1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,270 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,270 --> 00:00:18,220 at ocw.mit.edu. 8 00:00:30,194 --> 00:00:31,440 HERBERT GROSS: Hi. 9 00:00:31,440 --> 00:00:35,040 Today we tackle another facet of applications 10 00:00:35,040 --> 00:00:40,350 of complex numbers in our survey of our complex variable mini 11 00:00:40,350 --> 00:00:41,100 course. 12 00:00:41,100 --> 00:00:43,590 And in particular, what we're going to study today 13 00:00:43,590 --> 00:00:47,940 is the role of infinite series and sequences 14 00:00:47,940 --> 00:00:49,320 in complex numbers. 15 00:00:49,320 --> 00:00:52,110 And I call today's lesson, therefore, Sequences 16 00:00:52,110 --> 00:00:53,520 and Series. 17 00:00:53,520 --> 00:00:56,640 Now, among other things, let's take a look 18 00:00:56,640 --> 00:01:00,000 at a purely pedagogical problem. 19 00:01:00,000 --> 00:01:02,660 For example, at this stage of the game, 20 00:01:02,660 --> 00:01:05,099 notice that I can't even talk meaningfully 21 00:01:05,099 --> 00:01:10,260 about what I mean by e to the z, nor by what I mean by sine z. 22 00:01:10,260 --> 00:01:12,480 I can not visualize z as an angle, 23 00:01:12,480 --> 00:01:14,280 because it may be imaginary. 24 00:01:14,280 --> 00:01:19,470 I can't visualize it as a length in the traditional sense 25 00:01:19,470 --> 00:01:24,993 that we view functions from real variables into real variables. 26 00:01:24,993 --> 00:01:26,910 And the question that comes up is, how could I 27 00:01:26,910 --> 00:01:29,840 define e to the z and sine z? 28 00:01:29,840 --> 00:01:33,030 In fact, the cynic might say, why would you want to define e 29 00:01:33,030 --> 00:01:35,370 to the z and sine z? 30 00:01:35,370 --> 00:01:38,670 Now, one very cynical answer to the cynic 31 00:01:38,670 --> 00:01:41,190 would simply be that one of the most practical reasons 32 00:01:41,190 --> 00:01:45,120 for knowing how you define either the z and sine z 33 00:01:45,120 --> 00:01:47,820 is that they will be on the quiz at the end 34 00:01:47,820 --> 00:01:49,290 of this particular block. 35 00:01:49,290 --> 00:01:51,900 But that's hardly a fair motivation. 36 00:01:51,900 --> 00:01:55,180 I will go into this in more detail later in the lecture. 37 00:01:55,180 --> 00:01:57,870 Let me simply point out that, every time we 38 00:01:57,870 --> 00:02:02,820 have an analytic function, its real and imaginary parts define 39 00:02:02,820 --> 00:02:04,560 a conformal mapping. 40 00:02:04,560 --> 00:02:07,800 And consequently, every time we can meaningfully 41 00:02:07,800 --> 00:02:12,000 invent an analytic function, we have automatically 42 00:02:12,000 --> 00:02:14,910 invented a new conformal mapping. 43 00:02:14,910 --> 00:02:18,383 As I say, we'll go into that in more detail as we go along. 44 00:02:18,383 --> 00:02:19,800 But the question that now comes up 45 00:02:19,800 --> 00:02:22,500 is, how can I define e to the z? 46 00:02:22,500 --> 00:02:27,460 How can I define sine z, cosh z, all of these things? 47 00:02:27,460 --> 00:02:29,580 And one of the best ways of beginning 48 00:02:29,580 --> 00:02:31,710 is, again, to emphasize something 49 00:02:31,710 --> 00:02:33,720 that we said in one of our earlier elections 50 00:02:33,720 --> 00:02:35,190 on complex numbers-- 51 00:02:35,190 --> 00:02:37,680 that whatever definitions we invent, 52 00:02:37,680 --> 00:02:41,460 we want to make sure that the new definition does not 53 00:02:41,460 --> 00:02:43,260 contradict the old one. 54 00:02:43,260 --> 00:02:45,060 In other words, whatever definition 55 00:02:45,060 --> 00:02:48,600 I give of e to the z, I want to make sure 56 00:02:48,600 --> 00:02:52,170 that, if z happens to be real, if I replace z by x, then 57 00:02:52,170 --> 00:02:54,840 e to the x has the same meaning whether I 58 00:02:54,840 --> 00:02:57,780 view x as a real number or whether I view it 59 00:02:57,780 --> 00:02:59,890 as a complex number. 60 00:02:59,890 --> 00:03:02,670 Now, one of the standard ways of doing something like this 61 00:03:02,670 --> 00:03:04,090 is the following. 62 00:03:04,090 --> 00:03:06,930 We, in part one of our course, had 63 00:03:06,930 --> 00:03:11,070 shown that the power series representation for e to the x 64 00:03:11,070 --> 00:03:14,040 was given by this infinite series where 65 00:03:14,040 --> 00:03:16,800 this infinite series converged to e to the x 66 00:03:16,800 --> 00:03:19,800 for all real values of x. 67 00:03:19,800 --> 00:03:23,100 Now, what we can therefore say is, look at-- 68 00:03:23,100 --> 00:03:29,700 if I just verbatim, every place I see an x, replace that by a z 69 00:03:29,700 --> 00:03:33,570 and now make up the definition, that e to the z will 70 00:03:33,570 --> 00:03:36,750 be summation n goes from zero to infinity, z 71 00:03:36,750 --> 00:03:41,250 to the n over n factorial, then no matter what else happens, 72 00:03:41,250 --> 00:03:44,790 I'm sure that my definition of e to the z 73 00:03:44,790 --> 00:03:49,170 must agree with my definition of e to the x whenever z and x 74 00:03:49,170 --> 00:03:51,390 are equal-- in other words, whenever z happens 75 00:03:51,390 --> 00:03:53,220 to denote a real number. 76 00:03:53,220 --> 00:03:54,700 Why is that the case? 77 00:03:54,700 --> 00:03:58,950 Because I got the new definition by lifting it from the old, 78 00:03:58,950 --> 00:04:01,410 just replacing the x by a z. 79 00:04:01,410 --> 00:04:03,120 The question that comes up of course 80 00:04:03,120 --> 00:04:06,640 is, what do you mean by this expression? 81 00:04:06,640 --> 00:04:08,700 What do you mean by a series when you're 82 00:04:08,700 --> 00:04:10,680 dealing with complex numbers? 83 00:04:10,680 --> 00:04:12,570 In particular what it means is that, 84 00:04:12,570 --> 00:04:15,540 given a sequence of complex numbers, 85 00:04:15,540 --> 00:04:17,820 I must define what I mean by a limit 86 00:04:17,820 --> 00:04:21,000 and then go through this idea of defining a series 87 00:04:21,000 --> 00:04:26,820 to be a sequence of partial sums and going through the same bit 88 00:04:26,820 --> 00:04:30,330 structurally that I went through in the real case. 89 00:04:30,330 --> 00:04:32,070 Now let's do something that I think 90 00:04:32,070 --> 00:04:33,870 is very interesting over here. 91 00:04:33,870 --> 00:04:37,380 Let's just write down what the definition 92 00:04:37,380 --> 00:04:40,530 was for a limit of a sequence a sub n 93 00:04:40,530 --> 00:04:42,390 to equal l as n went to infinity. 