1 00:00:07,100 --> 00:00:09,070 GILBERT STRANG: OK, I thought I would talk 2 00:00:09,070 --> 00:00:13,270 today about power series. 3 00:00:13,270 --> 00:00:15,230 These are powers of x. 4 00:00:15,230 --> 00:00:17,050 I'm going to keep going. 5 00:00:17,050 --> 00:00:20,210 All powers, all those x to the fourth, x to the fifth, 6 00:00:20,210 --> 00:00:21,910 they'll all come in too. 7 00:00:21,910 --> 00:00:30,890 And my idea is combine them, add them up to get 8 00:00:30,890 --> 00:00:32,020 a function of x. 9 00:00:32,020 --> 00:00:38,320 So we're doing calculus, but a new part of it, with these 10 00:00:38,320 --> 00:00:39,980 infinite series. 11 00:00:39,980 --> 00:00:42,440 So what do I mean by combine them? 12 00:00:42,440 --> 00:00:46,780 I mean I'll multiply those powers by some numbers. 13 00:00:46,780 --> 00:00:50,000 Let me call those numbers a0. 14 00:00:50,000 --> 00:00:52,910 So this first guy will be an a0, and then I'll 15 00:00:52,910 --> 00:00:56,000 add on an a1 x. 16 00:00:56,000 --> 00:00:57,830 I'm out of x. 17 00:00:57,830 --> 00:01:06,540 I'll add on some a2 x squared, some a3 x cubed and onwards. 18 00:01:06,540 --> 00:01:09,320 So now I have a function. 19 00:01:09,320 --> 00:01:11,770 And that function, let me call it f of x. 20 00:01:14,860 --> 00:01:20,930 So here's my starting plan here. 21 00:01:20,930 --> 00:01:24,740 Well, we've seen this for e to the x. 22 00:01:24,740 --> 00:01:30,140 Let me remember how e to the x could come, the series for 23 00:01:30,140 --> 00:01:31,940 that particular function. 24 00:01:31,940 --> 00:01:33,730 So here's the plan. 25 00:01:33,730 --> 00:01:38,890 I'm going to choose those a's so as to match-- 26 00:01:38,890 --> 00:01:40,350 let me put these words down. 27 00:01:40,350 --> 00:01:44,110 I'll match at x equals 0. 28 00:01:48,640 --> 00:01:57,170 The function, its derivative, its next derivative, its third 29 00:01:57,170 --> 00:01:59,430 derivative, and onwards. 30 00:02:02,210 --> 00:02:08,430 Each a, like a3, will be chosen so that that right-hand 31 00:02:08,430 --> 00:02:13,010 side has the correct derivative, third derivative, 32 00:02:13,010 --> 00:02:14,400 at x equals 0. 33 00:02:14,400 --> 00:02:17,210 So this Taylor series-- 34 00:02:17,210 --> 00:02:23,410 Taylor's name is associated with series like this-- 35 00:02:23,410 --> 00:02:27,300 everything's happening at x equals 0. 36 00:02:27,300 --> 00:02:33,320 So in the case of e to the x, all its 37 00:02:33,320 --> 00:02:34,910 derivatives were the same. 38 00:02:34,910 --> 00:02:36,980 Still e to the x. 39 00:02:36,980 --> 00:02:41,230 And they all equal 1 at x equals 0. 40 00:02:41,230 --> 00:02:44,620 So I want that function to give me 1 for every 41 00:02:44,620 --> 00:02:46,330 derivative. 42 00:02:46,330 --> 00:02:51,320 That doesn't mean that the a's should all be 1. 43 00:02:51,320 --> 00:02:52,180 Why not? 44 00:02:52,180 --> 00:02:55,220 Because when I take the derivative, for example, of 45 00:02:55,220 --> 00:02:59,130 this guy, that x cubed, the first 46 00:02:59,130 --> 00:03:01,260 derivative, will be 3x squared. 47 00:03:01,260 --> 00:03:03,180 The next derivative 6x. 48 00:03:03,180 --> 00:03:04,620 The next derivative 6. 49 00:03:04,620 --> 00:03:06,260 That's the one I want. 50 00:03:06,260 --> 00:03:10,730 That third derivative, but it'll be 6 so a3 will have to 51 00:03:10,730 --> 00:03:14,630 be 1/6 to give me the correct answer 1. 52 00:03:14,630 --> 00:03:16,380 Let me write those things down. 53 00:03:19,790 --> 00:03:26,010 So what we just did is the derivative of x to the n-th. 54 00:03:26,010 --> 00:03:27,370 The n-th derivative. 55 00:03:30,120 --> 00:03:33,570 What's the n-th derivative of x to the n? 56 00:03:33,570 --> 00:03:37,490 We get to use our nice formula for derivatives. 57 00:03:37,490 --> 00:03:42,500 So the first derivative is nx to the n minus 1. 58 00:03:42,500 --> 00:03:47,500 The derivative of that will be n times n minus 1 x to one 59 00:03:47,500 --> 00:03:50,060 lower power. 60 00:03:50,060 --> 00:03:54,330 Keep going, do it n times, and what have you got? 61 00:03:54,330 --> 00:03:58,380 You finally got down to the 0-th power of x, a constant. 62 00:03:58,380 --> 00:04:00,150 But what is that constant? 