WEBVTT 00:00:07.100 --> 00:00:09.070 GILBERT STRANG: OK, I thought I would talk 00:00:09.070 --> 00:00:13.270 today about power series. 00:00:13.270 --> 00:00:15.230 These are powers of x. 00:00:15.230 --> 00:00:17.050 I'm going to keep going. 00:00:17.050 --> 00:00:20.210 All powers, all those x to the fourth, x to the fifth, 00:00:20.210 --> 00:00:21.910 they'll all come in too. 00:00:21.910 --> 00:00:30.890 And my idea is combine them, add them up to get 00:00:30.890 --> 00:00:32.020 a function of x. 00:00:32.020 --> 00:00:38.320 So we're doing calculus, but a new part of it, with these 00:00:38.320 --> 00:00:39.980 infinite series. 00:00:39.980 --> 00:00:42.440 So what do I mean by combine them? 00:00:42.440 --> 00:00:46.780 I mean I'll multiply those powers by some numbers. 00:00:46.780 --> 00:00:50.000 Let me call those numbers a0. 00:00:50.000 --> 00:00:52.910 So this first guy will be an a0, and then I'll 00:00:52.910 --> 00:00:56.000 add on an a1 x. 00:00:56.000 --> 00:00:57.830 I'm out of x. 00:00:57.830 --> 00:01:06.540 I'll add on some a2 x squared, some a3 x cubed and onwards. 00:01:06.540 --> 00:01:09.320 So now I have a function. 00:01:09.320 --> 00:01:11.770 And that function, let me call it f of x. 00:01:14.860 --> 00:01:20.930 So here's my starting plan here. 00:01:20.930 --> 00:01:24.740 Well, we've seen this for e to the x. 00:01:24.740 --> 00:01:30.140 Let me remember how e to the x could come, the series for 00:01:30.140 --> 00:01:31.940 that particular function. 00:01:31.940 --> 00:01:33.730 So here's the plan. 00:01:33.730 --> 00:01:38.890 I'm going to choose those a's so as to match-- 00:01:38.890 --> 00:01:40.350 let me put these words down. 00:01:40.350 --> 00:01:44.110 I'll match at x equals 0. 00:01:48.640 --> 00:01:57.170 The function, its derivative, its next derivative, its third 00:01:57.170 --> 00:01:59.430 derivative, and onwards. 00:02:02.210 --> 00:02:08.430 Each a, like a3, will be chosen so that that right-hand 00:02:08.430 --> 00:02:13.010 side has the correct derivative, third derivative, 00:02:13.010 --> 00:02:14.400 at x equals 0. 00:02:14.400 --> 00:02:17.210 So this Taylor series-- 00:02:17.210 --> 00:02:23.410 Taylor's name is associated with series like this-- 00:02:23.410 --> 00:02:27.300 everything's happening at x equals 0. 00:02:27.300 --> 00:02:33.320 So in the case of e to the x, all its 00:02:33.320 --> 00:02:34.910 derivatives were the same. 00:02:34.910 --> 00:02:36.980 Still e to the x. 00:02:36.980 --> 00:02:41.230 And they all equal 1 at x equals 0. 00:02:41.230 --> 00:02:44.620 So I want that function to give me 1 for every 00:02:44.620 --> 00:02:46.330 derivative. 00:02:46.330 --> 00:02:51.320 That doesn't mean that the a's should all be 1. 00:02:51.320 --> 00:02:52.180 Why not? 00:02:52.180 --> 00:02:55.220 Because when I take the derivative, for example, of 00:02:55.220 --> 00:02:59.130 this guy, that x cubed, the first 00:02:59.130 --> 00:03:01.260 derivative, will be 3x squared. 00:03:01.260 --> 00:03:03.180 The next derivative 6x. 00:03:03.180 --> 00:03:04.620 The next derivative 6. 00:03:04.620 --> 00:03:06.260 That's the one I want. 00:03:06.260 --> 00:03:10.730 That third derivative, but it'll be 6 so a3 will have to 00:03:10.730 --> 00:03:14.630 be 1/6 to give me the correct answer 1. 00:03:14.630 --> 00:03:16.380 Let me write those things down. 00:03:19.790 --> 00:03:26.010 So what we just did is the derivative of x to the n-th. 00:03:26.010 --> 00:03:27.370 The n-th derivative. 00:03:30.120 --> 00:03:33.570 What's the n-th derivative of x to the n? 00:03:33.570 --> 00:03:37.490 We get to use our nice formula for derivatives. 00:03:37.490 --> 00:03:42.500 So the first derivative is nx to the n minus 1. 00:03:42.500 --> 00:03:47.500 The derivative of that will be n times n minus 1 x to one 00:03:47.500 --> 00:03:50.060 lower power. 00:03:50.060 --> 00:03:54.330 Keep going, do it n times, and what have you got? 00:03:54.330 --> 00:03:58.380 You finally got down to the 0-th power of x, a constant. 00:03:58.380 --> 00:04:00.150 But what is that constant? 