94 00:04:42,390 --> 00:04:44,730 What was that definition in the real case? 95 00:04:44,730 --> 00:04:48,270 Limit as n approaches infinity a sub n equals l means, 96 00:04:48,270 --> 00:04:50,520 given any epsilon greater than 0, 97 00:04:50,520 --> 00:04:53,040 there exists a capital N such that, 98 00:04:53,040 --> 00:04:55,560 whenever a little n exceeds capital N, 99 00:04:55,560 --> 00:04:59,640 the absolute value of a sub n minus l is less than epsilon. 100 00:04:59,640 --> 00:05:01,930 I hope this is very familiar to you. 101 00:05:01,930 --> 00:05:05,180 If not, feel free to review it. 102 00:05:05,180 --> 00:05:07,540 But here is the interesting point. 103 00:05:07,540 --> 00:05:11,550 No place in here do I mention that a sub n and l have 104 00:05:11,550 --> 00:05:13,172 to be real numbers. 105 00:05:13,172 --> 00:05:15,130 In other words, when I made up this definition, 106 00:05:15,130 --> 00:05:16,330 I assumed that they were. 107 00:05:16,330 --> 00:05:19,400 But notice that no place in here in this definition 108 00:05:19,400 --> 00:05:21,940 do I say that a sub n and l are real. 109 00:05:21,940 --> 00:05:25,540 In fact, where are the only places that I use realness? 110 00:05:25,540 --> 00:05:28,600 That since epsilon is real, I would like to believe that, 111 00:05:28,600 --> 00:05:31,720 no matter what a sub n and l are, that the magnitude-- 112 00:05:31,720 --> 00:05:34,060 the absolute value of a sub n minus l-- 113 00:05:34,060 --> 00:05:37,240 must be less than epsilon means that the magnitude of a sub n 114 00:05:37,240 --> 00:05:40,210 minus l must be a real number, among other things-- 115 00:05:40,210 --> 00:05:42,680 in fact, preferably a positive real number. 116 00:05:42,680 --> 00:05:45,310 Notice that the magnitude of a complex number 117 00:05:45,310 --> 00:05:47,710 is still a non-negative real. 118 00:05:47,710 --> 00:05:51,640 Consequently, even if a sub n and l are complex, 119 00:05:51,640 --> 00:05:53,710 this definition still holds. 120 00:05:53,710 --> 00:05:56,680 And in fact, the pictorial interpretation 121 00:05:56,680 --> 00:05:59,530 of the definition holds precisely the same 122 00:05:59,530 --> 00:06:03,220 as it did in the real case, except that an epsilon 123 00:06:03,220 --> 00:06:08,290 neighborhood of l is an interval in the real case. 124 00:06:08,290 --> 00:06:10,698 And it's a circle, a disk-- 125 00:06:10,698 --> 00:06:11,740 I shouldn't say a circle. 126 00:06:11,740 --> 00:06:13,330 Remember, the circle was the thing 127 00:06:13,330 --> 00:06:14,470 we call the circumference. 128 00:06:14,470 --> 00:06:16,780 The disk is the inside of the circle. 129 00:06:16,780 --> 00:06:21,790 The neighborhood is a disk in the complex case. 130 00:06:21,790 --> 00:06:25,960 Namely, notice that to say that the absolute value of a sub n 131 00:06:25,960 --> 00:06:28,180 minus l is less than epsilon still 132 00:06:28,180 --> 00:06:31,480 means that a sub n is with an epsilon of l. 133 00:06:31,480 --> 00:06:33,550 The difference is that the domain of l 134 00:06:33,550 --> 00:06:36,880 is now the xy plane, the Argand diagram. 135 00:06:36,880 --> 00:06:39,490 So to be with an epsilon of l, I now 136 00:06:39,490 --> 00:06:44,500 take a circle centered at l with radius epsilon 137 00:06:44,500 --> 00:06:46,120 and draw that circle. 138 00:06:46,120 --> 00:06:48,630 And what I'm saying is that if the limit of the sequence 139 00:06:48,630 --> 00:06:51,850 is l, after a certain number of terms, 140 00:06:51,850 --> 00:06:55,160 all the remaining terms are inside the circle. 141 00:06:55,160 --> 00:06:58,840 In other words, just as in the real case, in the complex case, 142 00:06:58,840 --> 00:07:03,410 the limit replaces an infinite number of points-- 143 00:07:03,410 --> 00:07:05,410 numbers, you see, an infinite number of points-- 144 00:07:05,410 --> 00:07:08,402 by a finite number plus one dot. 145 00:07:08,402 --> 00:07:10,360 In other words, there's a bunch of numbers that 146 00:07:10,360 --> 00:07:12,610 may lie outside the circle. 147 00:07:12,610 --> 00:07:16,450 But after a point, after a certain term, 148 00:07:16,450 --> 00:07:20,770 all of the remaining terms lie inside this particular disk. 149 00:07:20,770 --> 00:07:22,420 And structurally, you see, that's 150 00:07:22,420 --> 00:07:25,240 exactly the same thing that happened in the real case. 151 00:07:25,240 --> 00:07:27,790 Consequently, since the structural definition 152 00:07:27,790 --> 00:07:30,220 is the same, since the geometric interpretation is 153 00:07:30,220 --> 00:07:34,030 the same except that disks have replaced intervals, 154 00:07:34,030 --> 00:07:37,300 I would expect that all the limit theorems held, 155 00:07:37,300 --> 00:07:38,470 just as in the real case. 156 00:07:38,470 --> 00:07:39,820 And indeed, they do. 157 00:07:39,820 --> 00:07:42,370 And we'll talk about those in the exercises. 158 00:07:42,370 --> 00:07:45,760 In a similar way, though, that we went from sequences 159 00:07:45,760 --> 00:07:49,270 to series in the real case, we can now do the same thing 160 00:07:49,270 --> 00:07:50,380 in the complex case. 161 00:07:50,380 --> 00:07:54,850 Namely, suppose c1, c2, c3, et cetera 162 00:07:54,850 --> 00:07:59,500 represent a sequence of complex numbers by the infinite sum 163 00:07:59,500 --> 00:08:01,960 and goes from 1 to infinitely c sub n. 164 00:08:01,960 --> 00:08:04,720 I simply mean the limit as n goes 165 00:08:04,720 --> 00:08:08,020 to infinity c1, plus et cetera, cn, 166 00:08:08,020 --> 00:08:09,310 noticing that this is-- what? 167 00:08:09,310 --> 00:08:12,020 One number, it depends on n. 168 00:08:12,020 --> 00:08:14,620 And that's what you'll call the n-th partial sum 169 00:08:14,620 --> 00:08:18,130 of the series that was added up c1 through cn. 170 00:08:18,130 --> 00:08:20,680 The next term would be obtained by adding 171 00:08:20,680 --> 00:08:23,890 on c sub n plus 1, et cetera. 