63 00:04:00,150 --> 00:04:08,010 It's n times n minus 1, so that n-th derivative will be n 64 00:04:08,010 --> 00:04:11,760 from the first, n minus 1 from the second. 65 00:04:11,760 --> 00:04:15,860 Keep multiply until you finally get down to 1. 66 00:04:15,860 --> 00:04:20,300 And of course, that's called because it comes up often 67 00:04:20,300 --> 00:04:22,600 enough to have its own special name. 68 00:04:22,600 --> 00:04:26,270 That name is n factorial. 69 00:04:26,270 --> 00:04:30,750 And it's written n with an exclamation mark. 70 00:04:30,750 --> 00:04:34,550 So that's n factorial and that's the n-th derivative of 71 00:04:34,550 --> 00:04:36,310 x to the n. 72 00:04:36,310 --> 00:04:42,060 So for the particular function e to the x, if I worked out 73 00:04:42,060 --> 00:04:48,730 its series, all the derivatives I'm trying to get 74 00:04:48,730 --> 00:04:50,990 are all 1's. 75 00:04:50,990 --> 00:04:54,290 But what the powers of x gives me these n factorials. 76 00:04:57,000 --> 00:05:00,680 The a's had better divide by the n factorial. 77 00:05:00,680 --> 00:05:05,180 So let me recall the series for e to the x, and then go 78 00:05:05,180 --> 00:05:08,120 onto new functions. 79 00:05:08,120 --> 00:05:09,740 That's the point of my lecture. 80 00:05:09,740 --> 00:05:14,060 So we're getting e to the x in a slightly different way from 81 00:05:14,060 --> 00:05:18,170 the original way, but this is a good way. 82 00:05:18,170 --> 00:05:22,760 e to the x at x equals 0 is 1. 83 00:05:22,760 --> 00:05:28,700 At the first derivative of e to the x is 1, so I divide by 84 00:05:28,700 --> 00:05:29,780 1 factorial. 85 00:05:29,780 --> 00:05:31,080 That's 1. 86 00:05:31,080 --> 00:05:36,340 But here I have to divide by 2 because the second derivative 87 00:05:36,340 --> 00:05:39,120 is 2 and I want those to cancel. 88 00:05:39,120 --> 00:05:41,860 And here I divide by-- 89 00:05:41,860 --> 00:05:44,730 what do I divide by? 90 00:05:44,730 --> 00:05:48,990 6 because the third derivative is 6. 91 00:05:48,990 --> 00:05:54,600 And a typical term is I have to divide by n factorial 92 00:05:54,600 --> 00:05:59,130 because when I take n derivatives I get n factorial. 93 00:06:01,640 --> 00:06:03,900 The n-th derivative of that thing we just 94 00:06:03,900 --> 00:06:05,910 worked out is n factorial. 95 00:06:05,910 --> 00:06:09,610 So I divide by n factorial and I've got the derivative to 96 00:06:09,610 --> 00:06:11,730 come out 1. 97 00:06:11,730 --> 00:06:14,760 And that's correct for e to the x. 98 00:06:14,760 --> 00:06:19,700 So that's the plan, matching derivatives at x equals 0 by 99 00:06:19,700 --> 00:06:22,450 each power of x. 100 00:06:22,450 --> 00:06:25,710 And now I'm ready for a new function. 101 00:06:25,710 --> 00:06:29,890 And a nice choice is sine x. 102 00:06:29,890 --> 00:06:35,960 So now on this board, if I can come here, I'm going take a 103 00:06:35,960 --> 00:06:36,800 different function. 104 00:06:36,800 --> 00:06:38,230 No longer e to the x. 105 00:06:38,230 --> 00:06:40,450 My function is going to be sine x. 106 00:06:43,320 --> 00:06:47,021 Well, I better figure out all its derivatives. 107 00:06:47,021 --> 00:06:48,970 And they're nice, of course. 108 00:06:48,970 --> 00:06:50,550 Sine x, its derivative. 109 00:06:50,550 --> 00:06:51,970 Can I just list them all? 110 00:06:51,970 --> 00:06:55,060 These are the things that I have to match. 111 00:06:55,060 --> 00:06:56,690 I'll plug in x equals 0. 112 00:06:56,690 --> 00:06:59,160 But let me first find the derivatives. 113 00:06:59,160 --> 00:07:01,840 The derivative of sine is cosine. 114 00:07:01,840 --> 00:07:04,590 The derivative of cosine is minus the sine. 115 00:07:04,590 --> 00:07:08,600 The derivative of minus the sine is minus the cosine. 116 00:07:08,600 --> 00:07:15,550 And then I'm back to sine again, and repeating forever. 117 00:07:15,550 --> 00:07:18,910 That's a list of the derivatives of sine x. 118 00:07:18,910 --> 00:07:21,220 This is my f of x here. 119 00:07:23,870 --> 00:07:27,660 This guy, first one. 120 00:07:27,660 --> 00:07:31,130 OK, now I plug in x equals 0 because I want all the 121 00:07:31,130 --> 00:07:33,120 derivatives at 0. 122 00:07:33,120 --> 00:07:36,270 The whole series is being built focused on that 123 00:07:36,270 --> 00:07:37,750 point x equals 0. 124 00:07:37,750 --> 00:07:41,890 So at x equals 0, that's easy to plug in. 125 00:07:41,890 --> 00:07:46,930 The sine is 0, the cosine is 1, the minus the sine is 0. 