00:04:00.150 --> 00:04:08.010 It's n times n minus 1, so that n-th derivative will be n 00:04:08.010 --> 00:04:11.760 from the first, n minus 1 from the second. 00:04:11.760 --> 00:04:15.860 Keep multiply until you finally get down to 1. 00:04:15.860 --> 00:04:20.300 And of course, that's called because it comes up often 00:04:20.300 --> 00:04:22.600 enough to have its own special name. 00:04:22.600 --> 00:04:26.270 That name is n factorial. 00:04:26.270 --> 00:04:30.750 And it's written n with an exclamation mark. 00:04:30.750 --> 00:04:34.550 So that's n factorial and that's the n-th derivative of 00:04:34.550 --> 00:04:36.310 x to the n. 00:04:36.310 --> 00:04:42.060 So for the particular function e to the x, if I worked out 00:04:42.060 --> 00:04:48.730 its series, all the derivatives I'm trying to get 00:04:48.730 --> 00:04:50.990 are all 1's. 00:04:50.990 --> 00:04:54.290 But what the powers of x gives me these n factorials. 00:04:57.000 --> 00:05:00.680 The a's had better divide by the n factorial. 00:05:00.680 --> 00:05:05.180 So let me recall the series for e to the x, and then go 00:05:05.180 --> 00:05:08.120 onto new functions. 00:05:08.120 --> 00:05:09.740 That's the point of my lecture. 00:05:09.740 --> 00:05:14.060 So we're getting e to the x in a slightly different way from 00:05:14.060 --> 00:05:18.170 the original way, but this is a good way. 00:05:18.170 --> 00:05:22.760 e to the x at x equals 0 is 1. 00:05:22.760 --> 00:05:28.700 At the first derivative of e to the x is 1, so I divide by 00:05:28.700 --> 00:05:29.780 1 factorial. 00:05:29.780 --> 00:05:31.080 That's 1. 00:05:31.080 --> 00:05:36.340 But here I have to divide by 2 because the second derivative 00:05:36.340 --> 00:05:39.120 is 2 and I want those to cancel. 00:05:39.120 --> 00:05:41.860 And here I divide by-- 00:05:41.860 --> 00:05:44.730 what do I divide by? 00:05:44.730 --> 00:05:48.990 6 because the third derivative is 6. 00:05:48.990 --> 00:05:54.600 And a typical term is I have to divide by n factorial 00:05:54.600 --> 00:05:59.130 because when I take n derivatives I get n factorial. 00:06:01.640 --> 00:06:03.900 The n-th derivative of that thing we just 00:06:03.900 --> 00:06:05.910 worked out is n factorial. 00:06:05.910 --> 00:06:09.610 So I divide by n factorial and I've got the derivative to 00:06:09.610 --> 00:06:11.730 come out 1. 00:06:11.730 --> 00:06:14.760 And that's correct for e to the x. 00:06:14.760 --> 00:06:19.700 So that's the plan, matching derivatives at x equals 0 by 00:06:19.700 --> 00:06:22.450 each power of x. 00:06:22.450 --> 00:06:25.710 And now I'm ready for a new function. 00:06:25.710 --> 00:06:29.890 And a nice choice is sine x. 00:06:29.890 --> 00:06:35.960 So now on this board, if I can come here, I'm going take a 00:06:35.960 --> 00:06:36.800 different function. 00:06:36.800 --> 00:06:38.230 No longer e to the x. 00:06:38.230 --> 00:06:40.450 My function is going to be sine x. 00:06:43.320 --> 00:06:47.021 Well, I better figure out all its derivatives. 00:06:47.021 --> 00:06:48.970 And they're nice, of course. 00:06:48.970 --> 00:06:50.550 Sine x, its derivative. 00:06:50.550 --> 00:06:51.970 Can I just list them all? 00:06:51.970 --> 00:06:55.060 These are the things that I have to match. 00:06:55.060 --> 00:06:56.690 I'll plug in x equals 0. 00:06:56.690 --> 00:06:59.160 But let me first find the derivatives. 00:06:59.160 --> 00:07:01.840 The derivative of sine is cosine. 00:07:01.840 --> 00:07:04.590 The derivative of cosine is minus the sine. 00:07:04.590 --> 00:07:08.600 The derivative of minus the sine is minus the cosine. 00:07:08.600 --> 00:07:15.550 And then I'm back to sine again, and repeating forever. 00:07:15.550 --> 00:07:18.910 That's a list of the derivatives of sine x. 00:07:18.910 --> 00:07:21.220 This is my f of x here. 00:07:23.870 --> 00:07:27.660 This guy, first one. 00:07:27.660 --> 00:07:31.130 OK, now I plug in x equals 0 because I want all the 00:07:31.130 --> 00:07:33.120 derivatives at 0. 00:07:33.120 --> 00:07:36.270 The whole series is being built focused on that 00:07:36.270 --> 00:07:37.750 point x equals 0. 