172 00:08:23,890 --> 00:08:26,380 But notice again the similar structure. 173 00:08:26,380 --> 00:08:28,810 And consequently, the usual theorems 174 00:08:28,810 --> 00:08:32,980 for series that apply to the real variable case 175 00:08:32,980 --> 00:08:35,350 apply to the complex variable case. 176 00:08:35,350 --> 00:08:36,950 Again, what's the only difference? 177 00:08:36,950 --> 00:08:40,240 The only difference will be that intervals of convergence 178 00:08:40,240 --> 00:08:44,920 will be replaced by disks or circles of convergence. 179 00:08:44,920 --> 00:08:47,860 In particular, if we let s denote 180 00:08:47,860 --> 00:08:52,180 the set of all complex numbers z for which the power 181 00:08:52,180 --> 00:08:55,660 series summation a nz to the n converges then, 182 00:08:55,660 --> 00:08:59,470 just as in the real case, one of three things must happen. 183 00:08:59,470 --> 00:09:03,220 Namely, either s consists solely of zero. 184 00:09:03,220 --> 00:09:06,720 See, obviously, if I replace z by zero, this thing converges. 185 00:09:06,720 --> 00:09:08,410 It's zero, in fact. 186 00:09:08,410 --> 00:09:11,440 Or it may be such that these terms 187 00:09:11,440 --> 00:09:14,200 go to zero so rapidly that this will converge 188 00:09:14,200 --> 00:09:16,390 for all complex numbers, c-- 189 00:09:16,390 --> 00:09:21,370 in other words, that the set S can be all complex numbers. 190 00:09:21,370 --> 00:09:24,310 Usually, you'll get something in between these two extremes. 191 00:09:24,310 --> 00:09:26,860 And that's the case, like just in the real case, 192 00:09:26,860 --> 00:09:31,060 there exists a number capital R greater than zero such 193 00:09:31,060 --> 00:09:36,640 that the set S consists of all of those z's such 194 00:09:36,640 --> 00:09:39,310 that the absolute value of z is less than R. 195 00:09:39,310 --> 00:09:42,070 And by the way, what does this mean geometrically? 196 00:09:42,070 --> 00:09:45,400 The absolute value of z is the magnitude of z. 197 00:09:45,400 --> 00:09:47,870 That's the distance of z from the origin. 198 00:09:47,870 --> 00:09:51,700 This says that z is within capital R of the origin. 199 00:09:51,700 --> 00:09:54,410 That means you're inside the circle, centered 200 00:09:54,410 --> 00:09:57,010 with the origin, with radius, R. 201 00:09:57,010 --> 00:10:00,130 At any rate, going on the convergences 202 00:10:00,130 --> 00:10:04,810 both absolute and uniform in any interior disk-- in other words, 203 00:10:04,810 --> 00:10:06,840 inside the disk absolute value of z 204 00:10:06,840 --> 00:10:09,780 is less than or equal to little r where a little r is 205 00:10:09,780 --> 00:10:13,140 less than capital R. Again, in terms of a picture, what we're 206 00:10:13,140 --> 00:10:16,410 saying is, if the first two conditions don't hold 207 00:10:16,410 --> 00:10:22,170 given a power series, at the origin, I draw a circle. 208 00:10:22,170 --> 00:10:23,120 I don't draw a circle. 209 00:10:23,120 --> 00:10:25,500 What I'm saying is there exists a circle such 210 00:10:25,500 --> 00:10:29,340 that the power series will converge 211 00:10:29,340 --> 00:10:31,350 every place inside here. 212 00:10:31,350 --> 00:10:34,440 It will diverge every place outside here. 213 00:10:34,440 --> 00:10:36,690 And what happens on the boundary must 214 00:10:36,690 --> 00:10:39,040 be investigated separately, again, 215 00:10:39,040 --> 00:10:41,370 just as in the real case. 216 00:10:41,370 --> 00:10:42,000 OK? 217 00:10:42,000 --> 00:10:43,680 That's what the key theorem is. 218 00:10:43,680 --> 00:10:45,960 And again, you'll have time to reread this 219 00:10:45,960 --> 00:10:50,970 and to digest it in your leisure before you try the exercises. 220 00:10:50,970 --> 00:10:55,650 Now, once we've define what we mean by a power series-- 221 00:10:55,650 --> 00:10:58,170 and notice, by the way, that absolute convergence 222 00:10:58,170 --> 00:11:01,560 goes through word for word in the complex value case 223 00:11:01,560 --> 00:11:04,170 because, after all, in terms of absolute convergence, 224 00:11:04,170 --> 00:11:06,060 we look at absolute values. 225 00:11:06,060 --> 00:11:09,432 And absolute values are non-negative reals, 226 00:11:09,432 --> 00:11:11,640 regardless of whether the thing that we're looking at 227 00:11:11,640 --> 00:11:14,670 happened to be complex numbers or real numbers to begin with. 228 00:11:14,670 --> 00:11:16,800 Magnitudes are non-negative. 229 00:11:16,800 --> 00:11:20,550 At any rate, one can then show, in precisely the same way 230 00:11:20,550 --> 00:11:23,790 that we did it in the real variable case, 231 00:11:23,790 --> 00:11:26,400 that this particular power series-- forget about this 232 00:11:26,400 --> 00:11:27,840 being e to the z right now. 233 00:11:27,840 --> 00:11:29,490 I make up this power series. 234 00:11:29,490 --> 00:11:31,230 I use the ratio test. 235 00:11:31,230 --> 00:11:33,240 And I can show, by the ratio test, 236 00:11:33,240 --> 00:11:37,170 that this series converges uniformly and absolutely 237 00:11:37,170 --> 00:11:40,590 for all real values of z, which-- 238 00:11:40,590 --> 00:11:42,998 I'm sorry, for all finite values of z. 239 00:11:42,998 --> 00:11:44,040 I don't mean real values. 240 00:11:44,040 --> 00:11:45,120 I mean finite values. 241 00:11:45,120 --> 00:11:47,640 Z does not have to be real. 242 00:11:47,640 --> 00:11:51,840 Now what I say is, because this is uniformly convergent, 243 00:11:51,840 --> 00:11:54,450 this power series behaves like a polynomial. 244 00:11:54,450 --> 00:11:56,520 I can differentiate it term by term. 245 00:11:56,520 --> 00:11:59,890 I can integrate it term by term, et cetera, et cetera, 246 00:11:59,890 --> 00:12:00,920 et cetera. 247 00:12:00,920 --> 00:12:02,610 I'll now give this a name. 248 00:12:02,610 --> 00:12:06,900 And being very, very judicious, what name do I pick? 249 00:12:06,900 --> 00:12:09,510 I pick e to the z, because that's 250 00:12:09,510 --> 00:12:12,840 what motivated my inventing this function, this series, 251 00:12:12,840 --> 00:12:14,250 in the first place. 