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 --> 00:00:05,770 Your support will help MIT OpenCourseWare 5 00:00:05,770 --> 00:00:09,370 continue to offer high-quality educational resources for free. 6 00:00:09,370 --> 00:00:12,530 To make a donation or to view additional materials 7 00:00:12,530 --> 00:00:16,170 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16,170 --> 00:00:20,490 at ocw.mit.edu. 9 00:00:20,490 --> 00:00:24,200 PROFESSOR STRANG: OK, so I've got Quiz 2 to give back to 10 00:00:24,200 --> 00:00:26,040 with good scores. 11 00:00:26,040 --> 00:00:29,050 So it's Christmas early. 12 00:00:29,050 --> 00:00:31,800 Or, good work on that exam. 13 00:00:31,800 --> 00:00:37,510 That's fine and so we all know Fourier series is 14 00:00:37,510 --> 00:00:42,450 like a central topic on the final third of the course, 15 00:00:42,450 --> 00:00:45,470 and I'll just keep going on Fourier series. 16 00:00:45,470 --> 00:00:51,360 I'll give you homework on 4.1 and 4.2. 17 00:00:51,360 --> 00:00:56,180 So 4.1 is what we complete today, the Fourier series. 18 00:00:56,180 --> 00:01:00,910 4.2 is the discrete Fourier series. 19 00:01:00,910 --> 00:01:03,990 So those are two major, major topics 20 00:01:03,990 --> 00:01:08,190 for this part of the course. 21 00:01:08,190 --> 00:01:12,960 This is the periodic one and 4.2 will be the finite one, 22 00:01:12,960 --> 00:01:15,910 with the Fourier matrix showing up. 23 00:01:15,910 --> 00:01:18,830 OK, so can I pick out, I've made a list 24 00:01:18,830 --> 00:01:25,570 of topics last time that were important for 4.1, 25 00:01:25,570 --> 00:01:26,780 for Fourier series. 26 00:01:26,780 --> 00:01:33,830 And I think these are the remaining entries on the list. 27 00:01:33,830 --> 00:01:42,970 I did the Fourier series for the odd square wave, the minus one 28 00:01:42,970 --> 00:01:45,090 stepping up to plus one. 29 00:01:45,090 --> 00:01:50,760 And you remember that, well just let me put that example down 30 00:01:50,760 --> 00:01:51,560 here. 31 00:01:51,560 --> 00:01:54,450 That was the step function; easier 32 00:01:54,450 --> 00:01:57,270 if I draw it than if I try to write equations, 33 00:01:57,270 --> 00:02:01,480 so it was minus one up to plus one, 34 00:02:01,480 --> 00:02:05,180 ending at-- of course period 2pi. 35 00:02:05,180 --> 00:02:11,890 And I guess we called that S(x), for S signaling step function, 36 00:02:11,890 --> 00:02:14,400 S signaling square wave. 37 00:02:14,400 --> 00:02:20,010 And S the general a signal that the function is odd. 38 00:02:20,010 --> 00:02:24,100 And that means that it goes with sine functions. 39 00:02:24,100 --> 00:02:31,450 And I think the numbers that we found from the formula was 40 00:02:31,450 --> 00:02:40,620 sin(x)/1, sin(3x)/3, sin(5x)/5, it's just a great one 41 00:02:40,620 --> 00:02:45,680 to remember. 42 00:02:45,680 --> 00:02:53,120 And that's the example that has the important, 43 00:02:53,120 --> 00:02:56,840 and I don't know if I wrote down Gibbs' name. 44 00:02:56,840 --> 00:03:02,945 He was the great physicist at Yale, well, 45 00:03:02,945 --> 00:03:10,520 a hundred years ago or more, and did this key idea 46 00:03:10,520 --> 00:03:14,260 that appears all the time, that any time you have a step 47 00:03:14,260 --> 00:03:20,380 function then the Fourier series, it does its best, 48 00:03:20,380 --> 00:03:23,910 I if I take a thousand term, it'll do its best 49 00:03:23,910 --> 00:03:27,960 but it will overshoot by an amount that Gibbs found, 50 00:03:27,960 --> 00:03:30,070 and then it will get really close 51 00:03:30,070 --> 00:03:36,480 and then it will overshoot again and then, symmetrically. 52 00:03:36,480 --> 00:03:39,010 Or anti-symmetrically, I should say. 53 00:03:39,010 --> 00:03:43,030 So that's the Gibbs phenomenon of great importance. 54 00:03:43,030 --> 00:03:46,120 I write this one out because first it's 55 00:03:46,120 --> 00:03:49,140 an important one to remember. 56 00:03:49,140 --> 00:03:51,410 Second it'll give us a good example 57 00:03:51,410 --> 00:03:58,620 for this important equality, that the energy in the function 58 00:03:58,620 --> 00:04:00,970 is the energy in the coefficient. 59 00:04:00,970 --> 00:04:02,250 That'll be good. 60 00:04:02,250 --> 00:04:05,270 OK, actually maybe I should do that one first 61 00:04:05,270 --> 00:04:09,900 because the delta function has got infinite energy 62 00:04:09,900 --> 00:04:12,560 and we don't learn anything from this equation. 63 00:04:12,560 --> 00:04:17,570 So let me jump to the energy in the function 64 00:04:17,570 --> 00:04:20,900 and the energy in the coefficient. 65 00:04:20,900 --> 00:04:23,910 So what do I mean by energy? 66 00:04:23,910 --> 00:04:25,300 Well, it's quadratic. 67 00:04:25,300 --> 00:04:26,640 Right? 68 00:04:26,640 --> 00:04:29,980 It's the length squared here. 69 00:04:29,980 --> 00:04:32,120 It's the length squared of the function. 70 00:04:32,120 --> 00:04:37,500 So let me compute, maybe I'll do it on this board underneath 71 00:04:37,500 --> 00:04:40,140 and leave space for the delta function. 72 00:04:40,140 --> 00:04:44,540 The energy in x space is just the integral-- 73 00:04:44,540 --> 00:04:45,630 It's the length squared. 74 00:04:45,630 --> 00:04:52,090 The integral of S(x) squared. dx. 75 00:04:52,090 --> 00:04:59,900 It's just what you would expect. 76 00:04:59,900 --> 00:05:02,990 We have a function, not a vector, 77 00:05:02,990 --> 00:05:07,180 so we can't sum coefficients squared. 78 00:05:07,180 --> 00:05:10,740 Instead we integrate all the values squared. 79 00:05:10,740 --> 00:05:14,550 And of course, this is a number that we can quickly 80 00:05:14,550 --> 00:05:16,280 compute for that function. 81 00:05:16,280 --> 00:05:18,800 So what does it turn out to be? 82 00:05:18,800 --> 00:05:23,200 Well, what is S(x) squared for that function? 83 00:05:23,200 --> 00:05:24,580 One, obviously. 84 00:05:24,580 --> 00:05:26,480 The function is one here. 85 00:05:26,480 --> 00:05:31,970 I'm looking at the original S(x), not the series. 86 00:05:31,970 --> 00:05:34,410 The function is one there and minus one there. 87 00:05:34,410 --> 00:05:37,100 When I squared those, S(x) squared 88 00:05:37,100 --> 00:05:40,570 is one everywhere so I'm integrating one everywhere 89 00:05:40,570 --> 00:05:42,450 from minus pi to pi. 90 00:05:42,450 --> 00:05:45,840 So I get the answer 2pi. 91 00:05:45,840 --> 00:05:47,900 So that's a case where the energy 92 00:05:47,900 --> 00:05:51,100 in the physical space, the x space, 93 00:05:51,100 --> 00:05:53,260 was totally easy to compute. 94 00:05:53,260 --> 00:05:56,660 Now, what about energy, what is this equality? 95 00:05:56,660 --> 00:06:03,630 This really neat easy to remember equality? 96 00:06:03,630 --> 00:06:09,480 I'm just going to find it by taking this thing squared. 97 00:06:09,480 --> 00:06:13,050 What's the integral of the right-hand side? 98 00:06:13,050 --> 00:06:20,590 The two are equal, so suppose I just fire away? 99 00:06:20,590 --> 00:06:23,980 I integrate the square of that infinite series. 