:06:05,830 And also u' at plus pi, and all time is zero. 99 00:06:05,830 --> 00:06:09,810 So up those lines, heat's going out. 100 00:06:09,810 --> 00:06:11,110 That's what that means. 101 00:06:11,110 --> 00:06:14,940 Is that what that means, or does that mean heat can't go out? 102 00:06:14,940 --> 00:06:18,050 No, so what's happening? 103 00:06:18,050 --> 00:06:19,370 Heat's not going out. 104 00:06:19,370 --> 00:06:21,640 Is that right? 105 00:06:21,640 --> 00:06:23,460 The slope is zero, right? 106 00:06:23,460 --> 00:06:25,812 The slope is the temperature gradient, 107 00:06:25,812 --> 00:06:27,520 we're requiring the temperature gradient. 108 00:06:27,520 --> 00:06:30,580 So the ends of the bar are insulated. 109 00:06:30,580 --> 00:06:32,390 So this is insulated. 110 00:06:32,390 --> 00:06:35,790 No passage through, right? 111 00:06:35,790 --> 00:06:36,940 Is that what that means? 112 00:06:36,940 --> 00:06:37,780 Yeah. 113 00:06:37,780 --> 00:06:43,300 It's like there's nobody, it's cut off there. 114 00:06:43,300 --> 00:06:46,170 The rod isn't extended for heat to go further. 115 00:06:46,170 --> 00:06:55,160 OK, so that tells us what the x and the t, 116 00:06:55,160 --> 00:06:57,230 what the A(x) and B(t) are. 117 00:06:57,230 --> 00:06:58,810 So here's the point. 118 00:06:58,810 --> 00:07:04,000 You plug that hoped-for solution into the equation, right? 119 00:07:04,000 --> 00:07:07,210 So I plug it in here. 120 00:07:07,210 --> 00:07:10,310 What do I have on the left side, just the time derivative? 121 00:07:10,310 --> 00:07:16,260 So that's A times-- A(x) B'(t), you see taking the time 122 00:07:16,260 --> 00:07:18,900 derivative doesn't touch A(x). 123 00:07:18,900 --> 00:07:22,130 On the right-hand side, I don't touch B(t), 124 00:07:22,130 --> 00:07:27,970 but I have a second derivative of the x part. 125 00:07:27,970 --> 00:07:30,470 So far, so good. 126 00:07:30,470 --> 00:07:35,090 Now, a little trick. 127 00:07:35,090 --> 00:07:39,080 If I divide both sides by A and by B, 128 00:07:39,080 --> 00:07:48,250 I get B'/B equaling A''/A. Right? 129 00:07:48,250 --> 00:07:52,160 Just, put the A under here, put the B under here. 130 00:07:52,160 --> 00:07:53,470 Now what? 131 00:07:53,470 --> 00:07:58,390 This is neat, because that function is only 132 00:07:58,390 --> 00:08:00,320 depending on time. 133 00:08:00,320 --> 00:08:05,770 This function depends only on x, so that the both sides have 134 00:08:05,770 --> 00:08:06,900 to be constant. 135 00:08:06,900 --> 00:08:11,890 One can't actually change with time, because this side is not 136 00:08:11,890 --> 00:08:13,410 changing with time. 137 00:08:13,410 --> 00:08:15,080 And this couldn't actually change 138 00:08:15,080 --> 00:08:17,590 with x, because that's not changing with x. 139 00:08:17,590 --> 00:08:20,030 So those are both constants. 140 00:08:20,030 --> 00:08:23,810 So both constants. 141 00:08:23,810 --> 00:08:27,090 Let's see, I'll maybe just put a constant. 142 00:08:27,090 --> 00:08:30,050 And various constants are possible. 143 00:08:30,050 --> 00:08:32,670 OK, so now you see the point here. 144 00:08:32,670 --> 00:08:35,580 Now I have two separate equations, 145 00:08:35,580 --> 00:08:40,630 I have an equation for the B part, dB/dt, B'. 146 00:08:40,630 --> 00:08:45,920 If I bring the B up there, I have equals, 147 00:08:45,920 --> 00:08:50,330 the constant times B. And I know the solution to that. 148 00:08:50,330 --> 00:08:55,730 B(t) is, as everybody knows, what's 149 00:08:55,730 --> 00:08:59,180 the solution to a first order constant coefficient equation? 150 00:08:59,180 --> 00:09:00,540 Just e^(ct)*B(0). 151 00:09:00,540 --> 00:09:03,480 152 00:09:03,480 --> 00:09:09,140 Good, we've got B. We've got a B(t) that works, 153 00:09:09,140 --> 00:09:11,820 and now what's the A that also works? 154 00:09:11,820 --> 00:09:14,200 That has A''. 155 00:09:14,200 --> 00:09:20,940 And I bring the A up, so now I have A''=cA, so the A that goes 156 00:09:20,940 --> 00:09:25,450 with it is? 157 00:09:25,450 --> 00:09:32,540 Oh, OK, I've used a c there, so what's the good? 158 00:09:32,540 --> 00:09:35,970 I want-- Two derivatives should bring down a c. 159 00:09:35,970 --> 00:09:42,100 Let me change c to minus lambda squared. 160 00:09:42,100 --> 00:09:46,640 How about if I look ahead, change this constant to minus 161 00:09:46,640 --> 00:09:49,430 lambda squared, because I want something 162 00:09:49,430 --> 00:09:52,840 where two derivatives bring down minus lambda squared, 163 00:09:52,840 --> 00:09:55,750 and what will do that? 164 00:09:55,750 --> 00:10:00,720 Any amount of cos(lambda*x), right? 165 00:10:00,720 --> 00:10:03,530 Because two derivatives will bring down a minus lambda 166 00:10:03,530 --> 00:10:04,230 squared. 167 00:10:04,230 --> 00:10:07,980 And any amount of sin(lambda*x). 168 00:10:07,980 --> 00:10:18,480 And this is now e to the minus lambda squared t. 169 00:10:18,480 --> 00:10:21,300 I'm doing this fast, but actually it's totally simple. 170 00:10:21,300 --> 00:10:25,400 The conclusion is that I now have a bunch of solutions 171 00:10:25,400 --> 00:10:28,880 of this special separated form. 172 00:10:28,880 --> 00:10:33,470 Where B(t) could be that and A(x) could be either of those 173 00:10:33,470 --> 00:10:35,690 or any combination of those. 174 00:10:35,690 --> 00:10:39,910 And I have to use the same lambda for each, 175 00:10:39,910 --> 00:10:46,110 so that the two equations will work in the original problem. 176 00:10:46,110 --> 00:10:47,500 Good. 177 00:10:47,500 --> 00:10:51,080 Now, so far, no boundary conditions. 178 00:10:51,080 --> 00:10:54,650 What I've got so far is just a lot of solutions. 179 00:10:54,650 --> 00:10:56,740 These times that. 180 00:10:56,740 --> 00:10:58,390 With any lambda. 181 00:10:58,390 --> 00:11:00,580 But of course the boundary conditions 182 00:11:00,580 --> 00:11:03,060 will tell me the lambdas, first of all. 183 00:11:03,060 --> 00:11:04,970 And how do they tell me? 184 00:11:04,970 --> 00:11:08,810 The only boundary condition I have is this free stuff. 185 00:11:08,810 --> 00:11:14,640 So it's free, the slope should be zero at pi, and zero at? 186 00:11:14,640 --> 00:11:16,980 So that's the x direction. 187 00:11:16,980 --> 00:11:19,710 So that's going to tell me-- I've forgotten. 188 00:11:19,710 --> 00:11:23,240 Do I want cosines or sines? 189 00:11:23,240 --> 00:11:24,640 Cosines. 