252 00:12:14,250 --> 00:12:15,960 And what am I now sure of? 253 00:12:15,960 --> 00:12:19,350 I'm sure of the fact that, if I replace z by x-- 254 00:12:19,350 --> 00:12:20,790 in other words, if z is real-- 255 00:12:20,790 --> 00:12:24,030 that this must still be the same as what e to the x was before. 256 00:12:24,030 --> 00:12:25,590 And what's the best proof of that? 257 00:12:25,590 --> 00:12:27,150 Replace z by x. 258 00:12:27,150 --> 00:12:30,390 What you get is summation. n goes from 0 to infinity, 259 00:12:30,390 --> 00:12:32,560 x to the n over n factorial. 260 00:12:32,560 --> 00:12:34,890 And we saw in part 1 of our course 261 00:12:34,890 --> 00:12:37,800 that that was precisely e to the x. 262 00:12:37,800 --> 00:12:40,260 Well, since exponentials may bother you a little bit more 263 00:12:40,260 --> 00:12:43,920 than sines and cosines, let's revisit the same problem 264 00:12:43,920 --> 00:12:46,650 in terms of sine z. 265 00:12:46,650 --> 00:12:52,800 What I do is I define sine z to be this uniformly, absolutely, 266 00:12:52,800 --> 00:12:55,560 convergent power series. 267 00:12:55,560 --> 00:13:01,320 Again, using hindsight-- knowing that, in the case of sine x, 268 00:13:01,320 --> 00:13:03,360 this was the convergent power series-- 269 00:13:03,360 --> 00:13:07,470 I simply replace x by z up here. 270 00:13:07,470 --> 00:13:10,020 All right, I replace x by z. 271 00:13:10,020 --> 00:13:13,260 I now use the ratio test on this to show that this 272 00:13:13,260 --> 00:13:16,170 does converge for all finite z. 273 00:13:16,170 --> 00:13:18,510 In fact, that's sort of redundant to do that. 274 00:13:18,510 --> 00:13:20,500 I know darn well that's going to happen, 275 00:13:20,500 --> 00:13:21,810 because it happened here. 276 00:13:21,810 --> 00:13:23,750 And structurally, these are the same. 277 00:13:23,750 --> 00:13:26,490 But I now then, after showing that, I say, 278 00:13:26,490 --> 00:13:28,920 OK, this converges for all z. 279 00:13:28,920 --> 00:13:29,950 Let's give it a name. 280 00:13:29,950 --> 00:13:31,320 I'll call it sine z. 281 00:13:31,320 --> 00:13:34,850 And what guarantee do I have that sine z is the same as sine 282 00:13:34,850 --> 00:13:36,930 x whenever z is real? 283 00:13:36,930 --> 00:13:38,880 And again, let me point out something. 284 00:13:38,880 --> 00:13:43,410 Notice, in the real case, we started knowing what sine x was 285 00:13:43,410 --> 00:13:46,170 but not knowing what its power series expansion was. 286 00:13:46,170 --> 00:13:48,120 And we developed the power series expansion. 287 00:13:48,120 --> 00:13:49,620 And by the way, notice that when you 288 00:13:49,620 --> 00:13:52,830 want to compute sine x to any number of decimal places, 289 00:13:52,830 --> 00:13:55,580 whether you're doing it by hand or by the computer, 290 00:13:55,580 --> 00:13:58,920 notice that it's the power series that you use. 291 00:13:58,920 --> 00:14:02,130 And you chop this off after a certain number of terms, 292 00:14:02,130 --> 00:14:02,850 if you wish. 293 00:14:02,850 --> 00:14:05,250 But one does use the power series idea. 294 00:14:05,250 --> 00:14:07,020 What I am saying though is, had I 295 00:14:07,020 --> 00:14:09,810 wished to, from a purely pedagogical point of view I 296 00:14:09,810 --> 00:14:12,030 could have said, let me now-- 297 00:14:12,030 --> 00:14:14,310 forget about this definition of sine x 298 00:14:14,310 --> 00:14:18,030 and define sine x just by this particular power series. 299 00:14:18,030 --> 00:14:20,550 That would have been artificial to do at that time. 300 00:14:20,550 --> 00:14:22,590 What I'm saying now, though, is this-- 301 00:14:22,590 --> 00:14:25,080 start with sine x in the real case. 302 00:14:25,080 --> 00:14:27,570 Develop its convergent power series. 303 00:14:27,570 --> 00:14:31,590 Take that power series, replace x by z, 304 00:14:31,590 --> 00:14:33,870 and define that new complex powers 305 00:14:33,870 --> 00:14:36,270 series to be what sine z is. 306 00:14:36,270 --> 00:14:38,640 And that gets the job done for you. 307 00:14:38,640 --> 00:14:42,180 Now, the interesting thing is that one can now 308 00:14:42,180 --> 00:14:44,550 show that all of the trigonometric functions, 309 00:14:44,550 --> 00:14:47,070 be they hyperbolic or circular functions, 310 00:14:47,070 --> 00:14:49,180 are related to e to the z. 311 00:14:49,180 --> 00:14:52,890 Well, why that's interesting is not very important right now. 312 00:14:52,890 --> 00:14:55,680 We'll mention that in more detail on other occasions, 313 00:14:55,680 --> 00:14:56,370 hopefully. 314 00:14:56,370 --> 00:14:58,050 But the idea is this. 315 00:14:58,050 --> 00:15:01,980 By using the fact that these various series are absolutely 316 00:15:01,980 --> 00:15:05,020 and uniformly convergent so I can manipulate them term 317 00:15:05,020 --> 00:15:05,890 by term-- 318 00:15:05,890 --> 00:15:08,080 and these things are done in detail in the text, 319 00:15:08,080 --> 00:15:11,290 and I will give you other ones to do for homework so that you 320 00:15:11,290 --> 00:15:12,640 get the drill on this-- 321 00:15:12,640 --> 00:15:17,350 one can show the very amazing results that e to the iz 322 00:15:17,350 --> 00:15:21,100 is cosine z plus i sine z, a very amazing result 323 00:15:21,100 --> 00:15:25,810 that relates the cosine and the sine to an exponential. 324 00:15:25,810 --> 00:15:28,810 In fact, to stress this from a different point of view, 325 00:15:28,810 --> 00:15:31,450 if I take the special case that z is real, 326 00:15:31,450 --> 00:15:34,150 notice that this says-- what? 327 00:15:34,150 --> 00:15:36,170 e to the ix-- 328 00:15:36,170 --> 00:15:37,780 so you replace z by x-- 329 00:15:37,780 --> 00:15:42,660 is cosine x plus i sine x. 330 00:15:42,660 --> 00:15:44,730 In particular, what this then says 331 00:15:44,730 --> 00:15:47,670 is, let's go back to the Argand diagram 332 00:15:47,670 --> 00:15:49,650 where we take the complex number, 333 00:15:49,650 --> 00:15:52,780 view it in polar coordinates as r comma theta-- 334 00:15:52,780 --> 00:15:53,670 which means what? 335 00:15:53,670 --> 00:15:57,270 r cosine theta plus ir sine theta. 