100 00:06:23,980 --> 00:06:28,050 You're going to say, well that's going to take a while. 101 00:06:28,050 --> 00:06:32,750 But what's going to be good? 102 00:06:32,750 --> 00:06:36,630 The key point, the first point in last time's lecture, 103 00:06:36,630 --> 00:06:40,180 the first point in every discussion of Fourier series, 104 00:06:40,180 --> 00:06:43,220 is orthogonality. 105 00:06:43,220 --> 00:06:47,530 Sines times other sines integrated are zero. 106 00:06:47,530 --> 00:06:49,790 So a whole lot of terms will go. 107 00:06:49,790 --> 00:06:52,090 So I take that thing, I square it. 108 00:06:52,090 --> 00:06:54,680 So let me let me do that one here. 109 00:06:54,680 --> 00:06:56,880 The interval from minus pi to pi. 110 00:06:56,880 --> 00:07:02,140 May I take out the (4/pi) squared? 111 00:07:02,140 --> 00:07:07,510 Just so it's not confusing. 112 00:07:07,510 --> 00:07:16,210 Now, this is the sin(x)/1, sin(3x)/3, sin(kx)/k, 113 00:07:16,210 --> 00:07:17,770 and so on. 114 00:07:17,770 --> 00:07:21,760 All squared. dx, and so what do I get? 115 00:07:21,760 --> 00:07:27,580 The (4/pi) squared. 116 00:07:27,580 --> 00:07:30,210 And now I've got a whole lot of terms. 117 00:07:30,210 --> 00:07:32,600 But the thing is, I can do this. 118 00:07:32,600 --> 00:07:35,660 Because when I square this, I'll have a lot of terms like 119 00:07:35,660 --> 00:07:40,870 sin(x)*sin(3x), and when I integrate those I get zero. 120 00:07:40,870 --> 00:07:43,290 So the only ones that I don't get zero 121 00:07:43,290 --> 00:07:46,540 are when sin(x) integrates against itself. 122 00:07:46,540 --> 00:07:49,070 And sin(3x) against itself. 123 00:07:49,070 --> 00:07:51,780 So when sin(x) integrates against itself, 124 00:07:51,780 --> 00:07:53,410 that's sine squared. 125 00:07:53,410 --> 00:07:56,655 Its integral is, you remember the integral 126 00:07:56,655 --> 00:08:03,260 of sine squared, which is, its average value is 1/2. 127 00:08:03,260 --> 00:08:06,750 We're over an interval-- I think I'm going to get pi, 128 00:08:06,750 --> 00:08:08,580 for sine squared. 129 00:08:08,580 --> 00:08:11,770 Because sine squared-- We could do that calculation separately. 130 00:08:11,770 --> 00:08:13,130 It's just a standard integral. 131 00:08:13,130 --> 00:08:16,600 The integral of sine squared is pi. 132 00:08:16,600 --> 00:08:19,580 Actually, yeah it just uses the fact 133 00:08:19,580 --> 00:08:25,460 that sine squared x is the same as whatever it is the same as. 134 00:08:25,460 --> 00:08:33,360 Is it 1-cos(2x) or something? 135 00:08:33,360 --> 00:08:35,220 Over two. 136 00:08:35,220 --> 00:08:37,230 Or plus, who cares? 137 00:08:37,230 --> 00:08:42,400 Because the integral of whichever, plus or minus, 138 00:08:42,400 --> 00:08:46,412 let me, well I suppose for history's sake 139 00:08:46,412 --> 00:08:47,370 we should get it right. 140 00:08:47,370 --> 00:08:49,190 Which is it? 141 00:08:49,190 --> 00:08:51,800 Is it a minus, so it looks OK now? 142 00:08:51,800 --> 00:08:55,690 OK, alright, if it's wrong I didn't say. 143 00:08:55,690 --> 00:08:57,490 OK, but I'm going to integrate. 144 00:08:57,490 --> 00:08:59,520 So the integral of the cosine is zero, 145 00:08:59,520 --> 00:09:04,240 and the integral of the 1/2 is the part I'm talking about. 146 00:09:04,240 --> 00:09:07,520 That 1/2 is there all the way from minus pi to pi. 147 00:09:07,520 --> 00:09:09,520 So I get a pi. 148 00:09:09,520 --> 00:09:11,300 From all these sines. 149 00:09:11,300 --> 00:09:14,410 And now, what are all the terms? 150 00:09:14,410 --> 00:09:21,170 Well, one over one squared, that just had a coefficient one, 151 00:09:21,170 --> 00:09:22,960 but what's the next guy? 152 00:09:22,960 --> 00:09:25,720 You remember I'm squaring it, I'm integrating. 153 00:09:25,720 --> 00:09:29,750 But I have a 1/3 squared. 154 00:09:29,750 --> 00:09:31,610 And 1/5 squared. 155 00:09:31,610 --> 00:09:34,940 And so on. 156 00:09:34,940 --> 00:09:37,110 And here's a great point. 157 00:09:37,110 --> 00:09:40,850 These two are equal. 158 00:09:40,850 --> 00:09:44,270 I've got the same function, expressed in x space, 159 00:09:44,270 --> 00:09:49,050 and here it's expressed in sine space, you could say. 160 00:09:49,050 --> 00:09:50,710 In harmonic space. 161 00:09:50,710 --> 00:09:52,210 OK, so that's equal. 162 00:09:52,210 --> 00:09:56,070 And that's going to be the fact in general. 163 00:09:56,070 --> 00:10:00,070 In general, that the integral of S(x) 164 00:10:00,070 --> 00:10:04,220 squared, so the general fact will be the integral of-- Well, 165 00:10:04,220 --> 00:10:05,590 I'll write it down below. 166 00:10:05,590 --> 00:10:08,540 But let's just see what we got for numbers here. 167 00:10:08,540 --> 00:10:11,530 So I had pi on both sides. 168 00:10:11,530 --> 00:10:14,990 And so if I lift that over there, I get something 169 00:10:14,990 --> 00:10:16,460 like-- what do I have? 170 00:10:16,460 --> 00:10:22,060 Pi squared over 16, maybe I have pi squared over eight. 171 00:10:22,060 --> 00:10:26,430 You just get a remarkable formula. 172 00:10:26,430 --> 00:10:28,820 Putting that up there would be pi squared over 16, 173 00:10:28,820 --> 00:10:30,550 and the two makes it an eight. 174 00:10:30,550 --> 00:10:36,640 And here I have the sum of 1/1 squared plus 1/3 squared 175 00:10:36,640 --> 00:10:39,700 plus 1/5 squared. 176 00:10:39,700 --> 00:10:43,830 So, that's an infinite sum that I would not 177 00:10:43,830 --> 00:10:50,460 have known how to do except it appears here. 178 00:10:50,460 --> 00:10:54,770 The sum of one over all those squares. 179 00:10:54,770 --> 00:10:57,600 If I picked another example, I could 180 00:10:57,600 --> 00:11:02,010 get the sum-- Oh, this was all the odd numbers squared. 181 00:11:02,010 --> 00:11:04,160 If I picked a different function, 182 00:11:04,160 --> 00:11:11,140 I could have got one that also had the sin(2x)/2 183 00:11:11,140 --> 00:11:12,800 and the sin(4x)/4. 184 00:11:12,800 --> 00:11:15,490 So this would have been the sum of all the squares. 185 00:11:15,490 --> 00:11:18,580 Do you happen to know what that comes out to be? 186 00:11:18,580 --> 00:11:21,070 I mean, here's a way to compute pi. 187 00:11:21,070 --> 00:11:24,320 We have a formula for pi. 188 00:11:24,320 --> 00:11:29,390 And we'd have another formula that involved all the sums. 189 00:11:29,390 --> 00:11:32,150 Maybe I have room for it up here. 190 00:11:32,150 --> 00:11:37,280 This would be the sum of 1/n squared, right? 191 00:11:37,280 --> 00:11:39,900 This here I have only the odd ones. 192 00:11:39,900 --> 00:11:41,620 And I get pi squared over eight. 193 00:11:41,620 --> 00:11:46,300 Do you happen to know what I get for all of them? 194 00:11:46,300 --> 00:11:48,810 So I'm also including 1/2 squared, 195 00:11:48,810 --> 00:11:51,160 there's a quarter also in here. 196 00:11:51,160 --> 00:11:52,960 And also a 16th. 197 00:11:52,960 --> 00:11:56,270 And also a 36th, in this one. 198 00:11:56,270 --> 00:12:01,080 And the answer happens to be pi squared over six. 199 00:12:01,080 --> 00:12:08,100 Pi squared over six. 200 00:12:08,100 --> 00:12:12,660 The important point about this energy equality 201 00:12:12,660 --> 00:12:19,010 is not being able to get a few very remarkable formulas 202 00:12:19,010 --> 00:12:21,800 for pi. 203 00:12:21,800 --> 00:12:26,050 There's another remarkable formula in the homework. 