126 00:07:46,930 --> 00:07:49,050 Minus the cosine is minus 1. 127 00:07:49,050 --> 00:07:50,260 The sine is 0. 128 00:07:50,260 --> 00:07:51,900 And repeat. 129 00:07:51,900 --> 00:07:57,740 0, 1, 0, minus 1 forever. 130 00:07:57,740 --> 00:08:01,720 OK, so I know the derivatives that I have to match. 131 00:08:01,720 --> 00:08:08,060 Now can I construct the power series that matches that? 132 00:08:08,060 --> 00:08:13,050 OK, so that power series will give me sine x, and 133 00:08:13,050 --> 00:08:15,070 what will it have? 134 00:08:15,070 --> 00:08:17,710 It starts with 0. 135 00:08:17,710 --> 00:08:22,210 The constant term is 0 because the sine of 0 when x is 0-- of 136 00:08:22,210 --> 00:08:24,600 course, we want to get the answer is 0. 137 00:08:24,600 --> 00:08:32,750 Then, the next term, the x, its coefficient is 1. 138 00:08:32,750 --> 00:08:34,000 1x. 139 00:08:35,770 --> 00:08:38,860 No x squared's in sine x. 140 00:08:38,860 --> 00:08:40,260 No x squared's. 141 00:08:40,260 --> 00:08:41,520 But now minus. 142 00:08:41,520 --> 00:08:43,950 Do I have minus 1x cubed? 143 00:08:43,950 --> 00:08:45,490 Not quite. 144 00:08:45,490 --> 00:08:53,460 Minus x cubed, but I have to divide by 6 because when I 145 00:08:53,460 --> 00:08:56,880 take that three derivatives, it will produce 6. 146 00:08:56,880 --> 00:09:01,520 So I have to divide by 6, which is 3 factorial. 147 00:09:01,520 --> 00:09:03,590 That's really the number that's there. 148 00:09:03,590 --> 00:09:05,360 3 times 2 times 1. 149 00:09:05,360 --> 00:09:06,900 6. 150 00:09:06,900 --> 00:09:10,740 Now the fourth degree term, the x to the 151 00:09:10,740 --> 00:09:13,950 fourth is not there. 152 00:09:13,950 --> 00:09:17,940 x to the fifth is going to come in with a plus. 153 00:09:17,940 --> 00:09:19,930 So there's a plus from this guy. 154 00:09:19,930 --> 00:09:26,040 This is x to the 0, 1, 2, 3, 4, 5. 155 00:09:26,040 --> 00:09:27,280 x to the fifth. 156 00:09:27,280 --> 00:09:31,230 And now what do I divide by now? 157 00:09:31,230 --> 00:09:32,480 5 factorial. 158 00:09:34,880 --> 00:09:36,270 120. 159 00:09:36,270 --> 00:09:38,090 And then minus and so on. 160 00:09:38,090 --> 00:09:42,210 Minus an x to the seventh over 7 factorial. 161 00:09:42,210 --> 00:09:50,530 We have created the power series around 0, focused on 0. 162 00:09:50,530 --> 00:09:55,580 And let me remove that 1 because just waste of space. 163 00:09:55,580 --> 00:09:57,160 x minus x cubed. 164 00:09:57,160 --> 00:09:59,960 All odd powers and that's because 165 00:09:59,960 --> 00:10:02,080 sine x is an odd function. 166 00:10:02,080 --> 00:10:07,600 If I change from x to minus x, everything will change sign. 167 00:10:11,080 --> 00:10:15,420 What would happen if I plugged in x equal pi? 168 00:10:15,420 --> 00:10:20,450 Suppose I took x equal pi in this formula for sine x. 169 00:10:20,450 --> 00:10:22,550 This infinite formula, keeps going forever. 170 00:10:26,250 --> 00:10:32,720 Well, I would get pi minus pi cubed over 6 plus pi to the 171 00:10:32,720 --> 00:10:34,040 fifth over 120. 172 00:10:34,040 --> 00:10:35,580 It would look ridiculous. 173 00:10:35,580 --> 00:10:38,870 But you and I know that the answer would have to come out. 174 00:10:38,870 --> 00:10:42,090 The correct sine of pi? 175 00:10:42,090 --> 00:10:44,200 0. 176 00:10:44,200 --> 00:10:46,980 I don't plan to do it, but it has to work. 177 00:10:46,980 --> 00:10:49,600 OK, so that's the sine. 178 00:10:49,600 --> 00:10:51,470 That's the sine. 179 00:10:51,470 --> 00:10:52,810 And it's an odd series. 180 00:10:52,810 --> 00:10:55,900 Now OK, good example. 181 00:10:55,900 --> 00:10:59,040 Its twin has got to show up here. 182 00:10:59,040 --> 00:11:00,820 The cosine. 183 00:11:00,820 --> 00:11:02,850 What's the series for the cosine? 184 00:11:02,850 --> 00:11:05,560 These are the two series that are worth knowing. 185 00:11:05,560 --> 00:11:09,780 You notice here that slope of 1, the big deal about the 186 00:11:09,780 --> 00:11:15,040 slope of sine x at x equals 0, the slope is 1. 187 00:11:15,040 --> 00:11:17,420 And that does have a slope of 1. 188 00:11:17,420 --> 00:11:18,990 OK, what about the cosine? 