336 00:15:57,270 --> 00:16:01,920 Factor out the r, so it's r times cosine theta plus i sine 337 00:16:01,920 --> 00:16:02,670 theta. 338 00:16:02,670 --> 00:16:06,540 But cosine theta plus i sine theta is e to the i theta. 339 00:16:06,540 --> 00:16:08,850 And what that says is, that if you really 340 00:16:08,850 --> 00:16:13,680 wanted a nice name for this, r comma theta does turn out 341 00:16:13,680 --> 00:16:16,470 in the language of polar coordinates 342 00:16:16,470 --> 00:16:20,100 using complex variables without Cartesian reference 343 00:16:20,100 --> 00:16:22,510 is re to the i theta. 344 00:16:22,510 --> 00:16:25,440 Which, by the way, would explain very nicely why, 345 00:16:25,440 --> 00:16:27,570 when you multiply complex numbers, 346 00:16:27,570 --> 00:16:31,140 you multiply the magnitudes and add the arguments. 347 00:16:31,140 --> 00:16:32,910 Namely, notice that the magnitudes 348 00:16:32,910 --> 00:16:36,150 are these scalar factors in front of the exponential. 349 00:16:36,150 --> 00:16:38,850 The arguments occur as exponents. 350 00:16:38,850 --> 00:16:41,540 And when you multiply two numbers to the base e, 351 00:16:41,540 --> 00:16:43,202 you add the exponents. 352 00:16:43,202 --> 00:16:45,160 In fact, when you multiply numbers to any base, 353 00:16:45,160 --> 00:16:46,950 you add the exponents. 354 00:16:46,950 --> 00:16:50,280 And that's why you add the angles over here. 355 00:16:50,280 --> 00:16:54,090 Again, a very interesting aside, which is worth absolutely 356 00:16:54,090 --> 00:16:55,367 nothing-- 357 00:16:55,367 --> 00:16:57,450 I think it was the first thing that ever turned me 358 00:16:57,450 --> 00:17:00,870 on in mathematics that got my metaphysical mathematical 359 00:17:00,870 --> 00:17:01,770 dander up-- 360 00:17:01,770 --> 00:17:07,050 was, if you replace x by pi over here, 361 00:17:07,050 --> 00:17:09,510 what does that mean if I replace x by pi? 362 00:17:09,510 --> 00:17:10,920 The magnitude is one. 363 00:17:10,920 --> 00:17:12,510 The angle is pi. 364 00:17:12,510 --> 00:17:14,940 e to the i pi-- 365 00:17:14,940 --> 00:17:22,400 e to the i pi is this another way of writing minus 1. 366 00:17:22,400 --> 00:17:24,170 What is e to the i pi? 367 00:17:24,170 --> 00:17:25,579 The argument is pi. 368 00:17:25,579 --> 00:17:27,170 That means 180 degrees. 369 00:17:27,170 --> 00:17:28,610 The argument is one. 370 00:17:28,610 --> 00:17:29,990 And the number, which is one unit 371 00:17:29,990 --> 00:17:33,380 from the origin in the Argand diagram at an angle 180 372 00:17:33,380 --> 00:17:35,870 degrees, is the real number minus 1. 373 00:17:35,870 --> 00:17:40,700 I've always been mystified by the three most 374 00:17:40,700 --> 00:17:41,960 remarkable numbers. 375 00:17:41,960 --> 00:17:45,140 Constants in mathematics seem to be ei and pi. 376 00:17:45,140 --> 00:17:47,900 And e to the i pi turns out to be minus 1. 377 00:17:47,900 --> 00:17:49,850 I just mentioned this to show you 378 00:17:49,850 --> 00:17:52,370 how complicated things can get if you try to read 379 00:17:52,370 --> 00:17:53,630 too much meaning into things. 380 00:17:53,630 --> 00:17:55,830 But don't worry about that right now. 381 00:17:55,830 --> 00:17:57,230 I couldn't resist the comment. 382 00:17:57,230 --> 00:17:58,700 What I would like to do though is 383 00:17:58,700 --> 00:18:03,020 to show you now how this is used to define other functions 384 00:18:03,020 --> 00:18:04,850 and get us other results. 385 00:18:04,850 --> 00:18:08,300 Among other things, notice that the first bad thing that 386 00:18:08,300 --> 00:18:10,940 happens with the polar coordinate representation 387 00:18:10,940 --> 00:18:14,720 is, if the complex number z can be written as re to the i 388 00:18:14,720 --> 00:18:20,240 theta, it can also be written as re to the i theta plus 2 pi k 389 00:18:20,240 --> 00:18:25,620 because, every time I changed my angle by 2 pi or 360 degrees, 390 00:18:25,620 --> 00:18:27,980 I come back to the same point again. 391 00:18:27,980 --> 00:18:30,500 At any rate, forgetting about that problem for the time 392 00:18:30,500 --> 00:18:33,890 being, notice how I can now define a logarithm 393 00:18:33,890 --> 00:18:35,270 for a complex number. 394 00:18:35,270 --> 00:18:37,990 In fact, the logarithm will turn out to be the inverse of e 395 00:18:37,990 --> 00:18:39,620 to the z, the same as before. 396 00:18:39,620 --> 00:18:41,610 But we won't worry about that here either. 397 00:18:41,610 --> 00:18:44,600 Let's see how we could define log z from this now. 398 00:18:44,600 --> 00:18:47,540 We would like the log to have the usual logarithmic 399 00:18:47,540 --> 00:18:48,350 properties. 400 00:18:48,350 --> 00:18:52,710 The log of a product should be the sum of the logs. 401 00:18:52,710 --> 00:18:57,620 So log z should be log r plus log e to the i theta plus 402 00:18:57,620 --> 00:18:59,240 2 pi k. 403 00:18:59,240 --> 00:19:02,120 Now, r is a positive real number. 404 00:19:02,120 --> 00:19:05,090 We already know that the natural log applies there. 405 00:19:05,090 --> 00:19:10,310 So what we say is, OK, the log of r is just natural log r. 406 00:19:10,310 --> 00:19:13,190 Log e to anything-- the log in the e-- 407 00:19:13,190 --> 00:19:15,330 are inverse of one another. 408 00:19:15,330 --> 00:19:16,700 So they should cancel. 409 00:19:16,700 --> 00:19:20,150 And this will just be i theta plus 2 pi k. 410 00:19:20,150 --> 00:19:23,660 In other words, to find the log of the complex number z, 411 00:19:23,660 --> 00:19:26,540 you simply take the log of its magnitude 412 00:19:26,540 --> 00:19:30,020 and add on, as the imaginary part, what? 413 00:19:30,020 --> 00:19:32,870 The argument-- noticing, by the way, 414 00:19:32,870 --> 00:19:35,480 that the argument is multi-valued. 415 00:19:35,480 --> 00:19:37,520 See, log z is multi-valued. 416 00:19:37,520 --> 00:19:40,850 Consequently, we must get into the idea of principal values. 