204 00:12:26,050 --> 00:12:27,480 This is a little famous. 205 00:12:27,480 --> 00:12:29,360 Do you know what this is? 206 00:12:29,360 --> 00:12:33,890 This is the famous Riemann zeta function. 207 00:12:33,890 --> 00:12:39,230 The sum of 1/n^x is the zeta function at x. 208 00:12:39,230 --> 00:12:41,840 Here's the zeta function at two. 209 00:12:41,840 --> 00:12:46,100 So if I could draws a zeta. 210 00:12:46,100 --> 00:12:49,700 Maybe? 211 00:12:49,700 --> 00:12:51,980 There's a Greek guy in this class who could 212 00:12:51,980 --> 00:12:54,470 do it properly, but anyway. 213 00:12:54,470 --> 00:12:55,650 I'll chicken out. 214 00:12:55,650 --> 00:13:02,530 Zeta of, at the value two, zeta(2). 215 00:13:02,530 --> 00:13:06,630 So we know zeta(2), we know zeta(4). 216 00:13:06,630 --> 00:13:10,495 I don't think we know zeta(3), I think 217 00:13:10,495 --> 00:13:18,000 it's not a special number like pi squared over six. 218 00:13:18,000 --> 00:13:23,440 So the zeta function, the sum of 1 over n to this thing 219 00:13:23,440 --> 00:13:28,350 is, actually that's the subject of a problem 220 00:13:28,350 --> 00:13:30,170 that Riemann did not solve. 221 00:13:30,170 --> 00:13:31,836 There's a problem Riemann did not solve, 222 00:13:31,836 --> 00:13:36,480 and nobody has succeeded to find it, to solve it since. 223 00:13:36,480 --> 00:13:40,830 There's a million-dollar prize for its solution. 224 00:13:40,830 --> 00:13:43,150 My neighbors think I should be working on this, 225 00:13:43,150 --> 00:13:45,760 but I know better. 226 00:13:45,760 --> 00:13:50,570 It's going to be solved one day, but it's pretty difficult. 227 00:13:50,570 --> 00:13:56,250 And that is to know where this zeta function, where it's zero. 228 00:13:56,250 --> 00:13:58,880 Of course, it isn't zero at two. 229 00:13:58,880 --> 00:14:01,260 Because it's pi squared over six. 230 00:14:01,260 --> 00:14:05,210 And actually the conjecture is that it's zero, 231 00:14:05,210 --> 00:14:10,140 all the zeroes are at points, complex numbers 232 00:14:10,140 --> 00:14:12,330 with real part 1/2. 233 00:14:12,330 --> 00:14:16,280 So they're on this famous line, the imaginary line 234 00:14:16,280 --> 00:14:18,480 with real part 1/2. 235 00:14:18,480 --> 00:14:25,250 And that's the most important problem in pure mathematics. 236 00:14:25,250 --> 00:14:26,130 So here we go. 237 00:14:26,130 --> 00:14:31,310 We got a formula for pi out of this energy identity. 238 00:14:31,310 --> 00:14:36,250 And I'll write it again, once I have the complex form. 239 00:14:36,250 --> 00:14:40,300 OK, but you see where it comes from. 240 00:14:40,300 --> 00:14:42,460 It just comes from orthogonality. 241 00:14:42,460 --> 00:14:44,710 The fact that we could integrate that square 242 00:14:44,710 --> 00:14:47,920 is what made it all work. 243 00:14:47,920 --> 00:14:53,650 OK, let's do the delta function. 244 00:14:53,650 --> 00:14:56,710 So that's an even function, the delta, right? 245 00:14:56,710 --> 00:14:58,760 Now I'm looking at the delta function. 246 00:14:58,760 --> 00:15:01,030 Minus pi to pi. 247 00:15:01,030 --> 00:15:05,360 It has the spike at zero and it's certainly 248 00:15:05,360 --> 00:15:07,990 even so we expect cosines. 249 00:15:07,990 --> 00:15:09,990 And what are the coefficients? 250 00:15:09,990 --> 00:15:15,580 So it's just an important one to know. 251 00:15:15,580 --> 00:15:17,760 Very important example. 252 00:15:17,760 --> 00:15:18,950 So what's a_0? 253 00:15:18,950 --> 00:15:22,900 In general, the coefficient a_0 in the Fourier series 254 00:15:22,900 --> 00:15:28,680 is the-- If I have a function, delta(x), S(x), 255 00:15:28,680 --> 00:15:34,460 whatever my function, the a_0 coefficient is the average. 256 00:15:34,460 --> 00:15:38,210 a for average, a_0 is the average value. 257 00:15:38,210 --> 00:15:46,000 So this is 1/(2pi) the integral from minus pi to pi 258 00:15:46,000 --> 00:15:49,920 of my function. 259 00:15:49,920 --> 00:15:51,400 Where did that come from? 260 00:15:51,400 --> 00:15:53,020 I just integrated. 261 00:15:53,020 --> 00:15:56,850 I just multiplied both sides by one. 262 00:15:56,850 --> 00:15:59,480 Or by 1/(2pi), and integrated. 263 00:15:59,480 --> 00:16:02,970 And those terms disappeared, and I was left with a_0, 264 00:16:02,970 --> 00:16:05,900 and what's the answer? 265 00:16:05,900 --> 00:16:08,730 Everybody knows that integral. 266 00:16:08,730 --> 00:16:11,560 The integral of the delta function is one, 267 00:16:11,560 --> 00:16:13,620 so I just get 1/(2pi). 268 00:16:13,620 --> 00:16:15,410 So 1/(2pi). 269 00:16:15,410 --> 00:16:17,600 OK, now ready for a_1. 270 00:16:17,600 --> 00:16:20,550 How much of cos(x) do I have? 271 00:16:20,550 --> 00:16:24,150 Can I just change this formula to give me a_1 272 00:16:24,150 --> 00:16:26,230 and you can tell me what it gives? 273 00:16:26,230 --> 00:16:28,150 Well let me do it here. 274 00:16:28,150 --> 00:16:29,540 Here's a_1. 275 00:16:29,540 --> 00:16:32,550 What's the formula for a_1? 276 00:16:32,550 --> 00:16:35,720 It's just like b_1, like the sine formulas. 277 00:16:35,720 --> 00:16:39,220 You have to remember you're only dividing by pi. 278 00:16:39,220 --> 00:16:43,010 Because that average value was 1/2, as we saw. 279 00:16:43,010 --> 00:16:46,780 And then you have the integral of whatever your function is. 280 00:16:46,780 --> 00:16:48,940 delta(x) in this case. 281 00:16:48,940 --> 00:16:52,440 Times the cos(1x). 282 00:16:52,440 --> 00:16:57,360 If we're looking for a_1 we've multiplied both sides by cos(x) 283 00:16:57,360 --> 00:16:58,460 and integrated. 284 00:16:58,460 --> 00:17:02,960 And what answer do we get? 285 00:17:02,960 --> 00:17:05,430 For a_1? 286 00:17:05,430 --> 00:17:11,520 What's the integral of delta(x) times cos(x)? 287 00:17:11,520 --> 00:17:14,110 dx, so I should put in a dx. 288 00:17:14,110 --> 00:17:15,340 And the answer is? 289 00:17:15,340 --> 00:17:16,860 One, also one. 290 00:17:16,860 --> 00:17:21,150 The delta function, this spike, picks out the value 291 00:17:21,150 --> 00:17:24,212 of this function at the spike. 292 00:17:24,212 --> 00:17:25,670 Because of course it doesn't matter 293 00:17:25,670 --> 00:17:27,620 what that function is away from the spike, 294 00:17:27,620 --> 00:17:31,357 because the other factor, delta, is zero. 295 00:17:31,357 --> 00:17:33,190 Everything is at that spike, all the action, 296 00:17:33,190 --> 00:17:35,690 and this happens to be one at the spike. 297 00:17:35,690 --> 00:17:39,080 So I get a 1/pi. 298 00:17:39,080 --> 00:17:43,010 And actually, the same formula for all these guys. 299 00:17:43,010 --> 00:17:48,280 This will be the same with cos(x) changed to cos(2x). 300 00:17:48,280 --> 00:17:51,550 And what will be the answer now? 301 00:17:51,550 --> 00:17:53,360 Again, one. 302 00:17:53,360 --> 00:17:54,030 Right? 