190 00:11:24,640 --> 00:11:29,990 I want the derivative to be zero at pi. 191 00:11:29,990 --> 00:11:32,910 Yeah, so I think I want cosines, good. 192 00:11:32,910 --> 00:11:35,690 And then lambdas can't be anything at all. 193 00:11:35,690 --> 00:11:41,570 Because, should lambda be an integer or something like that? 194 00:11:41,570 --> 00:11:43,850 I think maybe lambda should be an integer, 195 00:11:43,850 --> 00:11:47,090 because I want to plug in pi. 196 00:11:47,090 --> 00:11:50,560 So let me take the second derivative. 197 00:11:50,560 --> 00:11:54,670 Is minus lambda squared cos(lambda*x). 198 00:11:54,670 --> 00:11:58,910 And then I want to plug in x=pi. 199 00:11:58,910 --> 00:12:04,910 And I want this to be zero. 200 00:12:04,910 --> 00:12:10,010 So lambda should be an integer, is that right? 201 00:12:10,010 --> 00:12:13,900 At multiples of pi, the cosine is zero, yes. 202 00:12:13,900 --> 00:12:15,270 Is it? 203 00:12:15,270 --> 00:12:16,070 No. 204 00:12:16,070 --> 00:12:19,150 Did I want sine? 205 00:12:19,150 --> 00:12:23,800 Maybe I wanted sine. 206 00:12:23,800 --> 00:12:26,590 Oh, it's the first derivative I'm looking at, thank you. 207 00:12:26,590 --> 00:12:27,300 Thank you. 208 00:12:27,300 --> 00:12:31,740 OK, good. 209 00:12:31,740 --> 00:12:33,580 OK, now I've got it. 210 00:12:33,580 --> 00:12:34,410 Thank you. 211 00:12:34,410 --> 00:12:37,120 And now I see lambda should be an integer. 212 00:12:37,120 --> 00:12:41,090 Lambda should be, zero is, yeah zero's alright. 213 00:12:41,090 --> 00:12:43,150 That's the constant term, yeah, we need that. 214 00:12:43,150 --> 00:12:46,320 Zero, one, two, and so on. 215 00:12:46,320 --> 00:12:51,480 So do you see that I've now got, now I 216 00:12:51,480 --> 00:12:53,860 can take-- I've got a lot of solutions, 217 00:12:53,860 --> 00:12:55,510 and I have a linear problem. 218 00:12:55,510 --> 00:12:57,170 So I can take any combination. 219 00:12:57,170 --> 00:13:05,140 So finally I have final solution is that u(x,t) is any 220 00:13:05,140 --> 00:13:12,690 combination, with coefficients I'm free to choose, of A(x), 221 00:13:12,690 --> 00:13:18,570 which is cos(nx), because lambda had to be an n. 222 00:13:18,570 --> 00:13:22,700 And n could be anywhere from zero on up. 223 00:13:22,700 --> 00:13:31,070 Times e to the minus, lambda is n, so that's n squared t. 224 00:13:31,070 --> 00:13:32,150 Did I get that? 225 00:13:32,150 --> 00:13:34,970 Let me draw a circle and step back. 226 00:13:34,970 --> 00:13:35,620 What's up? 227 00:13:35,620 --> 00:13:37,380 AUDIENCE: [INAUDIBLE] 228 00:13:37,380 --> 00:13:40,280 PROFESSOR STRANG: I could have negative n's, 229 00:13:40,280 --> 00:13:42,820 they wouldn't give me anything new, right? 230 00:13:42,820 --> 00:13:47,510 I mean cos(nx) and cos(-nx) are just, 231 00:13:47,510 --> 00:13:51,070 one's just the negative of the other. 232 00:13:51,070 --> 00:13:53,040 So these are the good guys. 233 00:13:53,040 --> 00:13:58,360 I've got a cosine series because I've got free n's, right. 234 00:13:58,360 --> 00:14:00,570 Because of the boundary conditions. 235 00:14:00,570 --> 00:14:01,300 Do you see that? 236 00:14:01,300 --> 00:14:04,970 That's pretty nice. 237 00:14:04,970 --> 00:14:07,770 There's my A(x), there's my B(t), 238 00:14:07,770 --> 00:14:10,090 I can take any combination. 239 00:14:10,090 --> 00:14:11,600 Usual stuff. 240 00:14:11,600 --> 00:14:14,530 You get to that solution. 241 00:14:14,530 --> 00:14:18,950 OK, and now I have to meet the initial conditions, right? 242 00:14:18,950 --> 00:14:20,990 Boundary conditions are now built-in 243 00:14:20,990 --> 00:14:23,220 because I chose cosine. 244 00:14:23,220 --> 00:14:24,270 Or you did. 245 00:14:24,270 --> 00:14:30,810 Now, this will tell me what the c's are. 246 00:14:30,810 --> 00:14:33,030 I'm going to set t=0. 247 00:14:33,030 --> 00:14:38,380 At t=0, I'm given the initial condition, u(x,0), 248 00:14:38,380 --> 00:14:41,440 and I have the same sum of c_n*cos(nx)*e^0. 249 00:14:41,440 --> 00:14:45,460 250 00:14:45,460 --> 00:14:49,330 So this will tell me the c's are the cosine coefficients 251 00:14:49,330 --> 00:14:51,850 of the given initial conditions. 252 00:14:51,850 --> 00:14:56,890 So I expand, so here's where Fourier series has paid off. 253 00:14:56,890 --> 00:15:01,620 Expand the initial function in a cosine series. 254 00:15:01,620 --> 00:15:04,550 And then go forward in time. 255 00:15:04,550 --> 00:15:06,360 This is just the old e^(lambda*t). 256 00:15:06,360 --> 00:15:10,130 257 00:15:10,130 --> 00:15:14,300 Only the lambda we're talking about here is minus n squared. 258 00:15:14,300 --> 00:15:18,260 And you see what's happening here? 259 00:15:18,260 --> 00:15:21,260 What's going to happen for large time? 260 00:15:21,260 --> 00:15:23,780 So this is a very physical problem. 261 00:15:23,780 --> 00:15:28,220 That I think you cannot take 18.085 without seeing this 262 00:15:28,220 --> 00:15:29,600 problem. 263 00:15:29,600 --> 00:15:31,430 You can't learn about Fourier series 264 00:15:31,430 --> 00:15:36,410 without using it for the initial value. 265 00:15:36,410 --> 00:15:41,610 And then propagating in time with the usual exponentials 266 00:15:41,610 --> 00:15:43,150 for each one. 267 00:15:43,150 --> 00:15:46,690 And now as n increases what do I see? 268 00:15:46,690 --> 00:15:48,710 Faster and faster decay. 269 00:15:48,710 --> 00:15:54,040 For large n, these are going to zero extremely fast. 270 00:15:54,040 --> 00:16:00,090 So that what you see with a solid bar, which 271 00:16:00,090 --> 00:16:07,830 starts with the temperature u in some-- Oh, probably 272 00:16:07,830 --> 00:16:10,400 not negative unless it's a really cold bar. 273 00:16:10,400 --> 00:16:16,410 But, anyway, it starts with some initial temperature. 274 00:16:16,410 --> 00:16:18,880 That flattens out fast. 275 00:16:18,880 --> 00:16:21,990 The heat flows, to equilibrate. 276 00:16:21,990 --> 00:16:26,010 What I approach is the constant term, c_0. 277 00:16:26,010 --> 00:16:28,330 This approaches c_0. 278 00:16:28,330 --> 00:16:33,920 Because all these other n positives, they go to zero. 279 00:16:33,920 --> 00:16:38,690 So the heat distributes itself equally. 280 00:16:38,690 --> 00:16:43,570 OK, and now I guess the particular u(0) in the problem 281 00:16:43,570 --> 00:16:46,430 was a delta. 282 00:16:46,430 --> 00:16:52,470 OK, and so the particular u(0) was all, 283 00:16:52,470 --> 00:16:58,610 was from that really hot point. 284 00:16:58,610 --> 00:17:00,620 So we know the coefficients. 285 00:17:00,620 --> 00:17:04,910 We know the cosine coefficients for the delta function, 286 00:17:04,910 --> 00:17:09,240 we know these c's, and what were they? 287 00:17:09,240 --> 00:17:12,530 1/(2pi) was c_0. 