00:07:37.750 --> 00:07:41.890 So at x equals 0, that's easy to plug in. 00:07:41.890 --> 00:07:46.930 The sine is 0, the cosine is 1, the minus the sine is 0. 00:07:46.930 --> 00:07:49.050 Minus the cosine is minus 1. 00:07:49.050 --> 00:07:50.260 The sine is 0. 00:07:50.260 --> 00:07:51.900 And repeat. 00:07:51.900 --> 00:07:57.740 0, 1, 0, minus 1 forever. 00:07:57.740 --> 00:08:01.720 OK, so I know the derivatives that I have to match. 00:08:01.720 --> 00:08:08.060 Now can I construct the power series that matches that? 00:08:08.060 --> 00:08:13.050 OK, so that power series will give me sine x, and 00:08:13.050 --> 00:08:15.070 what will it have? 00:08:15.070 --> 00:08:17.710 It starts with 0. 00:08:17.710 --> 00:08:22.210 The constant term is 0 because the sine of 0 when x is 0-- of 00:08:22.210 --> 00:08:24.600 course, we want to get the answer is 0. 00:08:24.600 --> 00:08:32.750 Then, the next term, the x, its coefficient is 1. 00:08:32.750 --> 00:08:34.000 1x. 00:08:35.770 --> 00:08:38.860 No x squared's in sine x. 00:08:38.860 --> 00:08:40.260 No x squared's. 00:08:40.260 --> 00:08:41.520 But now minus. 00:08:41.520 --> 00:08:43.950 Do I have minus 1x cubed? 00:08:43.950 --> 00:08:45.490 Not quite. 00:08:45.490 --> 00:08:53.460 Minus x cubed, but I have to divide by 6 because when I 00:08:53.460 --> 00:08:56.880 take that three derivatives, it will produce 6. 00:08:56.880 --> 00:09:01.520 So I have to divide by 6, which is 3 factorial. 00:09:01.520 --> 00:09:03.590 That's really the number that's there. 00:09:03.590 --> 00:09:05.360 3 times 2 times 1. 00:09:05.360 --> 00:09:06.900 6. 00:09:06.900 --> 00:09:10.740 Now the fourth degree term, the x to the 00:09:10.740 --> 00:09:13.950 fourth is not there. 00:09:13.950 --> 00:09:17.940 x to the fifth is going to come in with a plus. 00:09:17.940 --> 00:09:19.930 So there's a plus from this guy. 00:09:19.930 --> 00:09:26.040 This is x to the 0, 1, 2, 3, 4, 5. 00:09:26.040 --> 00:09:27.280 x to the fifth. 00:09:27.280 --> 00:09:31.230 And now what do I divide by now? 00:09:31.230 --> 00:09:32.480 5 factorial. 00:09:34.880 --> 00:09:36.270 120. 00:09:36.270 --> 00:09:38.090 And then minus and so on. 00:09:38.090 --> 00:09:42.210 Minus an x to the seventh over 7 factorial. 00:09:42.210 --> 00:09:50.530 We have created the power series around 0, focused on 0. 00:09:50.530 --> 00:09:55.580 And let me remove that 1 because just waste of space. 00:09:55.580 --> 00:09:57.160 x minus x cubed. 00:09:57.160 --> 00:09:59.960 All odd powers and that's because 00:09:59.960 --> 00:10:02.080 sine x is an odd function. 00:10:02.080 --> 00:10:07.600 If I change from x to minus x, everything will change sign. 00:10:11.080 --> 00:10:15.420 What would happen if I plugged in x equal pi? 00:10:15.420 --> 00:10:20.450 Suppose I took x equal pi in this formula for sine x. 00:10:20.450 --> 00:10:22.550 This infinite formula, keeps going forever. 00:10:26.250 --> 00:10:32.720 Well, I would get pi minus pi cubed over 6 plus pi to the 00:10:32.720 --> 00:10:34.040 fifth over 120. 00:10:34.040 --> 00:10:35.580 It would look ridiculous. 00:10:35.580 --> 00:10:38.870 But you and I know that the answer would have to come out. 00:10:38.870 --> 00:10:42.090 The correct sine of pi? 00:10:42.090 --> 00:10:44.200 0. 00:10:44.200 --> 00:10:46.980 I don't plan to do it, but it has to work. 00:10:46.980 --> 00:10:49.600 OK, so that's the sine. 00:10:49.600 --> 00:10:51.470 That's the sine. 00:10:51.470 --> 00:10:52.810 And it's an odd series. 00:10:52.810 --> 00:10:55.900 Now OK, good example. 00:10:55.900 --> 00:10:59.040 Its twin has got to show up here. 00:10:59.040 --> 00:11:00.820 The cosine. 00:11:00.820 --> 00:11:02.850 What's the series for the cosine? 00:11:02.850 --> 00:11:05.560 These are the two series that are worth knowing. 00:11:05.560 --> 00:11:09.780 You notice here that slope of 1, the big deal about the 00:11:09.780 --> 00:11:15.040 slope of sine x at x equals 0, the slope is 1. 00:11:15.040 --> 00:11:17.420 And that does have a slope of 1. 00:11:17.420 --> 00:11:18.990 OK, what about the cosine? 