303 00:17:54,030 --> 00:17:57,740 It's the value of this integral. 304 00:17:57,740 --> 00:17:59,900 The formula for a_2 would have come 305 00:17:59,900 --> 00:18:05,520 by multiplying both sides by cos(2x), integrating. 306 00:18:05,520 --> 00:18:08,530 And the integral of cos squared would give me the pi, 307 00:18:08,530 --> 00:18:10,960 and I'm dividing by the pi, and I just 308 00:18:10,960 --> 00:18:12,970 need that integral and it's easy. 309 00:18:12,970 --> 00:18:15,090 It's also 1/pi. 310 00:18:15,090 --> 00:18:16,880 So all these are 1/pi. 311 00:18:16,880 --> 00:18:19,150 Let me just put in the formula. 312 00:18:19,150 --> 00:18:31,880 1/(2pi), and all the rest are 1/pi. cos(3x). 313 00:18:31,880 --> 00:18:33,970 All the cosines are there. 314 00:18:33,970 --> 00:18:36,940 All in the same amount. 315 00:18:36,940 --> 00:18:43,060 And the constant term is slightly different. 316 00:18:43,060 --> 00:18:49,330 OK, that's the formal Fourier series for the delta function. 317 00:18:49,330 --> 00:18:55,130 Formal meaning you can use it to compute, 318 00:18:55,130 --> 00:18:59,320 of course some things will fail, like what's the energy? 319 00:18:59,320 --> 00:19:03,120 If I integrate, if I try to do energy 320 00:19:03,120 --> 00:19:07,370 in x space of the delta function, or energy in k space, 321 00:19:07,370 --> 00:19:10,180 what answer would I get? 322 00:19:10,180 --> 00:19:13,720 Right, the integral of delta squared, its energy, 323 00:19:13,720 --> 00:19:15,320 is infinite, right? 324 00:19:15,320 --> 00:19:18,430 The integral of delta, if I have delta times delta 325 00:19:18,430 --> 00:19:20,800 then I'm really in trouble, right? 326 00:19:20,800 --> 00:19:24,860 Because this delta says, if I can speak informally, 327 00:19:24,860 --> 00:19:28,060 that delta says take the value of this function 328 00:19:28,060 --> 00:19:31,230 at zero, but of course that's infinite, 329 00:19:31,230 --> 00:19:33,450 so that would be infinite. 330 00:19:33,450 --> 00:19:39,940 That would be the energy in x space. 331 00:19:39,940 --> 00:19:42,620 What about the energy in k space? 332 00:19:42,620 --> 00:19:47,530 Well, let's think, what's the energy in k space? 333 00:19:47,530 --> 00:19:50,590 I'm going to do the squares of the coefficients, 334 00:19:50,590 --> 00:19:54,600 you remember that's what I had down here? 335 00:19:54,600 --> 00:19:56,450 And I'll have it again in a moment. 336 00:19:56,450 --> 00:20:00,620 It's the sum of the squares of the coefficients, fixed up 337 00:20:00,620 --> 00:20:03,800 by factors of pi. 338 00:20:03,800 --> 00:20:07,730 And here, all the coefficients are constants. 339 00:20:07,730 --> 00:20:09,590 So that sum is infinite. 340 00:20:09,590 --> 00:20:12,890 Again, it's the sum of squares, of constant, 341 00:20:12,890 --> 00:20:16,040 constant, constant, constants, and that series 342 00:20:16,040 --> 00:20:17,910 doesn't converge, it blows up. 343 00:20:17,910 --> 00:20:20,300 So the energy is infinite. 344 00:20:20,300 --> 00:20:21,910 That's OK. 345 00:20:21,910 --> 00:20:25,010 The key is that formula. 346 00:20:25,010 --> 00:20:29,620 OK, there's the formula to remember. 347 00:20:29,620 --> 00:20:31,510 There's the formula. 348 00:20:31,510 --> 00:20:32,800 On forever. 349 00:20:32,800 --> 00:20:40,370 Every frequency is in here to the same amount. 350 00:20:40,370 --> 00:20:42,520 OK, good. 351 00:20:42,520 --> 00:20:44,920 That's the delta function example. 352 00:20:44,920 --> 00:20:51,100 OK, ready to go to complex? 353 00:20:51,100 --> 00:20:55,990 Complex is no big deal because we know, actually, 354 00:20:55,990 --> 00:21:00,375 you can tell me the complex series for the delta function. 355 00:21:00,375 --> 00:21:02,720 I'll write it right underneath. 356 00:21:02,720 --> 00:21:06,000 The complex series for the delta function, 357 00:21:06,000 --> 00:21:11,890 just turn these guys into e^(i*theta)-- 358 00:21:11,890 --> 00:21:18,510 e^(i*x)'s, and e^(i*2x)'s, and e^(-i*2x)'s. 359 00:21:18,510 --> 00:21:21,460 Just term by term, just to see it. 360 00:21:21,460 --> 00:21:24,820 To see it clearly for this great example. 361 00:21:24,820 --> 00:21:28,780 So what does cos(2x) look like in terms 362 00:21:28,780 --> 00:21:34,600 of complex exponentials? 363 00:21:34,600 --> 00:21:45,800 Everybody knows cos(x) is the same as e^(ix) and e^(-ix), 364 00:21:45,800 --> 00:21:47,080 right? 365 00:21:47,080 --> 00:21:51,480 Divided by two, because this is cos(x)+i*sin(x), 366 00:21:51,480 --> 00:21:56,820 this is cos(x)-i*sin(x), when I add them I get two cos(x)'s. 367 00:21:56,820 --> 00:21:58,610 So I must divide by two. 368 00:21:58,610 --> 00:22:02,440 Let me divide by two all the way. 369 00:22:02,440 --> 00:22:05,670 So I'm dividing this 1/pi by two. 370 00:22:05,670 --> 00:22:07,750 You'll see it's so much nicer. 371 00:22:07,750 --> 00:22:11,470 So 1/(2pi) times the one, that's our first guy. 372 00:22:11,470 --> 00:22:15,020 And now I've got the next guy. 373 00:22:15,020 --> 00:22:15,520 Right? 374 00:22:15,520 --> 00:22:18,650 Because I need to divide by the two to get the cosine, 375 00:22:18,650 --> 00:22:20,870 and there's my two. 376 00:22:20,870 --> 00:22:22,340 OK, what's the next? 377 00:22:22,340 --> 00:22:25,710 What do I have next? 378 00:22:25,710 --> 00:22:27,740 From this guy. 379 00:22:27,740 --> 00:22:30,790 1/pi, still there. 380 00:22:30,790 --> 00:22:34,900 What's cos(2x), if I want to write it in terms 381 00:22:34,900 --> 00:22:37,560 of complex exponentials? 382 00:22:37,560 --> 00:22:51,330 So this guy now, I'm ready for him, is e^(i*2x) and e^(-i*2x). 383 00:22:51,330 --> 00:22:54,180 384 00:22:54,180 --> 00:22:57,990 Divided by two, and there's my two. 385 00:22:57,990 --> 00:23:01,690 Do you see what's happening? 386 00:23:01,690 --> 00:23:03,350 There's an example to show you why 387 00:23:03,350 --> 00:23:05,960 the complex case is so nice. 388 00:23:05,960 --> 00:23:12,140 Here we had to remember a different number for a_0. 389 00:23:12,140 --> 00:23:17,090 Here it's just, so the next one will be e^(i*3x), 390 00:23:17,090 --> 00:23:26,460 and e^(-i*3x), so that the delta function in the complex Fourier 391 00:23:26,460 --> 00:23:30,270 series, all the terms have coefficients one divided 392 00:23:30,270 --> 00:23:31,510 by the 2pi. 393 00:23:31,510 --> 00:23:35,180 That's a great example. 394 00:23:35,180 --> 00:23:40,850 And of course we see again that the sum of squares, oh yeah. 395 00:23:40,850 --> 00:23:47,740 So let's do the complex formula, by which I 396 00:23:47,740 --> 00:23:53,850 mean I'm taking any function F, not necessarily even, not 397 00:23:53,850 --> 00:23:57,390 necessarily odd, not necessarily real. 398 00:23:57,390 --> 00:24:00,550 Any function can now be a complex function, 399 00:24:00,550 --> 00:24:03,870 because we're going to use complex things here. 400 00:24:03,870 --> 00:24:12,950 So I'll have all the complex exponentials. 401 00:24:12,950 --> 00:24:14,220 For integer k. 402 00:24:14,220 --> 00:24:19,670 This k is an integer but it can go from minus infinity 403 00:24:19,670 --> 00:24:21,960 to infinity. 