189 00:11:18,990 --> 00:11:21,500 Well, now I have to plug in. 190 00:11:21,500 --> 00:11:24,460 All right, the cosine is going to start here. 191 00:11:24,460 --> 00:11:27,460 Cosine minus sine minus cosine. 192 00:11:27,460 --> 00:11:31,720 Now my f of x is going to be cosine x. 193 00:11:37,620 --> 00:11:41,920 And I need its derivatives. 194 00:11:41,920 --> 00:11:45,120 I'm going to have three lines again that are going to look 195 00:11:45,120 --> 00:11:47,650 just like these three lines. 196 00:11:47,650 --> 00:11:49,490 But they'll be for the cosine. 197 00:11:49,490 --> 00:11:51,400 So they start with a cosine. 198 00:11:51,400 --> 00:11:54,690 Its derivative is minus the sine. 199 00:11:54,690 --> 00:11:58,200 Its derivative is minus the cosine. 200 00:11:58,200 --> 00:11:59,600 Its derivative is what? 201 00:11:59,600 --> 00:12:09,430 Plus sine and then cosine, and forever, minus the sine. 202 00:12:09,430 --> 00:12:11,980 And let me plug in now at x equals 0. 203 00:12:16,650 --> 00:12:18,530 This is our system. 204 00:12:18,530 --> 00:12:21,470 Find the derivatives, plug in 0. 205 00:12:21,470 --> 00:12:24,640 So find the derivative at 0. 206 00:12:24,640 --> 00:12:28,780 Well, the function itself, the 0-th derivative is 1. 207 00:12:28,780 --> 00:12:30,780 The first derivative is 0. 208 00:12:30,780 --> 00:12:33,280 The second derivative is minus 1. 209 00:12:33,280 --> 00:12:35,000 The third is 0. 210 00:12:35,000 --> 00:12:40,560 The fourth derivative is plus 1, 0, and so on. 211 00:12:40,560 --> 00:12:47,270 It's the same line as we have, but just starting over by 1. 212 00:12:47,270 --> 00:12:50,770 Starting with the cosine. 213 00:12:50,770 --> 00:12:54,570 I know what derivatives I want, now I just have to 214 00:12:54,570 --> 00:13:03,660 create my series for cosine x, which matches these numbers. 215 00:13:03,660 --> 00:13:05,430 One more time. 216 00:13:05,430 --> 00:13:09,750 Just match those numbers with the coefficients that I 217 00:13:09,750 --> 00:13:16,050 originally called a0, a1, a2, a3, but now we have numbers. 218 00:13:16,050 --> 00:13:18,400 OK, at x equals 0. 219 00:13:18,400 --> 00:13:20,580 So how does this series start? 220 00:13:20,580 --> 00:13:24,060 At x equals 0, the cosine of 0 is 1. 221 00:13:24,060 --> 00:13:25,200 It starts with a 1. 222 00:13:25,200 --> 00:13:27,200 That's the constant term sitting there. 223 00:13:30,830 --> 00:13:38,140 The coefficient of x, the linear term is 0. 224 00:13:38,140 --> 00:13:43,560 Because the cosine has 0 slope at the start. 225 00:13:43,560 --> 00:13:46,270 Then we come to something that shows up. 226 00:13:46,270 --> 00:13:46,910 Minus. 227 00:13:46,910 --> 00:13:48,750 This will be-- now what are we in to? 228 00:13:48,750 --> 00:13:52,270 This is the constant, the first power is gone. 229 00:13:52,270 --> 00:13:56,200 The second power minus x squared. 230 00:13:56,200 --> 00:14:00,980 But you know if I'm looking to match the second derivative to 231 00:14:00,980 --> 00:14:03,280 make it b minus 1. 232 00:14:03,280 --> 00:14:06,000 Right now it's minus 2. 233 00:14:06,000 --> 00:14:09,280 Differentiating would give me a 2x and a 2. 234 00:14:09,280 --> 00:14:14,830 So I have to divide by that 2 or 2 factorial. 235 00:14:14,830 --> 00:14:16,360 Now it's good. 236 00:14:16,360 --> 00:14:20,430 Now it matches the correct second derivative minus 1. 237 00:14:20,430 --> 00:14:22,150 Then there's no third derivative. 238 00:14:22,150 --> 00:14:26,470 The fourth derivative is plus 1 x to the 239 00:14:26,470 --> 00:14:32,250 fourth over 4 factorial. 240 00:14:32,250 --> 00:14:36,580 And then minus and so on, x to the sixth over 6 factorial. 241 00:14:36,580 --> 00:14:44,720 All even powers, so this is an even powers. 242 00:14:44,720 --> 00:14:48,830 The 0-th, second, fourth, sixth power. 243 00:14:48,830 --> 00:14:53,700 So it's an even function. 244 00:14:53,700 --> 00:15:00,010 That means that the cosine of minus x is exactly the same as 245 00:15:00,010 --> 00:15:01,260 the cosine of x. 246 00:15:04,750 --> 00:15:12,380 We get a nice little insight on these two special groups 247 00:15:12,380 --> 00:15:17,040 for which the sine is the perfect example of an odd 248 00:15:17,040 --> 00:15:20,030 function and the cosine is the perfect 249 00:15:20,030 --> 00:15:23,660 example of an even function. 