00:11:18.990 --> 00:11:21.500 Well, now I have to plug in. 00:11:21.500 --> 00:11:24.460 All right, the cosine is going to start here. 00:11:24.460 --> 00:11:27.460 Cosine minus sine minus cosine. 00:11:27.460 --> 00:11:31.720 Now my f of x is going to be cosine x. 00:11:37.620 --> 00:11:41.920 And I need its derivatives. 00:11:41.920 --> 00:11:45.120 I'm going to have three lines again that are going to look 00:11:45.120 --> 00:11:47.650 just like these three lines. 00:11:47.650 --> 00:11:49.490 But they'll be for the cosine. 00:11:49.490 --> 00:11:51.400 So they start with a cosine. 00:11:51.400 --> 00:11:54.690 Its derivative is minus the sine. 00:11:54.690 --> 00:11:58.200 Its derivative is minus the cosine. 00:11:58.200 --> 00:11:59.600 Its derivative is what? 00:11:59.600 --> 00:12:09.430 Plus sine and then cosine, and forever, minus the sine. 00:12:09.430 --> 00:12:11.980 And let me plug in now at x equals 0. 00:12:16.650 --> 00:12:18.530 This is our system. 00:12:18.530 --> 00:12:21.470 Find the derivatives, plug in 0. 00:12:21.470 --> 00:12:24.640 So find the derivative at 0. 00:12:24.640 --> 00:12:28.780 Well, the function itself, the 0-th derivative is 1. 00:12:28.780 --> 00:12:30.780 The first derivative is 0. 00:12:30.780 --> 00:12:33.280 The second derivative is minus 1. 00:12:33.280 --> 00:12:35.000 The third is 0. 00:12:35.000 --> 00:12:40.560 The fourth derivative is plus 1, 0, and so on. 00:12:40.560 --> 00:12:47.270 It's the same line as we have, but just starting over by 1. 00:12:47.270 --> 00:12:50.770 Starting with the cosine. 00:12:50.770 --> 00:12:54.570 I know what derivatives I want, now I just have to 00:12:54.570 --> 00:13:03.660 create my series for cosine x, which matches these numbers. 00:13:03.660 --> 00:13:05.430 One more time. 00:13:05.430 --> 00:13:09.750 Just match those numbers with the coefficients that I 00:13:09.750 --> 00:13:16.050 originally called a0, a1, a2, a3, but now we have numbers. 00:13:16.050 --> 00:13:18.400 OK, at x equals 0. 00:13:18.400 --> 00:13:20.580 So how does this series start? 00:13:20.580 --> 00:13:24.060 At x equals 0, the cosine of 0 is 1. 00:13:24.060 --> 00:13:25.200 It starts with a 1. 00:13:25.200 --> 00:13:27.200 That's the constant term sitting there. 00:13:30.830 --> 00:13:38.140 The coefficient of x, the linear term is 0. 00:13:38.140 --> 00:13:43.560 Because the cosine has 0 slope at the start. 00:13:43.560 --> 00:13:46.270 Then we come to something that shows up. 00:13:46.270 --> 00:13:46.910 Minus. 00:13:46.910 --> 00:13:48.750 This will be-- now what are we in to? 00:13:48.750 --> 00:13:52.270 This is the constant, the first power is gone. 00:13:52.270 --> 00:13:56.200 The second power minus x squared. 00:13:56.200 --> 00:14:00.980 But you know if I'm looking to match the second derivative to 00:14:00.980 --> 00:14:03.280 make it b minus 1. 00:14:03.280 --> 00:14:06.000 Right now it's minus 2. 00:14:06.000 --> 00:14:09.280 Differentiating would give me a 2x and a 2. 00:14:09.280 --> 00:14:14.830 So I have to divide by that 2 or 2 factorial. 00:14:14.830 --> 00:14:16.360 Now it's good. 00:14:16.360 --> 00:14:20.430 Now it matches the correct second derivative minus 1. 00:14:20.430 --> 00:14:22.150 Then there's no third derivative. 00:14:22.150 --> 00:14:26.470 The fourth derivative is plus 1 x to the 00:14:26.470 --> 00:14:32.250 fourth over 4 factorial. 00:14:32.250 --> 00:14:36.580 And then minus and so on, x to the sixth over 6 factorial. 00:14:36.580 --> 00:14:44.720 All even powers, so this is an even powers. 00:14:44.720 --> 00:14:48.830 The 0-th, second, fourth, sixth power. 00:14:48.830 --> 00:14:53.700 So it's an even function. 00:14:53.700 --> 00:15:00.010 That means that the cosine of minus x is exactly the same as 00:15:00.010 --> 00:15:01.260 the cosine of x. 00:15:04.750 --> 00:15:12.380 We get a nice little insight on these two special groups 00:15:12.380 --> 00:15:17.040 for which the sine is the perfect example of an odd 00:15:17.040 --> 00:15:20.030 function and the cosine is the perfect 00:15:20.030 --> 00:15:23.660 example of an even function. 