288 00:17:12,530 --> 00:17:15,740 And the other c's were 1/pi, I think. 289 00:17:15,740 --> 00:17:18,320 Is that right? 290 00:17:18,320 --> 00:17:21,120 Maybe they're all 1/(2pi). 291 00:17:21,120 --> 00:17:22,030 Maybe. 292 00:17:22,030 --> 00:17:22,970 Yeah. 293 00:17:22,970 --> 00:17:24,140 Whatever. 294 00:17:24,140 --> 00:17:25,500 They disappear fast. 295 00:17:25,500 --> 00:17:28,210 And this is what we approach. 296 00:17:28,210 --> 00:17:31,230 So the heat from the delta function is, yeah. 297 00:17:31,230 --> 00:17:37,030 So is that everything the problem wanted? 298 00:17:37,030 --> 00:17:37,530 Yeah. 299 00:17:37,530 --> 00:17:38,690 Yeah. 300 00:17:38,690 --> 00:17:40,560 I think we've done it. 301 00:17:40,560 --> 00:17:44,680 We'll put in the c_n's to complete that picture. 302 00:17:44,680 --> 00:17:46,350 Into here. 303 00:17:46,350 --> 00:17:51,480 And then c_0 is the one that survives over time. 304 00:17:51,480 --> 00:17:54,820 Yeah. 305 00:17:54,820 --> 00:17:57,800 I guess you've, once I got rolling I couldn't stop 306 00:17:57,800 --> 00:18:01,870 and that's u. 307 00:18:01,870 --> 00:18:04,630 For investing time this afternoon 308 00:18:04,630 --> 00:18:10,350 you get a fast look at this classical, classical problem 309 00:18:10,350 --> 00:18:14,600 of separating the variables using the Fourier series 310 00:18:14,600 --> 00:18:16,350 for the initial function. 311 00:18:16,350 --> 00:18:24,190 And recognizing that we're doing this on a finite interval. 312 00:18:24,190 --> 00:18:31,670 If the bar was infinitely long, then we 313 00:18:31,670 --> 00:18:34,410 would be talking about Fourier integrals. 314 00:18:34,410 --> 00:18:37,890 And that's what's coming up a bit later. 315 00:18:37,890 --> 00:18:40,320 We would integrate instead of sum. 316 00:18:40,320 --> 00:18:40,900 Yeah. 317 00:18:40,900 --> 00:18:44,380 But the idea would not be different, 318 00:18:44,380 --> 00:18:46,720 if we had infinite bar then we would not 319 00:18:46,720 --> 00:18:49,150 be restricted to n equals zero, one, two, 320 00:18:49,150 --> 00:18:52,610 three; any n, any cosine, wouldn't 321 00:18:52,610 --> 00:18:54,440 have to be an integer at all. 322 00:18:54,440 --> 00:18:57,360 Any number, any frequency would be allowed. 323 00:18:57,360 --> 00:19:02,780 And so we would have to integrate that, yeah. 324 00:19:02,780 --> 00:19:08,330 And that's a classical problem too, again. 325 00:19:08,330 --> 00:19:13,210 It's come up in a modern way, that the famous Black-Scholes 326 00:19:13,210 --> 00:19:15,460 equation. 327 00:19:15,460 --> 00:19:17,200 So. 328 00:19:17,200 --> 00:19:21,480 The heat equation is for 18.086. 329 00:19:21,480 --> 00:19:26,490 Here, we brought it up because we could solve it fast. 330 00:19:26,490 --> 00:19:30,990 But the actual, yeah. 331 00:19:30,990 --> 00:19:33,290 The most important solution I could give you 332 00:19:33,290 --> 00:19:36,170 to the heat equation would be the one that starts 333 00:19:36,170 --> 00:19:39,650 from that point source of heat. 334 00:19:39,650 --> 00:19:42,730 But on the whole line. 335 00:19:42,730 --> 00:19:46,770 The one that would be integrals instead of sums. 336 00:19:46,770 --> 00:19:47,850 Yeah. 337 00:19:47,850 --> 00:19:52,320 So we came pretty close to solving the most important heat 338 00:19:52,320 --> 00:19:53,330 equation problem. 339 00:19:53,330 --> 00:19:58,170 But we're doing the periodic case, with just cosines, 340 00:19:58,170 --> 00:20:05,050 and the infinite line case would be the most famous of all. 341 00:20:05,050 --> 00:20:05,550 Yeah. 342 00:20:05,550 --> 00:20:07,820 And it has a beautiful form, and I 343 00:20:07,820 --> 00:20:12,190 was going to say that the heat equation's pretty classical. 344 00:20:12,190 --> 00:20:22,660 But let's see, where can I write the magic words, Black-Scholes. 345 00:20:22,660 --> 00:20:26,860 Next to the heat equation, so that's the heat equation 346 00:20:26,860 --> 00:20:36,880 but it's also-- do you know these names, Black and Scholes? 347 00:20:36,880 --> 00:20:40,180 Anybody in Mathematics of Finance? 348 00:20:40,180 --> 00:20:47,030 So these much-despised option, derivative options, 349 00:20:47,030 --> 00:20:49,760 so people on Wall Street, traders, 350 00:20:49,760 --> 00:20:55,560 will carry around a little calculator that 351 00:20:55,560 --> 00:20:57,720 solves the Black-Scholes equation 352 00:20:57,720 --> 00:20:59,820 so they can price the options that they're 353 00:20:59,820 --> 00:21:04,780 bidding to buy and sell, so they can price them fast. 354 00:21:04,780 --> 00:21:09,480 And that little calculator does a finite differences, 355 00:21:09,480 --> 00:21:14,650 or a Fourier series solution to this Black-Scholes equation, 356 00:21:14,650 --> 00:21:17,880 which actually, if you change variables on it, 357 00:21:17,880 --> 00:21:19,610 is the heat equation. 358 00:21:19,610 --> 00:21:23,540 So what you see here as is actually 359 00:21:23,540 --> 00:21:28,210 important on Wall Street except, it's probably not 360 00:21:28,210 --> 00:21:29,650 the right moment to mention it. 361 00:21:29,650 --> 00:21:34,010 Right? 362 00:21:34,010 --> 00:21:38,180 So you can blame the whole meltdown on mathematicians. 363 00:21:38,180 --> 00:21:41,900 But that wouldn't be entirely fair. 364 00:21:41,900 --> 00:21:43,820 They didn't mean it, anyway. 365 00:21:43,820 --> 00:21:49,940 But that's been the biggest source of employment, 366 00:21:49,940 --> 00:21:55,340 I guess, apart from teaching, in the last ten years. 367 00:21:55,340 --> 00:21:58,480 People who could work out the partial differential 368 00:21:58,480 --> 00:22:01,070 equations, and they get more complicated than the heat 369 00:22:01,070 --> 00:22:03,630 equation, you can be sure. 370 00:22:03,630 --> 00:22:09,650 And so the classical one, these guys 371 00:22:09,650 --> 00:22:13,620 are economists at MIT and Harvard, or they were. 372 00:22:13,620 --> 00:22:18,590 And I guess maybe the Nobel Prize in Economics 373 00:22:18,590 --> 00:22:20,420 came to part of that group. 374 00:22:20,420 --> 00:22:23,290 And also to Merton. 375 00:22:23,290 --> 00:22:28,750 Maybe Black, possibly Black died before the time 376 00:22:28,750 --> 00:22:31,210 of the Nobel Prize. 377 00:22:31,210 --> 00:22:32,690 Anyway, they were the first. 