250 00:15:23,660 --> 00:15:27,310 Well, there's so much here. 251 00:15:27,310 --> 00:15:33,630 What happens if I cut the series off? 252 00:15:33,630 --> 00:15:38,000 I just want to look at those first terms to see exactly 253 00:15:38,000 --> 00:15:39,260 what they represent. 254 00:15:39,260 --> 00:15:45,650 Suppose I stop here after the linear term. 255 00:15:45,650 --> 00:15:47,540 What do I have? 256 00:15:47,540 --> 00:15:52,250 What is that x just by itself? 257 00:15:52,250 --> 00:15:55,530 It's really 0 plus x because there was a 0 from the 258 00:15:55,530 --> 00:15:56,870 constant term. 259 00:15:56,870 --> 00:16:00,260 That is the linear approximation. 260 00:16:00,260 --> 00:16:03,940 That gives me the equation of the tangent line, y 261 00:16:03,940 --> 00:16:07,800 equals x, slope 1. 262 00:16:07,800 --> 00:16:11,050 More interesting, cut this one off. 263 00:16:11,050 --> 00:16:13,360 Cut this one off here. 264 00:16:13,360 --> 00:16:16,870 That's a very important estimate. 265 00:16:16,870 --> 00:16:19,810 It's not the exact cosine because the exact cosine has 266 00:16:19,810 --> 00:16:22,180 got all these later guys. 267 00:16:22,180 --> 00:16:25,290 But don't forget and I should have said this from the very 268 00:16:25,290 --> 00:16:34,270 beginning, these n factorials grow fast. And all the series 269 00:16:34,270 --> 00:16:38,930 that we're talking about, because those n factorials 270 00:16:38,930 --> 00:16:43,590 grow so fast and I'm dividing by them, I can take any x and 271 00:16:43,590 --> 00:16:46,520 I get a reasonable number. 272 00:16:46,520 --> 00:16:52,280 If I take x equaled pi, that's this sine series gave me 0. 273 00:16:52,280 --> 00:16:56,254 What do I get if I plug in x equals pi 274 00:16:56,254 --> 00:17:01,150 in the cosine series? 275 00:17:01,150 --> 00:17:05,020 So the cosine series, if I plugged in x equals pi and had 276 00:17:05,020 --> 00:17:09,290 patience to go pretty far, my numbers would be getting near 277 00:17:09,290 --> 00:17:11,920 the cosine of pi. 278 00:17:11,920 --> 00:17:14,230 Which would be minus 1. 279 00:17:14,230 --> 00:17:16,640 I don't see minus 1 coming out. 280 00:17:16,640 --> 00:17:20,109 Here is 1, minus 1/2 of pi squared. 281 00:17:20,109 --> 00:17:21,980 I don't know, that's around-- 282 00:17:21,980 --> 00:17:26,250 1/2 pi squared might be around 5 or something. 283 00:17:26,250 --> 00:17:28,319 But they knock each other off. 284 00:17:28,319 --> 00:17:32,390 They get very small and we get the answer minus 1. 285 00:17:32,390 --> 00:17:39,170 OK, so those are two important series and now I get to tell 286 00:17:39,170 --> 00:17:41,530 you about Euler's great formula. 287 00:17:46,730 --> 00:17:52,560 It connects these three series that you've seen. 288 00:17:52,560 --> 00:17:56,800 But to make that connection I have to bring in the 289 00:17:56,800 --> 00:18:00,280 imaginary number i. 290 00:18:00,280 --> 00:18:01,770 Is that OK? 291 00:18:01,770 --> 00:18:05,070 Just imagine a number i. 292 00:18:05,070 --> 00:18:09,270 And everybody knows what you're supposed to imagine. 293 00:18:09,270 --> 00:18:14,100 You're supposed to imagine that i squared is minus 1. 294 00:18:17,910 --> 00:18:22,880 And we all know there is no real number. 295 00:18:22,880 --> 00:18:26,370 The square of a real number is always going to be greater or 296 00:18:26,370 --> 00:18:27,590 equal to 0. 297 00:18:27,590 --> 00:18:35,400 So let's just create a symbol i with a rule, with the 298 00:18:35,400 --> 00:18:38,510 understanding that any time we see i squared, I'm entitled to 299 00:18:38,510 --> 00:18:41,750 write minus 1. 300 00:18:41,750 --> 00:18:41,860 OK. 301 00:18:41,860 --> 00:18:45,070 So now, what is Euler's great formula? 302 00:18:47,760 --> 00:18:54,590 Euler's great formula, his brilliant insight was make x 303 00:18:54,590 --> 00:19:00,120 in this e to the x series, make x imaginary. 304 00:19:00,120 --> 00:19:02,540 Change x to ix. 305 00:19:02,540 --> 00:19:05,410 So make it an imaginary number. 306 00:19:05,410 --> 00:19:11,870 So can I just take Euler's, take Taylor's series, or oh, 307 00:19:11,870 --> 00:19:14,460 maybe Euler's out of this too, because that 308 00:19:14,460 --> 00:19:16,770 letter e is his initial. 309 00:19:16,770 --> 00:19:18,330 Probably he did. 310 00:19:18,330 --> 00:19:23,460 So I guess that's why he found this lovely connection. 