00:15:23.660 --> 00:15:27.310 Well, there's so much here. 00:15:27.310 --> 00:15:33.630 What happens if I cut the series off? 00:15:33.630 --> 00:15:38.000 I just want to look at those first terms to see exactly 00:15:38.000 --> 00:15:39.260 what they represent. 00:15:39.260 --> 00:15:45.650 Suppose I stop here after the linear term. 00:15:45.650 --> 00:15:47.540 What do I have? 00:15:47.540 --> 00:15:52.250 What is that x just by itself? 00:15:52.250 --> 00:15:55.530 It's really 0 plus x because there was a 0 from the 00:15:55.530 --> 00:15:56.870 constant term. 00:15:56.870 --> 00:16:00.260 That is the linear approximation. 00:16:00.260 --> 00:16:03.940 That gives me the equation of the tangent line, y 00:16:03.940 --> 00:16:07.800 equals x, slope 1. 00:16:07.800 --> 00:16:11.050 More interesting, cut this one off. 00:16:11.050 --> 00:16:13.360 Cut this one off here. 00:16:13.360 --> 00:16:16.870 That's a very important estimate. 00:16:16.870 --> 00:16:19.810 It's not the exact cosine because the exact cosine has 00:16:19.810 --> 00:16:22.180 got all these later guys. 00:16:22.180 --> 00:16:25.290 But don't forget and I should have said this from the very 00:16:25.290 --> 00:16:34.270 beginning, these n factorials grow fast. And all the series 00:16:34.270 --> 00:16:38.930 that we're talking about, because those n factorials 00:16:38.930 --> 00:16:43.590 grow so fast and I'm dividing by them, I can take any x and 00:16:43.590 --> 00:16:46.520 I get a reasonable number. 00:16:46.520 --> 00:16:52.280 If I take x equaled pi, that's this sine series gave me 0. 00:16:52.280 --> 00:16:56.254 What do I get if I plug in x equals pi 00:16:56.254 --> 00:17:01.150 in the cosine series? 00:17:01.150 --> 00:17:05.020 So the cosine series, if I plugged in x equals pi and had 00:17:05.020 --> 00:17:09.290 patience to go pretty far, my numbers would be getting near 00:17:09.290 --> 00:17:11.920 the cosine of pi. 00:17:11.920 --> 00:17:14.230 Which would be minus 1. 00:17:14.230 --> 00:17:16.640 I don't see minus 1 coming out. 00:17:16.640 --> 00:17:20.109 Here is 1, minus 1/2 of pi squared. 00:17:20.109 --> 00:17:21.980 I don't know, that's around-- 00:17:21.980 --> 00:17:26.250 1/2 pi squared might be around 5 or something. 00:17:26.250 --> 00:17:28.319 But they knock each other off. 00:17:28.319 --> 00:17:32.390 They get very small and we get the answer minus 1. 00:17:32.390 --> 00:17:39.170 OK, so those are two important series and now I get to tell 00:17:39.170 --> 00:17:41.530 you about Euler's great formula. 00:17:46.730 --> 00:17:52.560 It connects these three series that you've seen. 00:17:52.560 --> 00:17:56.800 But to make that connection I have to bring in the 00:17:56.800 --> 00:18:00.280 imaginary number i. 00:18:00.280 --> 00:18:01.770 Is that OK? 00:18:01.770 --> 00:18:05.070 Just imagine a number i. 00:18:05.070 --> 00:18:09.270 And everybody knows what you're supposed to imagine. 00:18:09.270 --> 00:18:14.100 You're supposed to imagine that i squared is minus 1. 00:18:17.910 --> 00:18:22.880 And we all know there is no real number. 00:18:22.880 --> 00:18:26.370 The square of a real number is always going to be greater or 00:18:26.370 --> 00:18:27.590 equal to 0. 00:18:27.590 --> 00:18:35.400 So let's just create a symbol i with a rule, with the 00:18:35.400 --> 00:18:38.510 understanding that any time we see i squared, I'm entitled to 00:18:38.510 --> 00:18:41.750 write minus 1. 00:18:41.750 --> 00:18:41.860 OK. 00:18:41.860 --> 00:18:45.070 So now, what is Euler's great formula? 00:18:47.760 --> 00:18:54.590 Euler's great formula, his brilliant insight was make x 00:18:54.590 --> 00:19:00.120 in this e to the x series, make x imaginary. 00:19:00.120 --> 00:19:02.540 Change x to ix. 00:19:02.540 --> 00:19:05.410 So make it an imaginary number. 00:19:05.410 --> 00:19:11.870 So can I just take Euler's, take Taylor's series, or oh, 00:19:11.870 --> 00:19:14.460 maybe Euler's out of this too, because that 00:19:14.460 --> 00:19:16.770 letter e is his initial. 