378 00:22:32,690 --> 00:22:36,930 And it's a beautiful paper, beautiful paper too. 379 00:22:36,930 --> 00:22:39,850 Just to figure out how do you price 380 00:22:39,850 --> 00:22:44,410 the, what's the value of an option to buy, 381 00:22:44,410 --> 00:22:49,250 that allows you to buy or sell a stock at a later time? 382 00:22:49,250 --> 00:22:50,160 Yeah. 383 00:22:50,160 --> 00:22:54,770 So it's, well of course you have to make assumptions 384 00:22:54,770 --> 00:22:57,560 on what's going to happen over that time 385 00:22:57,560 --> 00:23:01,620 and that's where Wall Street came to grief, I guess. 386 00:23:01,620 --> 00:23:08,160 If you had to put it in a nutshell, 387 00:23:08,160 --> 00:23:12,020 I mean, the options, the standard straightforward 388 00:23:12,020 --> 00:23:13,790 options, those are fine. 389 00:23:13,790 --> 00:23:17,510 Using Black-Scholes, and then what's happened 390 00:23:17,510 --> 00:23:21,760 is they now price more and more complicated things. 391 00:23:21,760 --> 00:23:26,100 To the point that the banks were buying and selling 392 00:23:26,100 --> 00:23:32,030 credit default swaps, insurance swaps, that practically nobody 393 00:23:32,030 --> 00:23:33,950 understood what they were. 394 00:23:33,950 --> 00:23:37,190 They just assumed that if there was always a market for them, 395 00:23:37,190 --> 00:23:39,360 like insurance, somehow it wouldn't happen. 396 00:23:39,360 --> 00:23:41,060 And you get on. 397 00:23:41,060 --> 00:23:42,370 But it happened. 398 00:23:42,370 --> 00:23:46,290 So, now we're in trouble. 399 00:23:46,290 --> 00:23:50,560 OK, that's not 18.085, fortunately. 400 00:23:50,560 --> 00:23:55,280 Or math, but. 401 00:23:55,280 --> 00:23:58,320 Anyway, a lot of people got involved with things 402 00:23:58,320 --> 00:24:01,050 they didn't really know about. 403 00:24:01,050 --> 00:24:07,280 And then were selling them as well as of course the mortgage 404 00:24:07,280 --> 00:24:07,840 problems. 405 00:24:07,840 --> 00:24:11,650 Anyway. 406 00:24:11,650 --> 00:24:20,250 Ready for other questions on our homework, or these topics. 407 00:24:20,250 --> 00:24:21,710 Yeah. 408 00:24:21,710 --> 00:24:22,230 OK. 409 00:24:22,230 --> 00:24:23,837 AUDIENCE: [INAUDIBLE] 410 00:24:23,837 --> 00:24:24,670 PROFESSOR STRANG: OK 411 00:24:24,670 --> 00:24:28,860 AUDIENCE: [INAUDIBLE] 412 00:24:28,860 --> 00:24:35,230 PROFESSOR STRANG: OK, right, yeah. 413 00:24:35,230 --> 00:24:37,100 Have an image of waves, so -- 414 00:24:37,100 --> 00:24:38,860 AUDIENCE: [INAUDIBLE] 415 00:24:38,860 --> 00:24:41,600 PROFESSOR STRANG: Yeah. 416 00:24:41,600 --> 00:24:45,390 I suppose if I had to have a picture of the discrete thing, 417 00:24:45,390 --> 00:24:50,250 if my picture of the function case 418 00:24:50,250 --> 00:24:52,950 was a bunch of sines, and cosines, 419 00:24:52,950 --> 00:24:57,090 somehow adding up to my function. 420 00:24:57,090 --> 00:25:01,380 And if time, if I'm solving the heat equation 421 00:25:01,380 --> 00:25:09,030 then those separate waves are maybe decaying in time. 422 00:25:09,030 --> 00:25:09,920 Here. 423 00:25:09,920 --> 00:25:12,220 So that when I add them up at a later time 424 00:25:12,220 --> 00:25:13,530 I get something different. 425 00:25:13,530 --> 00:25:18,420 Or if it was the wave equation, which is probably your image, 426 00:25:18,420 --> 00:25:20,510 they're moving in time. 427 00:25:20,510 --> 00:25:23,780 So they add up to different things at different times, 428 00:25:23,780 --> 00:25:26,120 because they can move at different speeds. 429 00:25:26,120 --> 00:25:31,870 Yes, so a function is a sum of waves, right? 430 00:25:31,870 --> 00:25:34,520 Then what would the discrete guy be? 431 00:25:34,520 --> 00:25:38,640 I guess I would just have to imagine the function as only 432 00:25:38,640 --> 00:25:41,130 having those n values. 433 00:25:41,130 --> 00:25:44,750 And my wave would just be, a wave 434 00:25:44,750 --> 00:25:51,200 might be just n values there. 435 00:25:51,200 --> 00:25:57,630 But still, if I have a time-dependent problem, 436 00:25:57,630 --> 00:26:01,050 maybe that thing is pulsing up and down. 437 00:26:01,050 --> 00:26:05,400 It's just that it's only got a fixed number of points. 438 00:26:05,400 --> 00:26:09,230 And I'm not looking at the whole wave on a, yeah. 439 00:26:09,230 --> 00:26:15,270 But I don't think it's essentially different. 440 00:26:15,270 --> 00:26:19,090 And of course, the fast Fourier transform and the discrete case 441 00:26:19,090 --> 00:26:23,360 is used to approximate the continuous one. 442 00:26:23,360 --> 00:26:24,880 Yeah. 443 00:26:24,880 --> 00:26:26,700 You can look in Numerical Recipes 444 00:26:26,700 --> 00:26:31,080 for a discussion of approximating the Fourier 445 00:26:31,080 --> 00:26:33,300 series by discrete Fourier series. 446 00:26:33,300 --> 00:26:34,910 I mean, that's an important question. 447 00:26:34,910 --> 00:26:36,930 Because of course, Fourier series 448 00:26:36,930 --> 00:26:39,270 has got all these integrals. 449 00:26:39,270 --> 00:26:46,650 The coefficients come from an integral formula. 450 00:26:46,650 --> 00:26:49,300 We're not going to do those integrals exactly, 451 00:26:49,300 --> 00:26:53,000 so we have to do them some approximate way. 452 00:26:53,000 --> 00:26:58,220 And one way would be to use equally spaced points 453 00:26:58,220 --> 00:27:00,820 and do the DFT. 454 00:27:00,820 --> 00:27:04,360 Can you just remind me what that integral formula is? 455 00:27:04,360 --> 00:27:07,600 I don't want you to, it was on the board today. 456 00:27:07,600 --> 00:27:11,840 What's the formula for the coefficient c_k in the Fourier 457 00:27:11,840 --> 00:27:15,640 series. 458 00:27:15,640 --> 00:27:18,810 I'm really just asking you this because I 459 00:27:18,810 --> 00:27:24,410 think you should have it in some memory cache, 460 00:27:24,410 --> 00:27:28,830 you know in fast memory, rather than in the textbook. 461 00:27:28,830 --> 00:27:31,630 OK, so what's the formula for c_k, 462 00:27:31,630 --> 00:27:34,250 in the Fourier series case? 463 00:27:34,250 --> 00:27:36,910 Everybody think about it. 464 00:27:36,910 --> 00:27:39,130 It's going to be an integral, right? 465 00:27:39,130 --> 00:27:42,730 And I'll take it over zero to 2pi, I don't mind. 466 00:27:42,730 --> 00:27:44,650 Or minus pi, pi. 467 00:27:44,650 --> 00:27:46,160 And what do I integrate? 468 00:27:46,160 --> 00:27:49,760 This is the Fourier coefficients of the function f(x)? 469 00:27:49,760 --> 00:27:55,180 So I take f(x), I remember to divide by 2pi, 470 00:27:55,180 --> 00:27:59,050 I'm doing the continuous one, what do I multiply by 471 00:27:59,050 --> 00:28:03,340 to get the coefficients? 472 00:28:03,340 --> 00:28:05,220 e^(-ikx). 473 00:28:05,220 --> 00:28:12,350 474 00:28:12,350 --> 00:28:16,030 So I've forgotten whether I assigned some 475 00:28:16,030 --> 00:28:19,000 of these very early questions. 