417 00:19:40,850 --> 00:19:43,580 And one usually assumes, unless otherwise stated, 418 00:19:43,580 --> 00:19:47,030 that the argument is no greater than pi 419 00:19:47,030 --> 00:19:48,930 but greater than minus pi. 420 00:19:48,930 --> 00:19:51,350 If we don't mean that, we have to state that separately. 421 00:19:51,350 --> 00:19:54,140 Again, we will drill on that in the exercises. 422 00:19:54,140 --> 00:19:56,300 By the way, another application that 423 00:19:56,300 --> 00:20:00,920 relates to hyperbolic functions to the circular functions-- 424 00:20:00,920 --> 00:20:05,750 notice that, if we wanted to define hyperbolic cosine of ix 425 00:20:05,750 --> 00:20:07,860 mimicking what happened in the real case, 426 00:20:07,860 --> 00:20:12,680 this would be e to the ix plus e to the minus ix over 2. 427 00:20:12,680 --> 00:20:17,630 We already saw that e to the ix was cosine x plus i sine x. 428 00:20:17,630 --> 00:20:22,190 e to the minus ix is e to the i of minus x-- 429 00:20:22,190 --> 00:20:24,590 that's cosine minus x-- 430 00:20:24,590 --> 00:20:26,780 plus i sine minus x. 431 00:20:26,780 --> 00:20:28,940 By the way, I hope that's clear to you. 432 00:20:28,940 --> 00:20:32,430 When I say that e to the ix is cosine x plus i 433 00:20:32,430 --> 00:20:34,810 sine x, what I really mean is-- 434 00:20:34,810 --> 00:20:35,870 or even up here. 435 00:20:35,870 --> 00:20:38,690 What I really mean is that e to the i anything 436 00:20:38,690 --> 00:20:42,680 is cosine of that anything plus i sine of that anything. 437 00:20:42,680 --> 00:20:45,320 In particular, if I replace z by minus x, 438 00:20:45,320 --> 00:20:47,880 I get e to the i minus x. 439 00:20:47,880 --> 00:20:51,560 In other words, minus ix is cosine minus x plus i 440 00:20:51,560 --> 00:20:52,930 sine minus x. 441 00:20:55,880 --> 00:20:59,480 Cosine of minus x equals cosine x. 442 00:20:59,480 --> 00:21:01,910 Sine minus x is minus sine x. 443 00:21:01,910 --> 00:21:03,440 These terms cancel. 444 00:21:03,440 --> 00:21:05,345 This becomes 2 cosine x. 445 00:21:05,345 --> 00:21:09,140 2 cosine x divided by 2 is cosine x. 446 00:21:09,140 --> 00:21:11,690 And we have the rather amazing identity 447 00:21:11,690 --> 00:21:14,990 that cosh ix is cosine x-- 448 00:21:14,990 --> 00:21:17,480 in other words, a very interesting relationship 449 00:21:17,480 --> 00:21:20,790 between the hyperbolic and the circular functions. 450 00:21:20,790 --> 00:21:24,770 And by the way, this shouldn't be too surprising. 451 00:21:24,770 --> 00:21:26,900 Notice that the circular functions 452 00:21:26,900 --> 00:21:32,570 come from studying x squared plus y squared equals 1. 453 00:21:32,570 --> 00:21:35,930 The hyperbolic functions come from studying x squared 454 00:21:35,930 --> 00:21:38,150 minus y squared equals 1. 455 00:21:38,150 --> 00:21:40,580 And x squared minus y squared can 456 00:21:40,580 --> 00:21:45,240 be written as x squared plus the quantity iy squared. 457 00:21:45,240 --> 00:21:48,950 So what would be a hyperbole in the xy plane 458 00:21:48,950 --> 00:21:54,200 would be a circle in the x iy plane, whatever that might be. 459 00:21:54,200 --> 00:21:57,050 But again, that's just in the form of an aside. 460 00:21:57,050 --> 00:21:58,850 And now we'll come back to what we 461 00:21:58,850 --> 00:22:03,590 mean by saying that we can use these new analytic functions 462 00:22:03,590 --> 00:22:05,535 to do conforming mappings. 463 00:22:05,535 --> 00:22:07,160 Let's come back to the question that we 464 00:22:07,160 --> 00:22:11,390 raised at the very beginning of our lecture about e to the z. 465 00:22:11,390 --> 00:22:14,900 Certainly, because z is x plus iy, each of the z 466 00:22:14,900 --> 00:22:18,880 will be e to the x plus iy because of the properties 467 00:22:18,880 --> 00:22:20,475 that the exponential will have. 468 00:22:20,475 --> 00:22:22,600 And by the way, all of these exponential properties 469 00:22:22,600 --> 00:22:24,550 can be proven from the power series. 470 00:22:24,550 --> 00:22:27,070 You see, we don't need the real interpretation. 471 00:22:27,070 --> 00:22:29,440 The power series will give us all these results. 472 00:22:29,440 --> 00:22:32,200 But again, we'll drill on those in the exercises. 473 00:22:32,200 --> 00:22:33,910 I just want to get the highlights here. 474 00:22:33,910 --> 00:22:36,860 This is e to the x times e to the iy. 475 00:22:36,860 --> 00:22:39,893 But notice here, you see that y is real. 476 00:22:39,893 --> 00:22:42,310 Or even if it wasn't real, it doesn't make any difference. 477 00:22:42,310 --> 00:22:47,170 e to the iy is cosine y plus i sine y. 478 00:22:47,170 --> 00:22:49,060 Multiplying this out, each e to the z 479 00:22:49,060 --> 00:22:53,200 is e to the x cosine y plus ie to x sine y. 480 00:22:53,200 --> 00:22:56,050 The important thing being that, because y is real, 481 00:22:56,050 --> 00:23:01,060 e to the x cosine y and e to the x sine y are both real. 482 00:23:01,060 --> 00:23:04,390 And consequently, if I let you denote the real part and v 483 00:23:04,390 --> 00:23:07,780 denote the imaginary part, as usual, what I have is 484 00:23:07,780 --> 00:23:12,730 that u equals e to the x cosine y, v equals e to x sine y. 485 00:23:12,730 --> 00:23:15,310 Must be a real conformal mapping. 486 00:23:15,310 --> 00:23:15,880 Why? 487 00:23:15,880 --> 00:23:19,310 Because the real and the imaginary parts-- 488 00:23:19,310 --> 00:23:21,020 the real and the imaginary parts-- 489 00:23:21,020 --> 00:23:22,870 make up an analytic function. 490 00:23:22,870 --> 00:23:24,610 How do I know the function is analytic? 491 00:23:24,610 --> 00:23:26,920 Because its power series exists. 492 00:23:26,920 --> 00:23:29,500 And the fact that its power series exists 493 00:23:29,500 --> 00:23:32,430 means that all of the derivatives must exist. 494 00:23:32,430 --> 00:23:34,960 Now, again, I didn't write this in here. 495 00:23:34,960 --> 00:23:36,850 But maybe it's a worthwhile aside 496 00:23:36,850 --> 00:23:39,460 to show you what this conformal mapping looks like. 497 00:23:39,460 --> 00:23:43,780 For example, notice that, if I square both sides 498 00:23:43,780 --> 00:23:46,390 of each equation here and add-- 499 00:23:46,390 --> 00:23:48,970 if I square and add, what do I get? 500 00:23:48,970 --> 00:23:52,480 On one side, I get u squared plus v squared. 