00:19:16.770 --> 00:19:18.330 Probably he did. 00:19:18.330 --> 00:19:23.460 So I guess that's why he found this lovely connection. 00:19:23.460 --> 00:19:30.680 So if I take e to the ix and instead of x in this series I 00:19:30.680 --> 00:19:34.770 put in ix, just go for it. 00:19:34.770 --> 00:19:36.750 Let x be imaginary. 00:19:36.750 --> 00:19:41.370 OK, can I write out the series 1 plus-- 00:19:41.370 --> 00:19:44.330 instead of x I have ix. 00:19:44.330 --> 00:19:49.610 And then I have 1 over 2 factorial ix squared. 00:19:49.610 --> 00:19:54.890 And then I have 1 over 3 factorial ix cubes. 00:19:54.890 --> 00:19:58.810 And 1 over 4 factorial ix to the fourth. 00:19:58.810 --> 00:20:00.350 That's e to the ix. 00:20:03.880 --> 00:20:07.100 OK, you say, you just changed x to ix. 00:20:07.100 --> 00:20:08.710 That's all I did. 00:20:08.710 --> 00:20:11.380 Now, here's the point. 00:20:11.380 --> 00:20:17.850 Now I'm going to look at this mess and I'm going to separate 00:20:17.850 --> 00:20:25.140 out the part that is real from the part that's imaginary. 00:20:25.140 --> 00:20:27.820 I'm going to separate it into its real part and its 00:20:27.820 --> 00:20:29.220 imaginary part. 00:20:29.220 --> 00:20:33.760 So what is real in this thing? 00:20:33.760 --> 00:20:37.550 I see one is certainly a real number. 00:20:37.550 --> 00:20:41.260 Do you see the other one, the next one that's real? 00:20:41.260 --> 00:20:44.170 It comes from this i squared. 00:20:44.170 --> 00:20:47.790 That i squared I can replace by minus 1, perfectly real. 00:20:47.790 --> 00:20:53.760 So it's minus from the i squared 1 over 2 factorial. 00:20:53.760 --> 00:20:55.850 x squared is still there. 00:20:55.850 --> 00:20:57.360 The i squared was minus 1. 00:20:57.360 --> 00:20:58.720 That's all. 00:20:58.720 --> 00:21:05.600 And then would come something from the i to the fourth. 00:21:05.600 --> 00:21:08.540 Because what is i to the fourth? 00:21:08.540 --> 00:21:12.090 It's i squared squared minus 1 squared. 00:21:12.090 --> 00:21:13.620 We'd be back to plus 1. 00:21:13.620 --> 00:21:15.470 So plus sign. 00:21:15.470 --> 00:21:16.100 Good. 00:21:16.100 --> 00:21:24.070 Now comes the part that has an i in it and a single i I have 00:21:24.070 --> 00:21:25.430 to live with. 00:21:25.430 --> 00:21:29.780 So that i is multiplied by x. 00:21:29.780 --> 00:21:32.250 Now I have i cubed. 00:21:32.250 --> 00:21:34.600 How do I deal with i cubed? 00:21:34.600 --> 00:21:40.070 i cubed is i squared minus 1 times i. 00:21:40.070 --> 00:21:42.490 i squared times i is minus i. 00:21:42.490 --> 00:21:47.950 So I have a minus i. 00:21:47.950 --> 00:21:52.370 1 over 3 factorial and the x cubed and so on. 00:21:56.800 --> 00:21:59.230 Do you see what we have? 00:21:59.230 --> 00:22:04.440 Do you see what this real part of e to the ix is? 00:22:04.440 --> 00:22:06.970 It's the cosine. 00:22:06.970 --> 00:22:09.080 Right there, same thing. 00:22:09.080 --> 00:22:17.040 So I'm getting cosine x for the real part and then i times 00:22:17.040 --> 00:22:19.360 this series. 00:22:19.360 --> 00:22:22.740 And you can see what that series is. 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 00:22:30.030 --> 00:22:33.530 the fifth sine x. 00:22:33.530 --> 00:22:38.470 There is Euler's great formula that e to the ix-- 00:22:38.470 --> 00:22:42.020 oh, I better write it on a fresh board. 00:22:42.020 --> 00:22:43.645 Maybe I'll just write it over here. 00:22:46.550 --> 00:22:53.720 I'm going to copy from this board my Euler's great formula 00:22:53.720 --> 00:22:58.530 that e to the ix comes out to have a real part cos x. 00:22:58.530 --> 00:23:02.240 Imaginary part gives me the i sine x. 00:23:02.240 --> 00:23:03.420 And I'll write that down. 00:23:03.420 --> 00:23:05.610 Now let me work here. 00:23:05.610 --> 00:23:14.810 e to the ix is cos x plus i sine x. 00:23:14.810 --> 00:23:17.090 And I want to draw a picture. 00:23:17.090 --> 00:23:18.340 OK, here's a picture. 00:23:23.000 --> 00:23:28.110 Actually, Euler often wrote his formula, or we often write 00:23:28.110 --> 00:23:33.006 his formula because we're taking cosines and sines. 