311 00:19:23,460 --> 00:19:30,680 So if I take e to the ix and instead of x in this series I 312 00:19:30,680 --> 00:19:34,770 put in ix, just go for it. 313 00:19:34,770 --> 00:19:36,750 Let x be imaginary. 314 00:19:36,750 --> 00:19:41,370 OK, can I write out the series 1 plus-- 315 00:19:41,370 --> 00:19:44,330 instead of x I have ix. 316 00:19:44,330 --> 00:19:49,610 And then I have 1 over 2 factorial ix squared. 317 00:19:49,610 --> 00:19:54,890 And then I have 1 over 3 factorial ix cubes. 318 00:19:54,890 --> 00:19:58,810 And 1 over 4 factorial ix to the fourth. 319 00:19:58,810 --> 00:20:00,350 That's e to the ix. 320 00:20:03,880 --> 00:20:07,100 OK, you say, you just changed x to ix. 321 00:20:07,100 --> 00:20:08,710 That's all I did. 322 00:20:08,710 --> 00:20:11,380 Now, here's the point. 323 00:20:11,380 --> 00:20:17,850 Now I'm going to look at this mess and I'm going to separate 324 00:20:17,850 --> 00:20:25,140 out the part that is real from the part that's imaginary. 325 00:20:25,140 --> 00:20:27,820 I'm going to separate it into its real part and its 326 00:20:27,820 --> 00:20:29,220 imaginary part. 327 00:20:29,220 --> 00:20:33,760 So what is real in this thing? 328 00:20:33,760 --> 00:20:37,550 I see one is certainly a real number. 329 00:20:37,550 --> 00:20:41,260 Do you see the other one, the next one that's real? 330 00:20:41,260 --> 00:20:44,170 It comes from this i squared. 331 00:20:44,170 --> 00:20:47,790 That i squared I can replace by minus 1, perfectly real. 332 00:20:47,790 --> 00:20:53,760 So it's minus from the i squared 1 over 2 factorial. 333 00:20:53,760 --> 00:20:55,850 x squared is still there. 334 00:20:55,850 --> 00:20:57,360 The i squared was minus 1. 335 00:20:57,360 --> 00:20:58,720 That's all. 336 00:20:58,720 --> 00:21:05,600 And then would come something from the i to the fourth. 337 00:21:05,600 --> 00:21:08,540 Because what is i to the fourth? 338 00:21:08,540 --> 00:21:12,090 It's i squared squared minus 1 squared. 339 00:21:12,090 --> 00:21:13,620 We'd be back to plus 1. 340 00:21:13,620 --> 00:21:15,470 So plus sign. 341 00:21:15,470 --> 00:21:16,100 Good. 342 00:21:16,100 --> 00:21:24,070 Now comes the part that has an i in it and a single i I have 343 00:21:24,070 --> 00:21:25,430 to live with. 344 00:21:25,430 --> 00:21:29,780 So that i is multiplied by x. 345 00:21:29,780 --> 00:21:32,250 Now I have i cubed. 346 00:21:32,250 --> 00:21:34,600 How do I deal with i cubed? 347 00:21:34,600 --> 00:21:40,070 i cubed is i squared minus 1 times i. 348 00:21:40,070 --> 00:21:42,490 i squared times i is minus i. 349 00:21:42,490 --> 00:21:47,950 So I have a minus i. 350 00:21:47,950 --> 00:21:52,370 1 over 3 factorial and the x cubed and so on. 351 00:21:56,800 --> 00:21:59,230 Do you see what we have? 352 00:21:59,230 --> 00:22:04,440 Do you see what this real part of e to the ix is? 353 00:22:04,440 --> 00:22:06,970 It's the cosine. 354 00:22:06,970 --> 00:22:09,080 Right there, same thing. 355 00:22:09,080 --> 00:22:17,040 So I'm getting cosine x for the real part and then i times 356 00:22:17,040 --> 00:22:19,360 this series. 357 00:22:19,360 --> 00:22:22,740 And you can see what that series is. 358 00:22:22,740 --> 00:22:30,030 It's the sine series, x minus 1/6 x cubed plus 1/20 of x to 359 00:22:30,030 --> 00:22:33,530 the fifth sine x. 360 00:22:33,530 --> 00:22:38,470 There is Euler's great formula that e to the ix-- 361 00:22:38,470 --> 00:22:42,020 oh, I better write it on a fresh board. 362 00:22:42,020 --> 00:22:43,645 Maybe I'll just write it over here. 363 00:22:46,550 --> 00:22:53,720 I'm going to copy from this board my Euler's great formula 364 00:22:53,720 --> 00:22:58,530 that e to the ix comes out to have a real part cos x. 365 00:22:58,530 --> 00:23:02,240 Imaginary part gives me the i sine x. 366 00:23:02,240 --> 00:23:03,420 And I'll write that down. 367 00:23:03,420 --> 00:23:05,610 Now let me work here. 368 00:23:05,610 --> 00:23:14,810 e to the ix is cos x plus i sine x. 369 00:23:14,810 --> 00:23:17,090 And I want to draw a picture. 370 00:23:17,090 --> 00:23:18,340 OK, here's a picture. 371 00:23:23,000 --> 00:23:28,110 Actually, Euler often wrote his formula, or we often write 372 00:23:28,110 --> 00:23:33,006 his formula because we're taking cosines and sines. 