404 00:24:21,960 --> 00:24:23,820 That's the complex form. 405 00:24:23,820 --> 00:24:32,790 F(x) is a series again. 406 00:24:32,790 --> 00:24:37,030 The beauty is that every term looks the same. 407 00:24:37,030 --> 00:24:41,390 The thing you have to remember is that negative k is allowed, 408 00:24:41,390 --> 00:24:42,860 as well as positive k. 409 00:24:42,860 --> 00:24:46,010 You see that we needed the negative k 410 00:24:46,010 --> 00:24:52,950 to get cosines. k was minus 1 there, k was minus 2 there. 411 00:24:52,950 --> 00:24:55,540 And for sines we would also need them. 412 00:24:55,540 --> 00:24:59,850 So cosines go into it, sines go into it. 413 00:24:59,850 --> 00:25:01,870 F(x) could be complex. 414 00:25:01,870 --> 00:25:03,800 That's the complex series. 415 00:25:03,800 --> 00:25:07,520 So maybe we could have started with that series. 416 00:25:07,520 --> 00:25:12,030 But we didn't, we came to it here. 417 00:25:12,030 --> 00:25:15,400 But what's the formula for its coefficient? 418 00:25:15,400 --> 00:25:19,460 OK, actually, so the next half-hour now, 419 00:25:19,460 --> 00:25:21,830 we have to think complex. 420 00:25:21,830 --> 00:25:24,380 And that will bring a few changes. 421 00:25:24,380 --> 00:25:25,760 So watch for the changes. 422 00:25:25,760 --> 00:25:32,990 You see we're almost in the same ballpark, 423 00:25:32,990 --> 00:25:35,410 but there are a couple of things to notice. 424 00:25:35,410 --> 00:25:37,530 So let me write down that series again. 425 00:25:37,530 --> 00:25:43,310 Minus infinity to infinity of some coefficient e^(ikx). 426 00:25:43,310 --> 00:25:46,740 Now, what is c_k? 427 00:25:46,740 --> 00:25:54,800 What is the formula for c_k? 428 00:25:54,800 --> 00:25:56,830 As soon as we answer that question, 429 00:25:56,830 --> 00:26:02,690 you'll see the new aspect for complex. 430 00:26:02,690 --> 00:26:05,290 How do I find coefficients? 431 00:26:05,290 --> 00:26:09,320 I multiply by something, I integrate, 432 00:26:09,320 --> 00:26:11,970 and I use orthogonality. 433 00:26:11,970 --> 00:26:14,850 Same idea, just repeat after repeat. 434 00:26:14,850 --> 00:26:18,200 The question is, what do I multiply by? 435 00:26:18,200 --> 00:26:22,360 If I wanted to know c_3, suppose I 436 00:26:22,360 --> 00:26:26,430 want to know a formula for c_3, the coefficient. 437 00:26:26,430 --> 00:26:29,470 What am I going to multiply by that's 438 00:26:29,470 --> 00:26:31,860 going to give me the orthogonality I need, 439 00:26:31,860 --> 00:26:34,540 that all the other integrals are going to disappear, 440 00:26:34,540 --> 00:26:35,890 that's the key? 441 00:26:35,890 --> 00:26:39,740 I want to multiply by something so that when I integrate, 442 00:26:39,740 --> 00:26:42,020 all the other integrals are going to disappear. 443 00:26:42,020 --> 00:26:43,780 And let's just do it. 444 00:26:43,780 --> 00:26:48,750 Here, suppose I have e^(i5x). 445 00:26:48,750 --> 00:26:56,260 And I'm looking for c_3. 446 00:26:56,260 --> 00:26:58,820 So I'll look at e^(i3x). 447 00:26:58,820 --> 00:27:04,530 448 00:27:04,530 --> 00:27:06,460 So watch. 449 00:27:06,460 --> 00:27:09,840 This is the small point we have to make. 450 00:27:09,840 --> 00:27:13,760 So I'm looking for e^(i3x). 451 00:27:13,760 --> 00:27:15,460 So I'm going to multiply by something, 452 00:27:15,460 --> 00:27:16,900 and I'm going to integrate. 453 00:27:16,900 --> 00:27:21,070 And what would be the good thing to multiply by? 454 00:27:21,070 --> 00:27:24,840 Well, you would say, if you were just a real person, 455 00:27:24,840 --> 00:27:29,070 you would say multiply by e^(i3x), integrate. 456 00:27:29,070 --> 00:27:31,540 And hope for getting zero. 457 00:27:31,540 --> 00:27:33,230 You won't get zero. 458 00:27:33,230 --> 00:27:38,000 If I multiply e^(i3x)-- I'm sorry, you might say-- 459 00:27:38,000 --> 00:27:39,690 What am I doing here? 460 00:27:39,690 --> 00:27:42,670 I'm trying to check orthogonality. 461 00:27:42,670 --> 00:27:46,110 Let me instead of three use kx. 462 00:27:46,110 --> 00:27:49,650 463 00:27:49,650 --> 00:27:52,160 So that's a typical complex one. 464 00:27:52,160 --> 00:27:54,340 It's any one of these guys. 465 00:27:54,340 --> 00:28:00,850 And I want to see what's orthogonality. 466 00:28:00,850 --> 00:28:02,650 That's what I'm asking. 467 00:28:02,650 --> 00:28:04,840 Everything hinged on orthogonality. 468 00:28:04,840 --> 00:28:07,440 We've got to have orthogonality here. 469 00:28:07,440 --> 00:28:10,160 But let me show you what it is. 470 00:28:10,160 --> 00:28:18,670 The thing you multiply by to get the c_3 is not e^(i3x), it is? 471 00:28:18,670 --> 00:28:21,020 e^(-i3x). 472 00:28:21,020 --> 00:28:23,120 You take the conjugate. 473 00:28:23,120 --> 00:28:28,720 You change i to minus i in the complex case. 474 00:28:28,720 --> 00:28:34,880 So if I take e^(ikx) times e^(-ilx), 475 00:28:34,880 --> 00:28:40,350 and notice that minus, I claim that I get zero. 476 00:28:40,350 --> 00:28:43,700 Except if k is l. 477 00:28:43,700 --> 00:28:47,720 And when k is l, I probably get 2pi. 478 00:28:47,720 --> 00:28:50,540 If k is l. 479 00:28:50,540 --> 00:28:54,020 I get zero if k is not l. 480 00:28:54,020 --> 00:28:57,120 That's the beautiful orthogonality. 481 00:28:57,120 --> 00:29:02,750 I'm not too worried about the 2pi, I'll figure that out. 482 00:29:02,750 --> 00:29:08,110 So what I'm saying is, when you're taking inner products, 483 00:29:08,110 --> 00:29:11,260 dot products, and you've got complex stuff, 484 00:29:11,260 --> 00:29:17,090 one of the factors takes a complex conjugate. 485 00:29:17,090 --> 00:29:18,350 Change i to minus i. 486 00:29:18,350 --> 00:29:21,240 Do you see that we've got a completely easy integral now? 487 00:29:21,240 --> 00:29:23,750 What is the integral of this guy? 488 00:29:23,750 --> 00:29:27,430 How do I see that I really get zero? 489 00:29:27,430 --> 00:29:31,010 So this is the first time I'm actually doing an integration 490 00:29:31,010 --> 00:29:34,720 and seeing orthogonality. 491 00:29:34,720 --> 00:29:38,030 Anybody likes integrating these things, 492 00:29:38,030 --> 00:29:43,470 because they're so simple to integrate. 493 00:29:43,470 --> 00:29:45,330 Before I plug in the limits, what's 494 00:29:45,330 --> 00:29:46,460 the integral of this guy? 495 00:29:46,460 --> 00:29:53,160 How do I rewrite that to integrate it easily? 496 00:29:53,160 --> 00:29:55,150 I put the two exponents together. 497 00:29:55,150 --> 00:30:03,700 This is the same as e^(i(k-l)x), right? 498 00:30:03,700 --> 00:30:06,020 The exponentials follow that rule. 499 00:30:06,020 --> 00:30:10,580 If I multiply exponentials, I combine the exponents. 500 00:30:10,580 --> 00:30:12,730 And now I'm ready to integrate. 501 00:30:12,730 --> 00:30:15,610 And so what is the integral of e^(i(k-l))x? 502 00:30:15,610 --> 00:30:18,510 503 00:30:18,510 --> 00:30:22,235 When I integrate e to the something x, 504 00:30:22,235 --> 00:30:29,580 I get that same thing again divided by the something. 