476 00:28:19,000 --> 00:28:20,740 It just gave you the function and said 477 00:28:20,740 --> 00:28:22,440 what's the coefficient. 478 00:28:22,440 --> 00:28:24,590 If I just look at one or two. 479 00:28:24,590 --> 00:28:27,400 Suppose my function is f(x)=x. 480 00:28:27,400 --> 00:28:30,940 I guess in that problem, in Problem 1, 481 00:28:30,940 --> 00:28:33,700 I made it minus pi to pi. 482 00:28:33,700 --> 00:28:34,420 Suppose f(x)=x. 483 00:28:34,420 --> 00:28:40,220 484 00:28:40,220 --> 00:28:44,740 So you have to integrate x times e^(-ikx). 485 00:28:44,740 --> 00:28:48,150 486 00:28:48,150 --> 00:28:50,780 Well, you got an integral to do. 487 00:28:50,780 --> 00:28:55,920 OK, it's doable but not instantly doable. 488 00:28:55,920 --> 00:28:58,890 Let me ask you some questions and you tell me about it. 489 00:28:58,890 --> 00:29:01,530 So I draw the function. 490 00:29:01,530 --> 00:29:05,040 The function is x from minus pi to pi. 491 00:29:05,040 --> 00:29:08,330 And tell me about the coefficients, 492 00:29:08,330 --> 00:29:13,860 how quickly do they decay? 493 00:29:13,860 --> 00:29:18,790 This is like some constant over k to some power p, 494 00:29:18,790 --> 00:29:23,450 and what's p? 495 00:29:23,450 --> 00:29:27,620 What rate of decay are you expecting for the coefficients? 496 00:29:27,620 --> 00:29:29,910 Well you say to yourself, it looks like a pretty nice 497 00:29:29,910 --> 00:29:32,950 function, smooth as can be. 498 00:29:32,950 --> 00:29:37,500 But, what's the answer here? 499 00:29:37,500 --> 00:29:43,100 The rate of decay will be, what will that power be? 500 00:29:43,100 --> 00:29:44,530 One. 501 00:29:44,530 --> 00:29:49,090 Because the function jumps. 502 00:29:49,090 --> 00:29:53,630 The function has a jump there, and the Fourier coefficients 503 00:29:53,630 --> 00:29:55,170 have got to deal with it. 504 00:29:55,170 --> 00:29:57,980 So the Fourier series for this is going to be, 505 00:29:57,980 --> 00:30:01,550 if I took 100 terms it would be really close. 506 00:30:01,550 --> 00:30:09,240 And then it'll go down here to the, but it's got to get there. 507 00:30:09,240 --> 00:30:13,060 And got to start, and pick up below there. 508 00:30:13,060 --> 00:30:18,870 So it's got the same issue, the Gibbs phenomenon, 509 00:30:18,870 --> 00:30:21,440 that the square wave had. 510 00:30:21,440 --> 00:30:22,550 It's got that jump. 511 00:30:22,550 --> 00:30:26,780 So it'll go like 1/k. 512 00:30:26,780 --> 00:30:29,660 OK, coming back. 513 00:30:29,660 --> 00:30:31,650 Could you actually find those numbers? 514 00:30:31,650 --> 00:30:34,750 And do you remember how to do integral like that? 515 00:30:34,750 --> 00:30:38,710 Well, look it up, I guess is the best answer. 516 00:30:38,710 --> 00:30:43,830 But whoever did it the first time, well, 517 00:30:43,830 --> 00:30:46,490 it's integration by parts, somehow. 518 00:30:46,490 --> 00:30:49,640 The derivative of this makes it real simple, 519 00:30:49,640 --> 00:30:52,800 and this we can integrate really easily, right? 520 00:30:52,800 --> 00:30:55,810 So we integrate that, take the derivative of that. 521 00:30:55,810 --> 00:31:02,280 We get a boundary term, so I don't exactly 522 00:31:02,280 --> 00:31:03,830 remember the formula. 523 00:31:03,830 --> 00:31:09,960 But it'll have a couple of terms, but not bad. 524 00:31:09,960 --> 00:31:13,230 And we'll see this 1/k. 525 00:31:13,230 --> 00:31:16,930 OK, and that's the formula. 526 00:31:16,930 --> 00:31:21,180 If I changed x to something else, let's see. 527 00:31:21,180 --> 00:31:24,760 As long as I'm looking at number one, what if I took e^x? 528 00:31:24,760 --> 00:31:25,660 Oh, easy. 529 00:31:25,660 --> 00:31:31,900 Right? e^x, yeah, let's do e^x, because that's an easy one. 530 00:31:31,900 --> 00:31:36,580 Now, e^x looks like, so what's my graph of e^x? 531 00:31:36,580 --> 00:31:42,490 It's quite small here at minus pi, and pretty large at e^pi. 532 00:31:42,490 --> 00:31:44,750 What's the rate of decay of the Fourier 533 00:31:44,750 --> 00:31:48,830 coefficients for that guy? 534 00:31:48,830 --> 00:31:50,200 Same. 535 00:31:50,200 --> 00:31:51,960 Got to jump again. 536 00:31:51,960 --> 00:31:52,930 Drops down here. 537 00:31:52,930 --> 00:31:55,580 So let's find it. 538 00:31:55,580 --> 00:31:58,340 Let's find those Fourier, that integral. 539 00:31:58,340 --> 00:32:02,040 That's e^(x-ik). 540 00:32:02,040 --> 00:32:06,520 Sorry, let's put down what I'm integrating here. 541 00:32:06,520 --> 00:32:13,560 I'm integrating e to the x times (1-ik). 542 00:32:13,560 --> 00:32:16,230 Right? 543 00:32:16,230 --> 00:32:18,970 That's what I've got to integrate. 544 00:32:18,970 --> 00:32:22,460 The integral is the same thing divided by 1-ik. 545 00:32:22,460 --> 00:32:24,980 546 00:32:24,980 --> 00:32:28,711 I plug in the endpoints, minus pi and pi, 547 00:32:28,711 --> 00:32:29,710 and I've got the answer. 548 00:32:29,710 --> 00:32:30,890 And I divide by 2pi. 549 00:32:30,890 --> 00:32:33,500 550 00:32:33,500 --> 00:32:34,000 Yeah. 551 00:32:34,000 --> 00:32:36,000 So that's a totally doable one. 552 00:32:36,000 --> 00:32:37,221 Let's plug it in. 553 00:32:37,221 --> 00:32:37,720 1/(2pi). 554 00:32:37,720 --> 00:32:40,470 555 00:32:40,470 --> 00:32:45,460 And then the denominator is this 1-ik, yeah, well there you 556 00:32:45,460 --> 00:32:47,630 see it already, right? 557 00:32:47,630 --> 00:32:51,860 You see already the 1/k in the denominator. 558 00:32:51,860 --> 00:32:54,760 That's giving us the slow decay rate. 559 00:32:54,760 --> 00:33:00,190 And now I'm plugging in equal-- equal-- 560 00:33:00,190 --> 00:33:03,880 e^(pi(1-ik)) minus e^(-pi(1-ik)). 561 00:33:03,880 --> 00:33:10,590 562 00:33:10,590 --> 00:33:21,910 So as k gets big, this is slow decay. 563 00:33:21,910 --> 00:33:24,370 Now, what's happening here as k gets large? 564 00:33:24,370 --> 00:33:27,890 Oh, k is multiplied by i there. 565 00:33:27,890 --> 00:33:32,830 So this e^(pi*ik) is just sitting, 566 00:33:32,830 --> 00:33:36,210 it may even be just one or something, or minus one, 567 00:33:36,210 --> 00:33:36,820 or whatever. 568 00:33:36,820 --> 00:33:37,900 Is it? 569 00:33:37,900 --> 00:33:41,290 Yeah. k is just an integer, this is k*pi*i, 570 00:33:41,290 --> 00:33:43,440 that's just minus one to something. 571 00:33:43,440 --> 00:33:47,280 So all that numerator is minus one 572 00:33:47,280 --> 00:33:52,460 to the to the k-th power or something. 