501 00:23:52,480 --> 00:23:55,000 On the other side, I get e to the 2x cosine 502 00:23:55,000 --> 00:23:58,390 squared y plus e to the 2x sine squared y. 503 00:23:58,390 --> 00:24:00,610 I factor out e to the 2x. 504 00:24:00,610 --> 00:24:03,100 And that gives me e to the 2x sine 505 00:24:03,100 --> 00:24:06,170 squared y plus cosine squared y, which is 1. 506 00:24:06,170 --> 00:24:10,360 So this is just u squared plus v squared is e to the 2x. 507 00:24:10,360 --> 00:24:12,640 To eliminate x from the equations, 508 00:24:12,640 --> 00:24:15,760 I can divide the bottom by the top. 509 00:24:15,760 --> 00:24:24,460 And I get that v over u is equal to tangent y. 510 00:24:24,460 --> 00:24:27,370 And by the way, notice what this thing tells me? 511 00:24:27,370 --> 00:24:31,900 This tells me-- see, what are the basic lines in the xy 512 00:24:31,900 --> 00:24:35,530 plane that aligns x equal to constant, y equal to constant? 513 00:24:35,530 --> 00:24:37,390 Notice that, if x is a constant, this 514 00:24:37,390 --> 00:24:39,460 says that u squared plus v squared 515 00:24:39,460 --> 00:24:41,830 equals e to a constant power. 516 00:24:41,830 --> 00:24:44,380 That's u squared plus v square is a constant. 517 00:24:44,380 --> 00:24:46,790 That's a circle, centered at the origin. 518 00:24:46,790 --> 00:24:50,050 In other words, the lines x equal to constant map 519 00:24:50,050 --> 00:24:52,720 into concentric circles. 520 00:24:52,720 --> 00:24:55,120 On the other hand, if y is a constant, 521 00:24:55,120 --> 00:24:57,850 this says that v over u is a constant. 522 00:24:57,850 --> 00:25:01,060 In the uv plane, v over u is a constant 523 00:25:01,060 --> 00:25:04,420 is a straight line that goes through the origin. 524 00:25:04,420 --> 00:25:07,540 See, v is a constant times u. 525 00:25:07,540 --> 00:25:09,310 Now, what that means pictorially-- 526 00:25:09,310 --> 00:25:11,800 let's see if I have some room to squeeze this in here. 527 00:25:11,800 --> 00:25:14,530 What this means pictorially is that this kind 528 00:25:14,530 --> 00:25:19,240 of a network in the xy plane-- 529 00:25:19,240 --> 00:25:24,220 this kind of a network in the xy plane is mapped into what? 530 00:25:24,220 --> 00:25:27,010 This network is mapped into a what? 531 00:25:27,010 --> 00:25:30,760 The vertical lines go into concentric circles. 532 00:25:30,760 --> 00:25:36,730 And the horizontal lines go into lines through the origin here. 533 00:25:39,880 --> 00:25:43,450 Notice, by the way, that every line through the origin 534 00:25:43,450 --> 00:25:46,570 meets the circle at right angles. 535 00:25:46,570 --> 00:25:49,150 Notice that these lines were at right angles here. 536 00:25:49,150 --> 00:25:51,490 Notice the conformal property here. 537 00:25:51,490 --> 00:25:54,760 In other words, lines that mean at right angle in the xy plane 538 00:25:54,760 --> 00:25:58,610 must meet at right angles in the uv plane. 539 00:25:58,610 --> 00:26:00,370 And let me separate this out from anything 540 00:26:00,370 --> 00:26:02,680 else that I've done so that we don't get too mixed 541 00:26:02,680 --> 00:26:03,430 up with this. 542 00:26:03,430 --> 00:26:07,030 By the way, one of the ways that this is used in application 543 00:26:07,030 --> 00:26:08,860 is with the inverse mapping-- 544 00:26:08,860 --> 00:26:10,990 that, in certain cases where we have 545 00:26:10,990 --> 00:26:14,620 to deal with regions that have this kind of a shape, 546 00:26:14,620 --> 00:26:16,900 we back map this region. 547 00:26:16,900 --> 00:26:18,060 And what will it go into? 548 00:26:18,060 --> 00:26:21,860 It will go into a rectangle here. 549 00:26:21,860 --> 00:26:24,700 And for example, in solving Laplace's equation, 550 00:26:24,700 --> 00:26:28,180 it might be a lot easier to solve Laplace's equation 551 00:26:28,180 --> 00:26:31,840 on the boundary of a rectangle than on the boundary 552 00:26:31,840 --> 00:26:34,450 of a region of this particular type. 553 00:26:34,450 --> 00:26:36,850 Again, I don't want to go into too much detail 554 00:26:36,850 --> 00:26:39,790 here, because it obscures what the main overview is-- 555 00:26:39,790 --> 00:26:43,805 namely, every time we invent an analytic function like e 556 00:26:43,805 --> 00:26:47,410 to the z, sine z, cosh z, et cetera, 557 00:26:47,410 --> 00:26:50,680 we have invented a new conformal mapping. 558 00:26:50,680 --> 00:26:52,690 Whether that conformal mapping will help us 559 00:26:52,690 --> 00:26:56,240 in a particular application or not depends on the application. 560 00:26:56,240 --> 00:27:01,540 But what the meaning of it is is independent of any application. 561 00:27:01,540 --> 00:27:04,500 Now let's take one more important reason. 562 00:27:04,500 --> 00:27:06,610 You see, what I'm afraid of now is 563 00:27:06,610 --> 00:27:09,940 I'm giving you the impression that complex numbers have value 564 00:27:09,940 --> 00:27:12,370 only if you're doing conformal mappings. 565 00:27:12,370 --> 00:27:15,970 You see, complex numbers give you 566 00:27:15,970 --> 00:27:20,130 a degree of consciousness, so to speak, above the real numbers. 567 00:27:20,130 --> 00:27:22,380 Let me give you an example of another problem that 568 00:27:22,380 --> 00:27:25,800 used to hang me up when I was an undergraduate student learning 569 00:27:25,800 --> 00:27:26,310 calculus. 570 00:27:26,310 --> 00:27:30,750 I'm going to apply complex value results to real series now. 571 00:27:30,750 --> 00:27:34,320 Remember the geometric series 1 over 1 minus u 572 00:27:34,320 --> 00:27:36,750 is 1 plus u plus u squared, et cetera-- in other words, 573 00:27:36,750 --> 00:27:39,510 summation n goes from 0 to infinity, u to the n. 574 00:27:39,510 --> 00:27:41,520 And that converges for the absolute value 575 00:27:41,520 --> 00:27:43,360 of u less than 1. 576 00:27:43,360 --> 00:27:47,950 Let's suppose now I replace u by minus x squared. 577 00:27:47,950 --> 00:27:51,780 If I do that, I get 1 over 1 plus x squared 578 00:27:51,780 --> 00:27:54,330 is 1 minus x squared plus x to the fourth 579 00:27:54,330 --> 00:27:55,890 minus x to the sixth, et cetera. 