505 00:30:29,580 --> 00:30:33,710 So now I've integrated, and now I just want to go from zero 506 00:30:33,710 --> 00:30:40,610 to 2pi, plug in the limits from x=0 to x=2pi. 507 00:30:40,610 --> 00:30:44,200 This is what the integral is asking me for. 508 00:30:44,200 --> 00:30:47,400 So I'm actually doing the integral here. 509 00:30:47,400 --> 00:30:51,600 I put these together into that, I integrated, 510 00:30:51,600 --> 00:30:53,880 which just brought this term down below, 511 00:30:53,880 --> 00:31:01,900 because the derivative will bring that term above. 512 00:31:01,900 --> 00:31:03,340 Oh, they weren't meant to change. 513 00:31:03,340 --> 00:31:07,830 Actually, it's wrong but right. 514 00:31:07,830 --> 00:31:10,360 To change those integration limits. 515 00:31:10,360 --> 00:31:13,360 I mean, any 2pi would work, but thank you. 516 00:31:13,360 --> 00:31:18,010 You're totally right, I should have done minus pi to pi. 517 00:31:18,010 --> 00:31:18,970 Why do we get zero? 518 00:31:18,970 --> 00:31:20,240 That's the whole point. 519 00:31:20,240 --> 00:31:22,710 Here we actually did the integral, 520 00:31:22,710 --> 00:31:28,730 and we can just plug in x=pi and x=-pi. 521 00:31:28,730 --> 00:31:31,430 Or we could plug in zero and 2pi, 522 00:31:31,430 --> 00:31:35,630 or I could plug in any guys that were 2pi apart, 523 00:31:35,630 --> 00:31:37,380 any period of 2pi. 524 00:31:37,380 --> 00:31:43,590 Why do we get zero? 525 00:31:43,590 --> 00:31:48,040 Do you have to do the plugging in part to see it? 526 00:31:48,040 --> 00:31:49,860 You can, certainly. 527 00:31:49,860 --> 00:31:54,550 But the point is, this function is periodic. 528 00:31:54,550 --> 00:31:56,920 That's a function that has period 2pi. 529 00:31:56,920 --> 00:32:01,000 So it has to be the same at the lower and the upper limit. 530 00:32:01,000 --> 00:32:02,620 That's what it's coming to. 531 00:32:02,620 --> 00:32:05,120 That's a periodic function. 532 00:32:05,120 --> 00:32:07,430 It's equal at these two limits. 533 00:32:07,430 --> 00:32:10,990 And therefore, when I do the subtraction 534 00:32:10,990 --> 00:32:13,830 I take it at the top limit, minus the answer 535 00:32:13,830 --> 00:32:15,880 at the bottom limit, it's the same at both. 536 00:32:15,880 --> 00:32:17,240 So I get zero. 537 00:32:17,240 --> 00:32:22,740 So there is the actual check of orthogonality. 538 00:32:22,740 --> 00:32:27,110 So the key point was, orthogonality, 539 00:32:27,110 --> 00:32:30,480 or inner product, for complex functions, one of them 540 00:32:30,480 --> 00:32:33,610 has to take the complex conjugate. 541 00:32:33,610 --> 00:32:37,080 Let me just do for vectors, too. 542 00:32:37,080 --> 00:32:41,560 Complex vectors. 543 00:32:41,560 --> 00:32:44,680 I may have mentioned it, let me take the vector [1, i] 544 00:32:44,680 --> 00:32:48,240 as an extreme example. 545 00:32:48,240 --> 00:32:54,060 And suppose I wanted to find the inner product with itself. 546 00:32:54,060 --> 00:32:56,170 Which will be the length squared. 547 00:32:56,170 --> 00:32:59,740 What's the inner product of that vector, that complex vector 548 00:32:59,740 --> 00:33:01,750 [1, i], with itself? 549 00:33:01,750 --> 00:33:04,410 Let me just raise it up so we see it. 550 00:33:04,410 --> 00:33:06,540 Focus on this. 551 00:33:06,540 --> 00:33:12,930 Usually the length squared would be one squared plus i squared. 552 00:33:12,930 --> 00:33:13,560 No good. 553 00:33:13,560 --> 00:33:16,260 Why no good? 554 00:33:16,260 --> 00:33:17,660 Because it's zero. 555 00:33:17,660 --> 00:33:20,370 One squared plus i squared is zero. 556 00:33:20,370 --> 00:33:26,690 We want absolute values squared. 557 00:33:26,690 --> 00:33:29,560 Absolute values, we need squares. 558 00:33:29,560 --> 00:33:31,370 We need positive numbers here. 559 00:33:31,370 --> 00:33:35,060 So the length of this squared, I would-- So 560 00:33:35,060 --> 00:33:41,090 let me call that vector v. I want to take not v transpose 561 00:33:41,090 --> 00:33:50,130 v, that's the thing that would give me zero, if I did [1, i]. 562 00:33:50,130 --> 00:33:53,480 I have to take the complex conjugate. 563 00:33:53,480 --> 00:33:57,140 One of the two factors, and it's just a matter of convention 564 00:33:57,140 --> 00:34:00,490 which one you do, this is the thing that gives me 565 00:34:00,490 --> 00:34:03,610 the length of v squared. 566 00:34:03,610 --> 00:34:05,630 And it's the same thing for functions. 567 00:34:05,630 --> 00:34:10,430 It's the integral of F(x)-- Do you want me to write it 568 00:34:10,430 --> 00:34:16,930 as, it's the integral of F(x) times its conjugate. 569 00:34:16,930 --> 00:34:21,620 That gives me the length of F squared. 570 00:34:21,620 --> 00:34:27,300 And of course I can rewrite that as the integral of F. 571 00:34:27,300 --> 00:34:33,160 We have this handy notation for F times its conjugate. 572 00:34:33,160 --> 00:34:40,050 It's like re^(i*theta) times re^(-i*theta), 573 00:34:40,050 --> 00:34:51,420 the product of re^(i*theta) times re^(-i*theta) is what? 574 00:34:51,420 --> 00:34:55,150 This is the complex number, this is its conjugate, 575 00:34:55,150 --> 00:34:58,820 when I multiply, you notice that i went to minus i 576 00:34:58,820 --> 00:35:06,960 when I drop below the real axis, I just change i to minus i. 577 00:35:06,960 --> 00:35:11,100 So those cancel and I get r squared. 578 00:35:11,100 --> 00:35:14,910 The length, the size of the number. 579 00:35:14,910 --> 00:35:17,810 It's just, if you haven't thought complex 580 00:35:17,810 --> 00:35:22,090 just make this change and you're ready to go. 581 00:35:22,090 --> 00:35:29,430 Yeah just v bar transpose v, F bar transpose F, if you like, 582 00:35:29,430 --> 00:35:33,390 or F squared. 583 00:35:33,390 --> 00:35:35,320 And when we make the correct change 584 00:35:35,320 --> 00:35:39,810 we still have the orthogonality. 585 00:35:39,810 --> 00:35:41,820 OK? 586 00:35:41,820 --> 00:35:46,870 So that's what orthogonality is, now what's the coefficient? 587 00:35:46,870 --> 00:35:49,990 So now, after all that speech, tell me what 588 00:35:49,990 --> 00:35:51,930 do I multiply both sides by? 589 00:35:51,930 --> 00:35:55,830 Let me start again and remember here. 590 00:35:55,830 --> 00:36:03,440 So my F(x) is going to be sum of c_k*e^(ikx). 591 00:36:03,440 --> 00:36:07,940 And if I want to get a coefficient, 592 00:36:07,940 --> 00:36:12,480 I'm looking for a formula for c_3. 593 00:36:12,480 --> 00:36:15,870 So how do I find c_3? 594 00:36:15,870 --> 00:36:17,070 Let me slow down a second. 595 00:36:17,070 --> 00:36:18,210 How to find c_3? 596 00:36:18,210 --> 00:36:22,210 597 00:36:22,210 --> 00:36:27,430 Multiply both sides by what? 598 00:36:27,430 --> 00:36:31,680 And integrate. 599 00:36:31,680 --> 00:36:33,620 Minus pi to pi. 600 00:36:33,620 --> 00:36:36,200 What goes there? 601 00:36:36,200 --> 00:36:41,530 If I want c_3, I should multiply both sides by, should I 602 00:36:41,530 --> 00:36:45,110 multiply both sides by e^(i3x)? 603 00:36:45,110 --> 00:36:47,630 Nope. 604 00:36:47,630 --> 00:36:49,360 e^(-i3x). 