573 00:33:52,460 --> 00:33:56,480 Times e^pi. 574 00:33:56,480 --> 00:34:05,010 So e^pi is there. e^(-ik) at-- e^(-i*pi*k) is just minus one. 575 00:34:05,010 --> 00:34:10,750 And this is maybe e to the minus pi times the same, e^(+i*pi*k). 576 00:34:10,750 --> 00:34:14,100 577 00:34:14,100 --> 00:34:16,160 I think that's minus one to the k. 578 00:34:16,160 --> 00:34:17,760 Well, wait a minute. 579 00:34:17,760 --> 00:34:20,020 Maybe they're not both, whatever. 580 00:34:20,020 --> 00:34:22,500 It's a number. 581 00:34:22,500 --> 00:34:24,550 It's a number. 582 00:34:24,550 --> 00:34:28,090 That's just of this size. 583 00:34:28,090 --> 00:34:31,580 There's the big number, there's the small one. 584 00:34:31,580 --> 00:34:37,440 And divided by that, that's the thing that shows 585 00:34:37,440 --> 00:34:39,920 is yes, there is this jump. 586 00:34:39,920 --> 00:34:41,330 Right? 587 00:34:41,330 --> 00:34:45,920 OK, that's a couple of sets of Fourier coefficients. 588 00:34:45,920 --> 00:34:48,280 You could ask yourself, because on the quiz 589 00:34:48,280 --> 00:34:50,360 there'll probably be one, and I'll 590 00:34:50,360 --> 00:34:54,490 try to pick a function that's interesting. 591 00:34:54,490 --> 00:35:00,640 And I mean, I don't plan to pick xe^x or anything. 592 00:35:00,640 --> 00:35:03,690 Yeah. 593 00:35:03,690 --> 00:35:05,120 OK, good. 594 00:35:05,120 --> 00:35:05,970 Yes, thanks. 595 00:35:05,970 --> 00:35:11,167 AUDIENCE: [INAUDIBLE] Fourier series 596 00:35:11,167 --> 00:35:13,250 has twice the energy as another, what's that mean? 597 00:35:13,250 --> 00:35:13,680 PROFESSOR STRANG: I don't know. 598 00:35:13,680 --> 00:35:14,670 I guess. 599 00:35:14,670 --> 00:35:17,630 Hm. 600 00:35:17,630 --> 00:35:21,050 Maybe we're talking about power, and things 601 00:35:21,050 --> 00:35:29,010 like if we were dealing with electronics, 602 00:35:29,010 --> 00:35:33,860 I guess I would interpret that energy in terms of power. 603 00:35:33,860 --> 00:35:37,860 So that's what I'd be seeing. 604 00:35:37,860 --> 00:35:40,660 I'm not thinking of a really good answer 605 00:35:40,660 --> 00:35:44,070 to say well, why is that energy equality, 606 00:35:44,070 --> 00:35:46,950 but it's really useful. 607 00:35:46,950 --> 00:35:49,700 You know, so much of signal processing, 608 00:35:49,700 --> 00:35:51,880 and we'll do a bit of signal processing, 609 00:35:51,880 --> 00:35:57,920 is simply based on that energy equality 610 00:35:57,920 --> 00:36:04,170 there, and the moving into frequency space. 611 00:36:04,170 --> 00:36:05,940 And convolution. 612 00:36:05,940 --> 00:36:09,320 Actually, they would call it filtering. 613 00:36:09,320 --> 00:36:13,780 So we'll call it filtering when we use convolution. 614 00:36:13,780 --> 00:36:15,640 But it's pure convolution. 615 00:36:15,640 --> 00:36:22,710 Pure linear algebra, for these special bases. 616 00:36:22,710 --> 00:36:24,350 So, OK. 617 00:36:24,350 --> 00:36:29,670 I could try to come up with a better answer, or more focused 618 00:36:29,670 --> 00:36:42,650 answer than just to say power or, to use an electric power 619 00:36:42,650 --> 00:36:45,300 word is just one step. 620 00:36:45,300 --> 00:36:47,901 Yeah, thanks. 621 00:36:47,901 --> 00:36:48,400 Yes. 622 00:36:48,400 --> 00:36:51,530 AUDIENCE: [INAUDIBLE] 623 00:36:51,530 --> 00:36:53,180 PROFESSOR STRANG: 4.3, eight or nine. 624 00:36:53,180 --> 00:37:00,330 Let's just have a look and see what they're about. 625 00:37:00,330 --> 00:37:02,220 OK, just some regular guys. 626 00:37:02,220 --> 00:37:07,910 Yeah so that, OK, let me look at, 627 00:37:07,910 --> 00:37:12,180 you want me to do 4.3 eight? 628 00:37:12,180 --> 00:37:19,597 AUDIENCE: [INAUDIBLE] 629 00:37:19,597 --> 00:37:20,680 PROFESSOR STRANG: Ah, yes. 630 00:37:20,680 --> 00:37:22,190 Good, good question. 631 00:37:22,190 --> 00:37:25,370 OK, so 4.3 eight gave a couple of vectors, 632 00:37:25,370 --> 00:37:28,080 a little bit like the ones I did this morning. 633 00:37:28,080 --> 00:37:32,490 I mean here the c in 4.3 eight has a couple of ones, 634 00:37:32,490 --> 00:37:35,560 and this morning I just had a [1, 0, 0, 0]. 635 00:37:35,560 --> 00:37:37,210 But no big deal. 636 00:37:37,210 --> 00:37:39,850 Now, yeah. 637 00:37:39,850 --> 00:37:44,950 So if two vectors are orthogonal, 638 00:37:44,950 --> 00:37:49,990 are their transforms orthogonal? 639 00:37:49,990 --> 00:37:51,810 I think, yes. 640 00:37:51,810 --> 00:37:52,940 Yes. 641 00:37:52,940 --> 00:37:56,950 The Fourier, so the, yeah. 642 00:37:56,950 --> 00:38:00,380 So maybe this is worth a moment, here. 643 00:38:00,380 --> 00:38:01,560 Yeah. 644 00:38:01,560 --> 00:38:06,370 Because what do we, let me write down 645 00:38:06,370 --> 00:38:09,890 the letters for my question, and then answer the question. 646 00:38:09,890 --> 00:38:18,960 OK, so suppose I have two vectors, c and d. 647 00:38:18,960 --> 00:38:22,550 And they're orthogonal. 648 00:38:22,550 --> 00:38:26,430 And then I want to ask about their, 649 00:38:26,430 --> 00:38:33,670 if I multiply by the Fourier matrices, are those orthogonal? 650 00:38:33,670 --> 00:38:36,840 That's sort of the question. 651 00:38:36,840 --> 00:38:39,380 So here the vectors in frequency space, 652 00:38:39,380 --> 00:38:41,120 here they are in physical space. 653 00:38:41,120 --> 00:38:43,550 I don't mind if you started in physical space 654 00:38:43,550 --> 00:38:47,680 and went to frequency with F inverse, same question. 655 00:38:47,680 --> 00:38:53,430 Does the Fourier matrix preserve angles? 656 00:38:53,430 --> 00:38:55,360 Does it preserve angles? 657 00:38:55,360 --> 00:39:01,580 Do matrices, and F wouldn't be the only one 658 00:39:01,580 --> 00:39:06,590 with this property, do-- They preserve length, right? 659 00:39:06,590 --> 00:39:09,980 If you preserve length, do you preserve angles too? 660 00:39:09,980 --> 00:39:13,520 Preserve length, you're just looking at one vector. 661 00:39:13,520 --> 00:39:18,650 We know that we preserve length, and how do we know that? 662 00:39:18,650 --> 00:39:20,460 Let's just remember that. 663 00:39:20,460 --> 00:39:22,140 So here's the length question. 664 00:39:22,140 --> 00:39:24,030 And this'll be the angle question. 665 00:39:24,030 --> 00:39:26,940 So this is the energy inequality, 666 00:39:26,940 --> 00:39:28,500 coming back to that key thing. 667 00:39:28,500 --> 00:39:37,080 This is c bar transpose c, and over here we have (Fc,Fc) 668 00:39:37,080 --> 00:39:39,030 and this is what I did this morning. 669 00:39:39,030 --> 00:39:46,330 So that's c bar transpose F bar transpose F c, right? 670 00:39:46,330 --> 00:39:51,010 That's, why the bar as well as the transpose? 