373 00:23:33,006 --> 00:23:35,900 Somehow x isn't such-- 374 00:23:35,900 --> 00:23:37,150 those are angles. 375 00:23:40,540 --> 00:23:40,840 So it's more natural to write-- 376 00:23:40,840 --> 00:23:44,160 Now that we've showing up with sines and cosines, it's more 377 00:23:44,160 --> 00:23:48,360 natural to write a symbol that we think of as 378 00:23:48,360 --> 00:23:50,570 an angle like theta. 379 00:23:50,570 --> 00:23:55,360 So you would more often see it this way. 380 00:23:55,360 --> 00:24:00,750 I'm just changing letters from x to theta as a way of 381 00:24:00,750 --> 00:24:03,000 remembering that it's an angle. 382 00:24:03,000 --> 00:24:04,880 And now I'll draw it. 383 00:24:04,880 --> 00:24:09,040 So I have to draw that thing. 384 00:24:09,040 --> 00:24:13,560 OK, this is the real direction and that's 385 00:24:13,560 --> 00:24:15,440 the imaginary direction. 386 00:24:18,790 --> 00:24:20,670 I just go that's the real part. 387 00:24:20,670 --> 00:24:23,110 I go cos theta across here. 388 00:24:23,110 --> 00:24:25,860 So let that be cos theta. 389 00:24:25,860 --> 00:24:29,590 And then I go upwards in the imaginary up or down. 390 00:24:29,590 --> 00:24:33,970 So across is the real part, up/down is the imaginary part. 391 00:24:33,970 --> 00:24:36,930 Say sine theta I go up. 392 00:24:36,930 --> 00:24:44,390 That height is sine theta and that angle is theta. 393 00:24:44,390 --> 00:24:46,490 Fantastic. 394 00:24:46,490 --> 00:24:50,320 That's a picture of Euler's formula. 395 00:24:50,320 --> 00:24:53,350 Well, that's the best way to see it is 396 00:24:53,350 --> 00:24:55,880 that beautiful statement. 397 00:24:55,880 --> 00:24:58,040 And this is a picture to remind us. 398 00:25:04,860 --> 00:25:08,890 We would say that's the complex plane because points 399 00:25:08,890 --> 00:25:12,950 have two parts, a real part and an imaginary part. 400 00:25:12,950 --> 00:25:15,930 Nothing so complex about that. 401 00:25:15,930 --> 00:25:22,480 Now, before I stop, we've done three important series. 402 00:25:22,480 --> 00:25:28,290 Can I mention two more, just two more out of a long list of 403 00:25:28,290 --> 00:25:29,610 possibilities? 404 00:25:29,610 --> 00:25:33,410 One is the most important series of all, where the 405 00:25:33,410 --> 00:25:35,720 coefficients are all 1's. 406 00:25:38,980 --> 00:25:41,500 So the coefficients are all 1's. 407 00:25:41,500 --> 00:25:43,720 That's called the geometric series. 408 00:25:43,720 --> 00:25:45,110 Let me write its name here. 409 00:25:51,590 --> 00:25:53,080 That's a Taylor series. 410 00:25:53,080 --> 00:25:55,190 That's a power series. 411 00:25:55,190 --> 00:25:59,710 And the function it comes from happens to be 1 412 00:25:59,710 --> 00:26:02,670 over 1 minus x. 413 00:26:02,670 --> 00:26:04,220 That's the function. 414 00:26:04,220 --> 00:26:08,110 And you will see why, if you multiply both sides by 1 minus 415 00:26:08,110 --> 00:26:10,970 x, I'll get 1 here. 416 00:26:10,970 --> 00:26:14,360 If you watch, everything will cancel except the 1. 417 00:26:14,360 --> 00:26:16,090 So that's it. 418 00:26:16,090 --> 00:26:19,630 Now, there's a significant difference between that series 419 00:26:19,630 --> 00:26:20,940 and e to the x. 420 00:26:20,940 --> 00:26:24,180 The biggest difference is we're not dividing by n 421 00:26:24,180 --> 00:26:27,310 factorial anymore. 422 00:26:27,310 --> 00:26:32,790 And as a result, these terms don't get necessarily smaller 423 00:26:32,790 --> 00:26:34,430 and smaller and smaller. 424 00:26:34,430 --> 00:26:36,410 Unless x is below 1. 425 00:26:36,410 --> 00:26:42,640 So we're OK for x below 1. 426 00:26:42,640 --> 00:26:44,530 And x could be negative. 427 00:26:44,530 --> 00:26:49,270 I can even say absolute value of x below 1, then these terms 428 00:26:49,270 --> 00:26:49,930 gets smaller. 429 00:26:49,930 --> 00:26:52,700 But at x equals 1 we're dead. 430 00:26:52,700 --> 00:26:56,030 At x equals 1 I have 1 plus 1 plus 1 plus 1. 431 00:26:56,030 --> 00:26:57,340 All 1's. 432 00:26:57,340 --> 00:26:59,510 I'm getting infinity. 433 00:26:59,510 --> 00:27:02,240 And on the left side I'm getting infinity also. 434 00:27:02,240 --> 00:27:05,590 At x equals 1 blows up. 435 00:27:05,590 --> 00:27:08,030 OK, one more series, then we're done. 436 00:27:08,030 --> 00:27:10,630 One more. 437 00:27:10,630 --> 00:27:15,080 It's a neat one because it brings in the logarithm. 