605 00:36:49,360 --> 00:36:54,310 Multiply both sides by e^(-i3x) and integrate. 606 00:36:54,310 --> 00:36:57,700 e^(-i3x) d x. 607 00:36:57,700 --> 00:37:00,880 And then on the right side I get what? 608 00:37:00,880 --> 00:37:07,460 I get c_3, because that'll be e^(-i3x) times e^(+i3x), 609 00:37:07,460 --> 00:37:14,310 I'll be integrating one so I get a 2pi, and all zeroes. 610 00:37:14,310 --> 00:37:17,140 From all the other terms. 611 00:37:17,140 --> 00:37:19,960 Orthogonality doing its job again. 612 00:37:19,960 --> 00:37:22,290 All these zeroes are by orthogonality. 613 00:37:22,290 --> 00:37:28,580 By exactly this integration that we did. 614 00:37:28,580 --> 00:37:35,960 If k is three and l is seven or k is seven and l is three, 615 00:37:35,960 --> 00:37:37,340 the integral gives zero. 616 00:37:37,340 --> 00:37:39,480 So what's the formula then? 617 00:37:39,480 --> 00:37:47,190 This is all gone, divide by 2pi and you've got it. 618 00:37:47,190 --> 00:37:48,360 There you are. 619 00:37:48,360 --> 00:37:51,200 That's the coefficient. 620 00:37:51,200 --> 00:37:53,760 OK. 621 00:37:53,760 --> 00:37:55,780 We have to move to complex numbers, 622 00:37:55,780 --> 00:38:01,760 and now in those 20 minutes-- The only change to make 623 00:38:01,760 --> 00:38:04,730 is conjugate one of them. 624 00:38:04,730 --> 00:38:11,180 One of the things when it can be complex. 625 00:38:11,180 --> 00:38:16,470 So that's the complex series. 626 00:38:16,470 --> 00:38:18,330 What's the energy inequality now? 627 00:38:18,330 --> 00:38:25,010 Let me do the energy equality in x space and in k space. 628 00:38:25,010 --> 00:38:30,940 If I take this, now I'd like-- So that's 629 00:38:30,940 --> 00:38:33,090 an equality of two functions. 630 00:38:33,090 --> 00:38:38,200 Now I'd like to get it, I want to do energy. 631 00:38:38,200 --> 00:38:39,530 This is fantastic. 632 00:38:39,530 --> 00:38:42,160 And extremely practical and useful. 633 00:38:42,160 --> 00:38:45,490 The fact that the energy is going to come out beautifully, 634 00:38:45,490 --> 00:38:45,990 too. 635 00:38:45,990 --> 00:38:48,280 So what am I going to do for energy? 636 00:38:48,280 --> 00:38:55,190 I'm going to integrate F(x) squared, dx. 637 00:38:55,190 --> 00:38:56,070 That's the energy. 638 00:38:56,070 --> 00:39:00,730 That's the energy in x space, in the function. 639 00:39:00,730 --> 00:39:03,190 What do I do now? 640 00:39:03,190 --> 00:39:11,310 I integrate this series. c_k*e^(ikx) squared. 641 00:39:11,310 --> 00:39:13,470 Oh no, whoa. 642 00:39:13,470 --> 00:39:18,220 If I put a square there, right, I'm fired. 643 00:39:18,220 --> 00:39:18,775 Right? 644 00:39:18,775 --> 00:39:19,650 What do I have to do? 645 00:39:19,650 --> 00:39:21,840 I will put a square there, but I have 646 00:39:21,840 --> 00:39:24,190 to straighten out something. 647 00:39:24,190 --> 00:39:26,000 What do I do? 648 00:39:26,000 --> 00:39:29,260 Those curvy lines, which just meant 649 00:39:29,260 --> 00:39:35,460 take the thing and square, should be changed to? 650 00:39:35,460 --> 00:39:36,480 Straight. 651 00:39:36,480 --> 00:39:38,420 Just a matter of getting straight. 652 00:39:38,420 --> 00:39:39,860 Straightening this out. 653 00:39:39,860 --> 00:39:41,700 OK, straight. 654 00:39:41,700 --> 00:39:42,750 There we are. 655 00:39:42,750 --> 00:39:44,120 OK. 656 00:39:44,120 --> 00:39:47,570 Now, that means the thing times its conjugate. 657 00:39:47,570 --> 00:39:49,380 That's the thing times its conjugate. 658 00:39:49,380 --> 00:39:51,520 So what do I get? 659 00:39:51,520 --> 00:39:54,540 All the terms disappear except the perfect, 660 00:39:54,540 --> 00:39:56,450 the ones that are squared. 661 00:39:56,450 --> 00:39:58,870 So what do I get here? 662 00:39:58,870 --> 00:40:01,900 And the ones that are squared I'm integrating one, 663 00:40:01,900 --> 00:40:04,050 so I get a 2pi. 664 00:40:04,050 --> 00:40:05,290 So there's a 2pi. 665 00:40:05,290 --> 00:40:08,910 This whole subject's full of these pi's and 2pi's, it's 666 00:40:08,910 --> 00:40:11,510 just part of the deal. 667 00:40:11,510 --> 00:40:15,230 Now, what's left? 668 00:40:15,230 --> 00:40:17,510 And now I'm looking only at the terms where 669 00:40:17,510 --> 00:40:22,560 I'm integrating something by its conjugate, 670 00:40:22,560 --> 00:40:24,320 by its own conjugate. 671 00:40:24,320 --> 00:40:28,040 And then I'm getting c_k times its own conjugate, 672 00:40:28,040 --> 00:40:30,380 so I'm getting all the terms. 673 00:40:30,380 --> 00:40:32,880 I'm getting no cross terms, just the terms 674 00:40:32,880 --> 00:40:36,620 that come from that times it's own conjugate. 675 00:40:36,620 --> 00:40:40,340 Which is c_k squared. 676 00:40:40,340 --> 00:40:46,470 That's the energy in the coefficients. 677 00:40:46,470 --> 00:40:50,400 That's the energy in k space, and of course that sum 678 00:40:50,400 --> 00:40:52,670 goes minus infinity to infinity. 679 00:40:52,670 --> 00:40:54,330 I've got them all. 680 00:40:54,330 --> 00:41:00,220 That's the energy in k space, here's the energy in x space. 681 00:41:00,220 --> 00:41:06,410 You can expect that this orthogonality 682 00:41:06,410 --> 00:41:11,760 is going to give you something nice for the energy. 683 00:41:11,760 --> 00:41:13,850 For the integral of the square. 684 00:41:13,850 --> 00:41:16,780 Alright, so you saw that. 685 00:41:16,780 --> 00:41:18,150 And let's see. 686 00:41:18,150 --> 00:41:21,000 So we saw it for, we actually got a good number 687 00:41:21,000 --> 00:41:23,880 for the square wave. 688 00:41:23,880 --> 00:41:28,280 We got infinity for the delta function. 689 00:41:28,280 --> 00:41:32,170 I've put on here as a last topic, 690 00:41:32,170 --> 00:41:36,730 to just give the word function. 691 00:41:36,730 --> 00:41:42,000 Part of what this course is doing is to speak the language, 692 00:41:42,000 --> 00:41:49,410 teach the language of applied math. 693 00:41:49,410 --> 00:41:52,960 So that when you see something, you see this, 694 00:41:52,960 --> 00:41:56,910 you recognize hey I've seen that word before. 695 00:41:56,910 --> 00:42:00,980 So I want to know, what's the right space of functions? 696 00:42:00,980 --> 00:42:07,530 This is my measure for the length squared of a function. 697 00:42:07,530 --> 00:42:11,580 My square wave is great. 698 00:42:11,580 --> 00:42:14,180 It's length squared was whatever it was, 699 00:42:14,180 --> 00:42:16,760 pi squared over something. 700 00:42:16,760 --> 00:42:22,060 And that's its length squared. 701 00:42:22,060 --> 00:42:26,580 The delta function gave infinity. 702 00:42:26,580 --> 00:42:30,470 So it's not going to be allowed into the function space. 703 00:42:30,470 --> 00:42:33,370 It's a vector of infinite length. 704 00:42:33,370 --> 00:42:36,640 Like the vector 1, 1, 1, 1, 1 forever. 705 00:42:36,640 --> 00:42:38,550 Too long. 706 00:42:38,550 --> 00:42:42,830 Pythagoras fails, because we sum the squares, we get infinite. 707 00:42:42,830 --> 00:42:49,350 So what functions, what vectors should we allow in our space? 