00:23:33.006 --> 00:23:35.900 Somehow x isn't such-- 00:23:35.900 --> 00:23:37.150 those are angles. 00:23:40.540 --> 00:23:40.840 So it's more natural to write-- 00:23:40.840 --> 00:23:44.160 Now that we've showing up with sines and cosines, it's more 00:23:44.160 --> 00:23:48.360 natural to write a symbol that we think of as 00:23:48.360 --> 00:23:50.570 an angle like theta. 00:23:50.570 --> 00:23:55.360 So you would more often see it this way. 00:23:55.360 --> 00:24:00.750 I'm just changing letters from x to theta as a way of 00:24:00.750 --> 00:24:03.000 remembering that it's an angle. 00:24:03.000 --> 00:24:04.880 And now I'll draw it. 00:24:04.880 --> 00:24:09.040 So I have to draw that thing. 00:24:09.040 --> 00:24:13.560 OK, this is the real direction and that's 00:24:13.560 --> 00:24:15.440 the imaginary direction. 00:24:18.790 --> 00:24:20.670 I just go that's the real part. 00:24:20.670 --> 00:24:23.110 I go cos theta across here. 00:24:23.110 --> 00:24:25.860 So let that be cos theta. 00:24:25.860 --> 00:24:29.590 And then I go upwards in the imaginary up or down. 00:24:29.590 --> 00:24:33.970 So across is the real part, up/down is the imaginary part. 00:24:33.970 --> 00:24:36.930 Say sine theta I go up. 00:24:36.930 --> 00:24:44.390 That height is sine theta and that angle is theta. 00:24:44.390 --> 00:24:46.490 Fantastic. 00:24:46.490 --> 00:24:50.320 That's a picture of Euler's formula. 00:24:50.320 --> 00:24:53.350 Well, that's the best way to see it is 00:24:53.350 --> 00:24:55.880 that beautiful statement. 00:24:55.880 --> 00:24:58.040 And this is a picture to remind us. 00:25:04.860 --> 00:25:08.890 We would say that's the complex plane because points 00:25:08.890 --> 00:25:12.950 have two parts, a real part and an imaginary part. 00:25:12.950 --> 00:25:15.930 Nothing so complex about that. 00:25:15.930 --> 00:25:22.480 Now, before I stop, we've done three important series. 00:25:22.480 --> 00:25:28.290 Can I mention two more, just two more out of a long list of 00:25:28.290 --> 00:25:29.610 possibilities? 00:25:29.610 --> 00:25:33.410 One is the most important series of all, where the 00:25:33.410 --> 00:25:35.720 coefficients are all 1's. 00:25:38.980 --> 00:25:41.500 So the coefficients are all 1's. 00:25:41.500 --> 00:25:43.720 That's called the geometric series. 00:25:43.720 --> 00:25:45.110 Let me write its name here. 00:25:51.590 --> 00:25:53.080 That's a Taylor series. 00:25:53.080 --> 00:25:55.190 That's a power series. 00:25:55.190 --> 00:25:59.710 And the function it comes from happens to be 1 00:25:59.710 --> 00:26:02.670 over 1 minus x. 00:26:02.670 --> 00:26:04.220 That's the function. 00:26:04.220 --> 00:26:08.110 And you will see why, if you multiply both sides by 1 minus 00:26:08.110 --> 00:26:10.970 x, I'll get 1 here. 00:26:10.970 --> 00:26:14.360 If you watch, everything will cancel except the 1. 00:26:14.360 --> 00:26:16.090 So that's it. 00:26:16.090 --> 00:26:19.630 Now, there's a significant difference between that series 00:26:19.630 --> 00:26:20.940 and e to the x. 00:26:20.940 --> 00:26:24.180 The biggest difference is we're not dividing by n 00:26:24.180 --> 00:26:27.310 factorial anymore. 00:26:27.310 --> 00:26:32.790 And as a result, these terms don't get necessarily smaller 00:26:32.790 --> 00:26:34.430 and smaller and smaller. 00:26:34.430 --> 00:26:36.410 Unless x is below 1. 00:26:36.410 --> 00:26:42.640 So we're OK for x below 1. 00:26:42.640 --> 00:26:44.530 And x could be negative. 00:26:44.530 --> 00:26:49.270 I can even say absolute value of x below 1, then these terms 00:26:49.270 --> 00:26:49.930 gets smaller. 00:26:49.930 --> 00:26:52.700 But at x equals 1 we're dead. 00:26:52.700 --> 00:26:56.030 At x equals 1 I have 1 plus 1 plus 1 plus 1. 00:26:56.030 --> 00:26:57.340 All 1's. 00:26:57.340 --> 00:26:59.510 I'm getting infinity. 00:26:59.510 --> 00:27:02.240 And on the left side I'm getting infinity also. 00:27:02.240 --> 00:27:05.590 At x equals 1 blows up. 00:27:05.590 --> 00:27:08.030 OK, one more series, then we're done. 00:27:08.030 --> 00:27:10.630 One more. 00:27:10.630 --> 00:27:15.080 It's a neat one because it brings in the logarithm. 