438 00:27:15,080 --> 00:27:17,440 How am I going to get it? 439 00:27:17,440 --> 00:27:22,120 I'm going to start with this series, which is the big one, 440 00:27:22,120 --> 00:27:23,840 the geometric series. 441 00:27:23,840 --> 00:27:28,170 And I'm going to take the integral of every term. 442 00:27:28,170 --> 00:27:32,160 So if I integrate 1 I get x. 443 00:27:32,160 --> 00:27:35,460 If I integrate x I get x squared over 2. 444 00:27:35,460 --> 00:27:39,210 If I integrate x squared I get x cube over 3. 445 00:27:39,210 --> 00:27:41,530 x fourth over 4 and so on. 446 00:27:44,800 --> 00:27:47,690 Not 3 factorial, just 3. 447 00:27:47,690 --> 00:27:52,880 And if I integrate this, well, let me put the answer down and 448 00:27:52,880 --> 00:27:57,080 then we can take its derivative and say, yep, it 449 00:27:57,080 --> 00:27:58,310 does give that. 450 00:27:58,310 --> 00:28:02,260 So the answer is minus. 451 00:28:02,260 --> 00:28:06,020 This minus sign shows up here as a minus the 452 00:28:06,020 --> 00:28:09,010 logarithm of 1 minus x. 453 00:28:12,140 --> 00:28:17,540 Because if I take the derivative of that the 454 00:28:17,540 --> 00:28:21,190 logarithm always puts this inside function down to the 455 00:28:21,190 --> 00:28:24,110 bottom, and then the derivative of the inside 456 00:28:24,110 --> 00:28:28,350 function, the chain rule brings out a minus 1, and the 457 00:28:28,350 --> 00:28:31,720 minus 1's go away, and beautiful. 458 00:28:31,720 --> 00:28:37,230 So just have a look at that series then for the logarithm. 459 00:28:37,230 --> 00:28:39,170 The logarithm of 1 minus x. 460 00:28:39,170 --> 00:28:43,300 Again, we're matching at x equals 0. 461 00:28:43,300 --> 00:28:46,040 At x equals 0, this function is OK. 462 00:28:46,040 --> 00:28:48,750 In fact, at x equals 0, what is that function? 463 00:28:48,750 --> 00:28:52,670 Logarithm of 1, which is 0, and there's no constant term. 464 00:28:52,670 --> 00:28:54,360 Good. 465 00:28:54,360 --> 00:29:00,570 OK, what comments to make about this final example? 466 00:29:00,570 --> 00:29:04,490 This one was OK for x smaller than 1. 467 00:29:04,490 --> 00:29:07,750 But then it died at x equals 1. 468 00:29:07,750 --> 00:29:13,890 This one, well, it's getting a little help dividing 469 00:29:13,890 --> 00:29:16,120 by 3 and 4 and 5. 470 00:29:16,120 --> 00:29:19,670 But that's puny help. 471 00:29:19,670 --> 00:29:24,150 That's no way compared to dividing by 3 factorial, 4 472 00:29:24,150 --> 00:29:27,270 factorial, and so on, which will really help. 473 00:29:27,270 --> 00:29:32,730 So actually, this series is also only OK 474 00:29:32,730 --> 00:29:35,350 out to x equals 1. 475 00:29:35,350 --> 00:29:39,760 At x equals 1, it fails again. 476 00:29:39,760 --> 00:29:42,800 At x equals 1, what do I have? 477 00:29:42,800 --> 00:29:49,240 When x is 1, I have the log of 0 minus infinity. 478 00:29:49,240 --> 00:29:51,850 I've got infinity at x equals 1. 479 00:29:51,850 --> 00:29:55,020 At x equals 1, this is 1 plus 1/2 plus 1/3 480 00:29:55,020 --> 00:29:57,230 plus 1/4 plus 1/5. 481 00:29:57,230 --> 00:30:00,640 Getting smaller, but not very fast and 482 00:30:00,640 --> 00:30:03,060 adding up to infinity. 483 00:30:03,060 --> 00:30:05,430 So there's a whole discussion. 484 00:30:05,430 --> 00:30:12,170 We could spend hours on that famous series, 1 plus 1/2 plus 485 00:30:12,170 --> 00:30:16,150 1/3 plus a quarter and other series of numbers. 486 00:30:16,150 --> 00:30:21,960 I wanted to do calculus, derivatives, integrals, so I 487 00:30:21,960 --> 00:30:26,840 took functions and series of powers, not series of numbers 488 00:30:26,840 --> 00:30:28,830 to illustrate this. 489 00:30:28,830 --> 00:30:29,660 OK, good. 490 00:30:29,660 --> 00:30:31,520 Thanks. 491 00:30:31,520 --> 00:30:33,310 ANNOUNCER: This has been a production of MIT 492 00:30:33,310 --> 00:30:35,710 OpenCourseWare and Gilbert Strang. 493 00:30:35,710 --> 00:30:37,980 Funding for this video was provided by the Lord 494 00:30:37,980 --> 00:30:39,190 Foundation. 495 00:30:39,190 --> 00:30:42,330 To help OCW continue to provide free and open access 496 00:30:42,330 --> 00:30:45,400 to MIT courses, please make a donation at 497 00:30:45,400 --> 00:30:46,960 ocw.mit.edu/donate.