671 00:39:51,010 --> 00:39:53,450 Because we're doing complex. 672 00:39:53,450 --> 00:39:59,600 OK, let me make a little more space. 673 00:39:59,600 --> 00:40:01,920 And what did we do this morning? 674 00:40:01,920 --> 00:40:07,700 We replaced that by, well I wish it were the identity. 675 00:40:07,700 --> 00:40:11,670 But it's a multiple of the identity, that's what matters. 676 00:40:11,670 --> 00:40:17,620 So this was N, this is the identity with an N, 677 00:40:17,620 --> 00:40:19,500 c bar transpose c. 678 00:40:19,500 --> 00:40:25,290 So the conclusion was that this thing is just N times 679 00:40:25,290 --> 00:40:27,500 this thing. 680 00:40:27,500 --> 00:40:31,040 Well, and times this one. 681 00:40:31,040 --> 00:40:34,460 I'm expecting oh, but over here we got a zero. 682 00:40:34,460 --> 00:40:36,780 So my N is going to wash out. 683 00:40:36,780 --> 00:40:39,970 OK, let's just do the same thing for angle. 684 00:40:39,970 --> 00:40:46,020 This is the key energy equality for length. 685 00:40:46,020 --> 00:40:51,770 And all I want to say is, this Fourier matrix, 686 00:40:51,770 --> 00:40:57,230 like other orthogonal matrices, is just rotating the space. 687 00:40:57,230 --> 00:41:00,130 It's sort of amazing to think you 688 00:41:00,130 --> 00:41:05,340 have one space, physical space, and you rotate it. 689 00:41:05,340 --> 00:41:07,300 I'll use that word. 690 00:41:07,300 --> 00:41:10,160 Because somehow that's what an orthogonal matrix does. 691 00:41:10,160 --> 00:41:13,210 Well it's complex N-dimensional space. 692 00:41:13,210 --> 00:41:18,860 Sorry, so it's not so easy to visualize rotations in C^N, 693 00:41:18,860 --> 00:41:21,020 but it's a rotation. 694 00:41:21,020 --> 00:41:24,350 Angles don't change, and let's just see why? 695 00:41:24,350 --> 00:41:26,100 The inner product of this with this 696 00:41:26,100 --> 00:41:29,920 is c bar transpose, F bar transpose, 697 00:41:29,920 --> 00:41:31,590 because I have to transpose that. 698 00:41:31,590 --> 00:41:36,960 Times Fd, what do I do now? 699 00:41:36,960 --> 00:41:43,790 This one was c bar transpose d, with zero. 700 00:41:43,790 --> 00:41:45,980 You see you have it? 701 00:41:45,980 --> 00:41:50,210 Again, this fact that this rotation and inverse, 702 00:41:50,210 --> 00:41:56,090 and the transpose is the identity, apart from an N. So 703 00:41:56,090 --> 00:41:59,300 again, inner products are multiplied 704 00:41:59,300 --> 00:42:02,750 by N. Not only the length squared, 705 00:42:02,750 --> 00:42:05,070 which is inner product with itself, 706 00:42:05,070 --> 00:42:08,480 but all inner products are just multiplied by N. 707 00:42:08,480 --> 00:42:11,500 So if they start zero they end up zero. 708 00:42:11,500 --> 00:42:14,340 Yeah. 709 00:42:14,340 --> 00:42:15,980 That was your question, right? 710 00:42:15,980 --> 00:42:17,540 Yep, right. 711 00:42:17,540 --> 00:42:19,860 So if I have a couple of vectors, 712 00:42:19,860 --> 00:42:22,100 as I guess that problem proposed, 713 00:42:22,100 --> 00:42:24,700 it happened to propose two inputs that 714 00:42:24,700 --> 00:42:27,290 happen to be orthogonal, then you 715 00:42:27,290 --> 00:42:30,460 should be able to see that the transforms are, too. 716 00:42:30,460 --> 00:42:31,300 Yeah. 717 00:42:31,300 --> 00:42:33,950 Yep. 718 00:42:33,950 --> 00:42:37,660 Can I ask you about question two, was that on the homework? 719 00:42:37,660 --> 00:42:41,470 Problem two in Section 4.3? 720 00:42:41,470 --> 00:42:43,950 Was it? 721 00:42:43,950 --> 00:42:49,880 All I want to say is that's a simple fact. 722 00:42:49,880 --> 00:42:56,200 Let me just write that fact down so we're looking at that fact. 723 00:42:56,200 --> 00:43:01,750 If I look at this F matrix, it's so simple 724 00:43:01,750 --> 00:43:05,790 but it leads to lots of good, it's just a key fact. 725 00:43:05,790 --> 00:43:12,690 But if I look at my F matrix, am I looking at rows here? 726 00:43:12,690 --> 00:43:14,351 Yeah, I happened to look at rows, 727 00:43:14,351 --> 00:43:16,100 columns would be the same, it's symmetric. 728 00:43:16,100 --> 00:43:23,520 So here's row one, here's row two, I'm sorry that's row zero. 729 00:43:23,520 --> 00:43:26,830 This is row one. 730 00:43:26,830 --> 00:43:29,850 And let me look at row N-1. 731 00:43:29,850 --> 00:43:38,700 This is w, w^(N-1), this was w^2, this is w^(2(N-1)). 732 00:43:38,700 --> 00:43:40,070 And so on. 733 00:43:40,070 --> 00:43:41,910 Everybody's got the idea of the-- 734 00:43:41,910 --> 00:43:48,110 And then all these in-between rows, two, three, et cetera. 735 00:43:48,110 --> 00:43:55,580 And the question asks, show that this and this 736 00:43:55,580 --> 00:43:57,640 are complex conjugates. 737 00:43:57,640 --> 00:44:03,590 That that row and that row are the same, except conjugates. 738 00:44:03,590 --> 00:44:08,430 So if we look at the row number one in F, 739 00:44:08,430 --> 00:44:14,040 we'll be looking at row number N-1 in F bar. 740 00:44:14,040 --> 00:44:16,920 Now, why is that row the conjugate of that row? 741 00:44:16,920 --> 00:44:19,320 Why is this the conjugate of this? 742 00:44:19,320 --> 00:44:23,940 Why are those two conjugates? 743 00:44:23,940 --> 00:44:29,690 So I'm asking you to explain to me why w bar is the, why 744 00:44:29,690 --> 00:44:40,180 is the conjugate of this, this? 745 00:44:40,180 --> 00:44:46,410 So it's just one more neat fact about these important, crucial, 746 00:44:46,410 --> 00:44:48,000 numbers w. 747 00:44:48,000 --> 00:44:50,460 And so how do I see this? 748 00:44:50,460 --> 00:44:54,650 Well, this is w-- This is one way to do it. 749 00:44:54,650 --> 00:44:57,990 This is w^N times w^(-1). 750 00:44:57,990 --> 00:45:00,590 And what's w^N? 751 00:45:00,590 --> 00:45:01,140 One. 752 00:45:01,140 --> 00:45:04,640 Everybody knows, w^N is one. 753 00:45:04,640 --> 00:45:07,650 So I have w, so I'm trying to show that. 754 00:45:07,650 --> 00:45:12,910 Well, we know that this is true. 755 00:45:12,910 --> 00:45:18,240 So we know that the conjugate of w, right? 756 00:45:18,240 --> 00:45:20,450 There's w. 757 00:45:20,450 --> 00:45:25,770 Here's its conjugate, and it's also the inverse. 758 00:45:25,770 --> 00:45:31,100 This w is some e^(i*theta), this is some e to the the i some 759 00:45:31,100 --> 00:45:32,470 angle. 760 00:45:32,470 --> 00:45:35,450 Then always, it's sitting on the unit circle. 761 00:45:35,450 --> 00:45:40,840 So its reciprocal is also on the unit circle. 762 00:45:40,840 --> 00:45:46,780 The reciprocal has e^(-i*theta), so it's just the conjugate. 