708 00:42:49,350 --> 00:42:51,800 So we're going to have a space that's 709 00:42:51,800 --> 00:42:54,980 going to be infinite-dimensional because our coefficients, 710 00:42:54,980 --> 00:42:57,800 we've got infinitely many coefficients. 711 00:42:57,800 --> 00:43:01,070 Our functions have infinitely many values. 712 00:43:01,070 --> 00:43:03,930 So we've moved up from n-dimensional space 713 00:43:03,930 --> 00:43:05,910 to infinite-dimensional space. 714 00:43:05,910 --> 00:43:13,880 And everybody calls it after the guy who, Hilbert space. 715 00:43:13,880 --> 00:43:17,420 So I don't know if you've seen that word before, that name 716 00:43:17,420 --> 00:43:19,350 before, Hilbert space. 717 00:43:19,350 --> 00:43:24,110 It's the space of functions with finite energy. 718 00:43:24,110 --> 00:43:26,110 Finite length. 719 00:43:26,110 --> 00:43:31,780 So this function is in it, the delta function is not in it. 720 00:43:31,780 --> 00:43:37,270 And the point is that this space of functions, we've 721 00:43:37,270 --> 00:43:40,830 got-- These guys are a great basis 722 00:43:40,830 --> 00:43:42,540 for the space of functions. 723 00:43:42,540 --> 00:43:44,900 The sines and cosines are another basis 724 00:43:44,900 --> 00:43:46,410 for this space of functions. 725 00:43:46,410 --> 00:43:48,650 We just have a whole lot of functions, 726 00:43:48,650 --> 00:43:55,420 and all the facts of n-dimensional space. 727 00:43:55,420 --> 00:43:59,630 So what are important facts about n-dimensional space? 728 00:43:59,630 --> 00:44:07,090 One that comes to mind that involves length is the-- A 729 00:44:07,090 --> 00:44:16,580 key fact about length is, length and angle, 730 00:44:16,580 --> 00:44:19,510 I could say actually-- Many people 731 00:44:19,510 --> 00:44:22,660 would say this is the most important inequality 732 00:44:22,660 --> 00:44:23,940 in mathematics. 733 00:44:23,940 --> 00:44:36,310 That the dot product of two vectors, 734 00:44:36,310 --> 00:44:39,350 it's called the Schwarz inequality. 735 00:44:39,350 --> 00:44:43,990 Several people found it independently, 736 00:44:43,990 --> 00:44:50,560 Schwarz is the single name most often used. 737 00:44:50,560 --> 00:44:55,980 What do you know about the dot product of two vectors? 738 00:44:55,980 --> 00:44:58,740 Somehow it tells you the angle between them, right? 739 00:44:58,740 --> 00:45:03,800 Somehow the dot product of two vectors 740 00:45:03,800 --> 00:45:09,490 is, if I divide by the length of the vectors, 741 00:45:09,490 --> 00:45:13,140 so the dot product of vectors divided by the length, 742 00:45:13,140 --> 00:45:16,750 do you know what this is, in geometry? 743 00:45:16,750 --> 00:45:17,830 It's a cosine. 744 00:45:17,830 --> 00:45:21,650 It's the cosine of the angle between them. 745 00:45:21,650 --> 00:45:25,010 And cosines are never larger than one. 746 00:45:25,010 --> 00:45:28,400 So this quantity here is never larger than one; 747 00:45:28,400 --> 00:45:30,570 in other words, this is never larger 748 00:45:30,570 --> 00:45:33,620 than the length of one vector times the length 749 00:45:33,620 --> 00:45:35,890 of the other vector. 750 00:45:35,890 --> 00:45:37,140 I could do an example. 751 00:45:37,140 --> 00:45:42,250 Let v be [1, 3]. 752 00:45:42,250 --> 00:45:46,730 And let w be [1, 6]. 753 00:45:46,730 --> 00:45:49,070 I don't know how this is going to work. 754 00:45:49,070 --> 00:45:51,020 What's the dot product of those two vectors? 755 00:45:51,020 --> 00:45:53,090 Oh, it's 19. 756 00:45:53,090 --> 00:45:54,420 Sorry about that. 757 00:45:54,420 --> 00:46:02,720 Let's change this, I'd like a nice number here. 758 00:46:02,720 --> 00:46:06,890 What do you suggest? 759 00:46:06,890 --> 00:46:08,640 Make it five somewhere? 760 00:46:08,640 --> 00:46:12,670 Five wouldn't be bad. 761 00:46:12,670 --> 00:46:19,000 It wouldn't be too good either, but. 762 00:46:19,000 --> 00:46:20,050 Ah, OK. 763 00:46:20,050 --> 00:46:22,250 What's the dot product of those? 764 00:46:22,250 --> 00:46:23,490 16. 765 00:46:23,490 --> 00:46:25,400 And now what am I claiming, that that's 766 00:46:25,400 --> 00:46:29,560 length less than the length of this vector, which is what? 767 00:46:29,560 --> 00:46:32,620 What's the length of [1, 3]? 768 00:46:32,620 --> 00:46:33,440 Square root of ten. 769 00:46:33,440 --> 00:46:36,170 It's good to do these small ones, just to remember. 770 00:46:36,170 --> 00:46:37,910 The length of that vector is the sum 771 00:46:37,910 --> 00:46:39,830 of the square root of the sum of the squares. 772 00:46:39,830 --> 00:46:41,160 Square root of ten. 773 00:46:41,160 --> 00:46:44,030 And the length of this guy? 774 00:46:44,030 --> 00:46:48,450 Is the square root of 26. 775 00:46:48,450 --> 00:46:53,180 And so I hope, and Schwarz hopes, that 16 776 00:46:53,180 --> 00:46:54,760 is less than that square root. 777 00:46:54,760 --> 00:46:58,300 Can we check it? 778 00:46:58,300 --> 00:47:02,000 Let's square both sides, that would make it easier. 779 00:47:02,000 --> 00:47:07,220 So the right-hand side when I square both sides will be? 780 00:47:07,220 --> 00:47:08,900 260. 781 00:47:08,900 --> 00:47:14,270 When I square both sides, and what's the square of 16? 782 00:47:14,270 --> 00:47:16,350 256. 783 00:47:16,350 --> 00:47:18,470 That was close. 784 00:47:18,470 --> 00:47:24,510 But, it worked. 785 00:47:24,510 --> 00:47:27,660 I'll admit to you-- Oops not equal. 786 00:47:27,660 --> 00:47:28,930 Ah. 787 00:47:28,930 --> 00:47:32,930 Less or equal, Schwarz would say. 788 00:47:32,930 --> 00:47:37,090 And it's actually less than because these vectors are not 789 00:47:37,090 --> 00:47:38,260 in the same direction. 790 00:47:38,260 --> 00:47:40,350 If they were exactly in the same direction, 791 00:47:40,350 --> 00:47:43,620 or opposite directions, the cosine would be one 792 00:47:43,620 --> 00:47:45,220 and we would have equal. 793 00:47:45,220 --> 00:47:49,420 But since the angle, you see the angle between those two vectors 794 00:47:49,420 --> 00:47:53,350 is a pretty small angle. 795 00:47:53,350 --> 00:47:55,010 The cosine is quite near one. 796 00:47:55,010 --> 00:47:58,160 But it's not exactly one. 797 00:47:58,160 --> 00:48:00,690 So I'm glad somebody knew 16 squared. 798 00:48:00,690 --> 00:48:05,080 Does anybody know 99 squared? 799 00:48:05,080 --> 00:48:09,320 The reason I ask that is, or 999. 800 00:48:09,320 --> 00:48:12,780 I'll make it sound harder. 801 00:48:12,780 --> 00:48:16,200 The hope, when I was about 11 or something, 802 00:48:16,200 --> 00:48:17,630 I was I was always hoping somebody 803 00:48:17,630 --> 00:48:19,950 would ask me 999 squared. 804 00:48:19,950 --> 00:48:24,030 Because I was all ready with the answer. 805 00:48:24,030 --> 00:48:25,690 Nobody ever asked. 806 00:48:25,690 --> 00:48:26,330 Anyway. 807 00:48:26,330 --> 00:48:28,520 But you've asked, I think. 808 00:48:28,520 --> 00:48:31,350 So 998,001. 809 00:48:31,350 --> 00:48:36,140 And now I I've finally got a chance to show that I know it. 810 00:48:36,140 --> 00:48:39,480 OK, have a great weekend and see you. 811 00:48:39,480 --> 00:48:40,853