00:27:15.080 --> 00:27:17.440 How am I going to get it? 00:27:17.440 --> 00:27:22.120 I'm going to start with this series, which is the big one, 00:27:22.120 --> 00:27:23.840 the geometric series. 00:27:23.840 --> 00:27:28.170 And I'm going to take the integral of every term. 00:27:28.170 --> 00:27:32.160 So if I integrate 1 I get x. 00:27:32.160 --> 00:27:35.460 If I integrate x I get x squared over 2. 00:27:35.460 --> 00:27:39.210 If I integrate x squared I get x cube over 3. 00:27:39.210 --> 00:27:41.530 x fourth over 4 and so on. 00:27:44.800 --> 00:27:47.690 Not 3 factorial, just 3. 00:27:47.690 --> 00:27:52.880 And if I integrate this, well, let me put the answer down and 00:27:52.880 --> 00:27:57.080 then we can take its derivative and say, yep, it 00:27:57.080 --> 00:27:58.310 does give that. 00:27:58.310 --> 00:28:02.260 So the answer is minus. 00:28:02.260 --> 00:28:06.020 This minus sign shows up here as a minus the 00:28:06.020 --> 00:28:09.010 logarithm of 1 minus x. 00:28:12.140 --> 00:28:17.540 Because if I take the derivative of that the 00:28:17.540 --> 00:28:21.190 logarithm always puts this inside function down to the 00:28:21.190 --> 00:28:24.110 bottom, and then the derivative of the inside 00:28:24.110 --> 00:28:28.350 function, the chain rule brings out a minus 1, and the 00:28:28.350 --> 00:28:31.720 minus 1's go away, and beautiful. 00:28:31.720 --> 00:28:37.230 So just have a look at that series then for the logarithm. 00:28:37.230 --> 00:28:39.170 The logarithm of 1 minus x. 00:28:39.170 --> 00:28:43.300 Again, we're matching at x equals 0. 00:28:43.300 --> 00:28:46.040 At x equals 0, this function is OK. 00:28:46.040 --> 00:28:48.750 In fact, at x equals 0, what is that function? 00:28:48.750 --> 00:28:52.670 Logarithm of 1, which is 0, and there's no constant term. 00:28:52.670 --> 00:28:54.360 Good. 00:28:54.360 --> 00:29:00.570 OK, what comments to make about this final example? 00:29:00.570 --> 00:29:04.490 This one was OK for x smaller than 1. 00:29:04.490 --> 00:29:07.750 But then it died at x equals 1. 00:29:07.750 --> 00:29:13.890 This one, well, it's getting a little help dividing 00:29:13.890 --> 00:29:16.120 by 3 and 4 and 5. 00:29:16.120 --> 00:29:19.670 But that's puny help. 00:29:19.670 --> 00:29:24.150 That's no way compared to dividing by 3 factorial, 4 00:29:24.150 --> 00:29:27.270 factorial, and so on, which will really help. 00:29:27.270 --> 00:29:32.730 So actually, this series is also only OK 00:29:32.730 --> 00:29:35.350 out to x equals 1. 00:29:35.350 --> 00:29:39.760 At x equals 1, it fails again. 00:29:39.760 --> 00:29:42.800 At x equals 1, what do I have? 00:29:42.800 --> 00:29:49.240 When x is 1, I have the log of 0 minus infinity. 00:29:49.240 --> 00:29:51.850 I've got infinity at x equals 1. 00:29:51.850 --> 00:29:55.020 At x equals 1, this is 1 plus 1/2 plus 1/3 00:29:55.020 --> 00:29:57.230 plus 1/4 plus 1/5. 00:29:57.230 --> 00:30:00.640 Getting smaller, but not very fast and 00:30:00.640 --> 00:30:03.060 adding up to infinity. 00:30:03.060 --> 00:30:05.430 So there's a whole discussion. 00:30:05.430 --> 00:30:12.170 We could spend hours on that famous series, 1 plus 1/2 plus 00:30:12.170 --> 00:30:16.150 1/3 plus a quarter and other series of numbers. 00:30:16.150 --> 00:30:21.960 I wanted to do calculus, derivatives, integrals, so I 00:30:21.960 --> 00:30:26.840 took functions and series of powers, not series of numbers 00:30:26.840 --> 00:30:28.830 to illustrate this. 00:30:28.830 --> 00:30:29.660 OK, good. 00:30:29.660 --> 00:30:31.520 Thanks. 00:30:31.520 --> 00:30:33.310 ANNOUNCER: This has been a production of MIT 00:30:33.310 --> 00:30:35.710 OpenCourseWare and Gilbert Strang. 00:30:35.710 --> 00:30:37.980 Funding for this video was provided by the Lord 00:30:37.980 --> 00:30:39.190 Foundation. 00:30:39.190 --> 00:30:42.330 To help OCW continue to provide free and open access 00:30:42.330 --> 00:30:45.400 to MIT courses, please make a donation at 00:30:45.400 --> 00:30:46.960 ocw.mit.edu/donate.