763 00:45:46,780 --> 00:45:50,090 That's a great fact. 764 00:45:50,090 --> 00:45:54,810 That's a beautiful fact about all these w's. 765 00:45:54,810 --> 00:45:55,740 And their powers. 766 00:45:55,740 --> 00:45:59,020 Conjugate and inverse the same. 767 00:45:59,020 --> 00:46:01,450 Right. 768 00:46:01,450 --> 00:46:04,630 So that was the key to problem two. 769 00:46:04,630 --> 00:46:08,250 I don't think I asked you to go through all 770 00:46:08,250 --> 00:46:12,070 the steps of problem three, but just in case 771 00:46:12,070 --> 00:46:14,580 you didn't read problem three, let me tell you 772 00:46:14,580 --> 00:46:18,410 in one tiny space what happens. 773 00:46:18,410 --> 00:46:22,210 In problem three you discover that the fourth power of F 774 00:46:22,210 --> 00:46:25,670 is the identity. 775 00:46:25,670 --> 00:46:34,200 Except there is an N squared, because from two F's we got 776 00:46:34,200 --> 00:46:38,700 an N, so from four F's-- So that's another fantastic fact 777 00:46:38,700 --> 00:46:41,320 about the Fourier matrix. 778 00:46:41,320 --> 00:46:45,640 That its fourth power, it must be somehow, 779 00:46:45,640 --> 00:46:50,070 the Fourier matrix is rotating, yeah. 780 00:46:50,070 --> 00:46:53,080 Somehow, I don't know, how do you 781 00:46:53,080 --> 00:46:56,580 understand that the Fourier matrix is, 782 00:46:56,580 --> 00:46:59,870 its fourth power brings you back to the identity. 783 00:46:59,870 --> 00:47:03,050 Apart from, we just didn't normalize it. 784 00:47:03,050 --> 00:47:07,850 So that we wouldn't, to avoid that N squared, 785 00:47:07,850 --> 00:47:09,340 so we had to use it. 786 00:47:09,340 --> 00:47:10,590 Yeah. 787 00:47:10,590 --> 00:47:13,880 That's pretty amazing. 788 00:47:13,880 --> 00:47:15,110 Pretty amazing. 789 00:47:15,110 --> 00:47:18,020 So if I had normalized it right, if I 790 00:47:18,020 --> 00:47:22,900 took F over the square root of N, that's 791 00:47:22,900 --> 00:47:28,120 the exact normalization to give me a, 792 00:47:28,120 --> 00:47:32,780 to put the N's where they belong, you could say. 793 00:47:32,780 --> 00:47:35,460 Then the fourth power of that matrix is the identity. 794 00:47:35,460 --> 00:47:37,420 Yeah. 795 00:47:37,420 --> 00:47:43,230 New math and new applications keep coming for these things. 796 00:47:43,230 --> 00:47:45,750 I'll tell you, actually. 797 00:47:45,750 --> 00:47:48,950 You could tell me eigenvalues of this matrix. 798 00:47:48,950 --> 00:47:50,950 Yeah, this is a good question. 799 00:47:50,950 --> 00:47:53,700 What are the eigenvalues of a matrix whose fourth 800 00:47:53,700 --> 00:47:57,375 power is the identity? 801 00:47:57,375 --> 00:47:58,250 AUDIENCE: [INAUDIBLE] 802 00:47:58,250 --> 00:47:59,416 PROFESSOR STRANG: Yeah, one. 803 00:47:59,416 --> 00:48:01,660 What else could the eigenvalue be? 804 00:48:01,660 --> 00:48:05,200 If the fourth power of the matrix is I, 805 00:48:05,200 --> 00:48:07,810 what are the possible eigenvalues? 806 00:48:07,810 --> 00:48:16,430 So let me, so this matrix is, can I call it M for the-- Or 807 00:48:16,430 --> 00:48:30,180 maybe U for the matrix F-- So if U^4 is the identity, 808 00:48:30,180 --> 00:48:32,570 what are the eigenvalues? 809 00:48:32,570 --> 00:48:34,890 What could, well of course u could be the identity, 810 00:48:34,890 --> 00:48:36,040 but it's not. 811 00:48:36,040 --> 00:48:41,060 It's the Fourier matrix. 812 00:48:41,060 --> 00:48:46,330 So the eigenvalues could be one, what else could they be? 813 00:48:46,330 --> 00:48:48,410 Minus one is possible. 814 00:48:48,410 --> 00:48:55,000 Because minus the identity, that'd be fine. i, and minus i. 815 00:48:55,000 --> 00:48:57,660 Four possible eigenvalues. 816 00:48:57,660 --> 00:49:00,540 And when the matrix reaches, I think 817 00:49:00,540 --> 00:49:06,510 if you put the four by four Fourier-- I don't know. 818 00:49:06,510 --> 00:49:09,680 If you, say, put the four by four Fourier 819 00:49:09,680 --> 00:49:13,910 matrix into MATLAB, and see what you get for eigenvalues, 820 00:49:13,910 --> 00:49:16,820 I've forgotten whether you get one of each of these. 821 00:49:16,820 --> 00:49:20,790 Or whether somebody's repeated at the level four, 822 00:49:20,790 --> 00:49:23,030 but then go up to five or six you'll 823 00:49:23,030 --> 00:49:25,430 see these guys start showing up. 824 00:49:25,430 --> 00:49:28,070 Different multiplicity. 825 00:49:28,070 --> 00:49:28,900 Right? 826 00:49:28,900 --> 00:49:34,500 The 1,024 matrix has got these guys. 827 00:49:34,500 --> 00:49:37,610 Some number of times, adding up to a 1,024. 828 00:49:37,610 --> 00:49:38,760 Right, yeah. 829 00:49:38,760 --> 00:49:41,300 So it's quite neat. 830 00:49:41,300 --> 00:49:44,020 Now, here's a question, which I actually 831 00:49:44,020 --> 00:49:46,950 just learned a good answer to. 832 00:49:46,950 --> 00:49:49,970 What are the eigenvectors? 833 00:49:49,970 --> 00:49:51,520 What are the eigenvectors? 834 00:49:51,520 --> 00:49:55,270 You could give a sort of half-baked description. 835 00:49:55,270 --> 00:49:57,740 Because once you know the eigenvalues. 836 00:49:57,740 --> 00:50:01,400 But to really get a handle on the eigenvectors, that's has 837 00:50:01,400 --> 00:50:08,370 been a problem that was studied in IEEE transactions papers. 838 00:50:08,370 --> 00:50:12,670 But not really the right, not a nice specific description 839 00:50:12,670 --> 00:50:13,770 of the eigenvectors. 840 00:50:13,770 --> 00:50:18,980 And somebody was in my office, this fall, a guy, 841 00:50:18,980 --> 00:50:22,400 post-doc at Berkeley who's seen the right way 842 00:50:22,400 --> 00:50:26,620 to look at that problem and describe the eigenvectors 843 00:50:26,620 --> 00:50:28,010 of the Fourier matrix. 844 00:50:28,010 --> 00:50:31,850 So that's, like, amazing to me that such 845 00:50:31,850 --> 00:50:34,210 a fundamental question was waiting. 846 00:50:34,210 --> 00:50:40,020 And turned out to involve quite important, deep ideas. 847 00:50:40,020 --> 00:50:42,580 OK, ready for one more for today? 848 00:50:42,580 --> 00:50:45,720 Anything? 849 00:50:45,720 --> 00:50:48,340 Or not. 850 00:50:48,340 --> 00:50:52,750 OK, so I hope you're getting the good stuff now 851 00:50:52,750 --> 00:50:54,030 on the Fourier series. 852 00:50:54,030 --> 00:50:57,030 Fourier discrete transform. 853 00:50:57,030 --> 00:50:59,590 Please come on Friday, because Friday will 854 00:50:59,590 --> 00:51:02,300 be the big day for convolution. 855 00:51:02,300 --> 00:51:09,620 And that's the essential thing that we still have to do. 856 00:51:09,620 --> 00:51:11,670 OK, see you Friday.