580 00:27:55,890 --> 00:27:59,380 In other words, summation n goes from 0 to infinity. 581 00:27:59,380 --> 00:28:01,410 I want my sines to alternate with the first one 582 00:28:01,410 --> 00:28:02,590 being positive. 583 00:28:02,590 --> 00:28:05,580 So I put in minus 1 to the n, x to the 2n. 584 00:28:05,580 --> 00:28:07,470 And that converges for the absolute value 585 00:28:07,470 --> 00:28:09,520 of x less than 1. 586 00:28:09,520 --> 00:28:12,160 Now let me just focus my attention on a question for you 587 00:28:12,160 --> 00:28:12,660 here. 588 00:28:12,660 --> 00:28:14,160 And I'll come back to this. 589 00:28:14,160 --> 00:28:15,790 I want to come back here in a second. 590 00:28:15,790 --> 00:28:20,390 But the idea is, what goes wrong when x equals plus or minus 1? 591 00:28:20,390 --> 00:28:22,740 You see, let me show you what I mean by that. 592 00:28:22,740 --> 00:28:25,830 When I saw this series for the first time, and somebody says, 593 00:28:25,830 --> 00:28:28,890 look at, you got to be careful when the absolute value of u 594 00:28:28,890 --> 00:28:30,690 is equal to 1-- in other words, be careful 595 00:28:30,690 --> 00:28:33,280 when u equals 1 or minus 1. 596 00:28:33,280 --> 00:28:35,220 Well, minus 1 didn't bother me that much. 597 00:28:35,220 --> 00:28:38,340 But 1, I could see right away what went wrong. 598 00:28:38,340 --> 00:28:41,430 Namely, when u is 1 here, 1 over 1 minus u 599 00:28:41,430 --> 00:28:43,530 is 1 over 0, which is undefined. 600 00:28:43,530 --> 00:28:45,990 And I expect trouble over here. 601 00:28:45,990 --> 00:28:48,390 On the other hand, when the absolute value of x 602 00:28:48,390 --> 00:28:51,390 equals 1 here, that means that x squared is 1. 603 00:28:51,390 --> 00:28:54,590 I mean, whether x is 1 or minus 1, x squared is 1. 604 00:28:54,590 --> 00:28:56,760 Look at-- nothing goes wrong over here. 605 00:28:56,760 --> 00:28:59,690 1 over 1 plus x squared is perfectly well 606 00:28:59,690 --> 00:29:03,060 defined at x equals 1. 607 00:29:03,060 --> 00:29:05,670 I would expect-- because my upbringing was such 608 00:29:05,670 --> 00:29:08,850 that I always thought that the bad points occurred 609 00:29:08,850 --> 00:29:11,530 where the denominator blew up. 610 00:29:11,530 --> 00:29:13,980 But see, this denominator never blows up, 611 00:29:13,980 --> 00:29:19,410 because 1 plus x squared is always at least as big as 1. 612 00:29:19,410 --> 00:29:20,890 Provided what, of course? 613 00:29:20,890 --> 00:29:23,100 That x is real, because a square of a real number 614 00:29:23,100 --> 00:29:24,030 can't be negative. 615 00:29:24,030 --> 00:29:25,320 That's the hint, by the way. 616 00:29:25,320 --> 00:29:28,800 The square of a complex number could certainly be negative. 617 00:29:28,800 --> 00:29:30,330 And here's what I'm saying. 618 00:29:30,330 --> 00:29:34,260 If I now take this real-valued series 619 00:29:34,260 --> 00:29:37,830 and convert it to a complex-valued series 620 00:29:37,830 --> 00:29:44,610 by replacing x by z using the same ratio test, et cetera, 621 00:29:44,610 --> 00:29:49,260 that we did before, I can prove that this infinite series, 622 00:29:49,260 --> 00:29:52,470 this power series, converges to 1 over 1 623 00:29:52,470 --> 00:29:57,000 plus z squared provided the absolute value of z 624 00:29:57,000 --> 00:29:57,780 is less than 1. 625 00:29:57,780 --> 00:29:59,280 By the way, what does it mean to say 626 00:29:59,280 --> 00:30:01,170 that the absolute value of z is less than 1? 627 00:30:01,170 --> 00:30:03,870 It means that z is less than 1 unit from the origin. 628 00:30:03,870 --> 00:30:05,970 It means that, in the Argand diagram, 629 00:30:05,970 --> 00:30:09,600 z is inside the circle, centered at the origin, with a radius 630 00:30:09,600 --> 00:30:10,620 equal to 1. 631 00:30:10,620 --> 00:30:12,240 But here's the key point. 632 00:30:12,240 --> 00:30:15,060 When I look at this, I'm not at all upset 633 00:30:15,060 --> 00:30:17,580 that something can go wrong on this circle. 634 00:30:17,580 --> 00:30:21,690 Namely, 1 over 1 plus z squared factors into 1 635 00:30:21,690 --> 00:30:24,720 over z plus i times z minus i. 636 00:30:24,720 --> 00:30:27,720 And certainly, I am leery of letting 637 00:30:27,720 --> 00:30:31,170 z either equal minus i or i, because it's going 638 00:30:31,170 --> 00:30:33,060 to make my denominator vanish. 639 00:30:33,060 --> 00:30:35,790 And even in the complex numbers, as you recall from our first 640 00:30:35,790 --> 00:30:39,600 lecture on complex numbers, you cannot divide by 0. 641 00:30:39,600 --> 00:30:41,910 In other words, at a glance, I recognize 642 00:30:41,910 --> 00:30:45,570 that z equals plus or minus i is trouble. 643 00:30:45,570 --> 00:30:48,870 In other words, going to the Argand diagram in this picture, 644 00:30:48,870 --> 00:30:52,770 I know I'm in trouble at these two points on the circle, 645 00:30:52,770 --> 00:30:54,460 absolute value of z equals 1. 646 00:30:54,460 --> 00:30:55,980 Remember, this is a circle. 647 00:30:55,980 --> 00:30:59,160 It's the locus of all points in the Argand diagram, 648 00:30:59,160 --> 00:31:00,900 one unit from the origin. 649 00:31:00,900 --> 00:31:03,950 Now, remember what we said about the circle of convergence-- 650 00:31:03,950 --> 00:31:07,140 that that set S, where the power series converges-- 651 00:31:07,140 --> 00:31:10,500 there is a circle such that, inside the circle, 652 00:31:10,500 --> 00:31:12,360 the thing always converges. 653 00:31:12,360 --> 00:31:14,520 Outside, it always diverges. 654 00:31:14,520 --> 00:31:17,850 Consequently, the fact that I have two bad spots over here 655 00:31:17,850 --> 00:31:22,740 tells me that I cannot go beyond the circle in talking about 656 00:31:22,740 --> 00:31:26,580 what happened with the power series expansion for 1 over 1 657 00:31:26,580 --> 00:31:27,970 plus z squared. 658 00:31:27,970 --> 00:31:31,080 Notice that all I require is that something go wrong 659 00:31:31,080 --> 00:31:32,160 on the circle. 660 00:31:32,160 --> 00:31:34,800 There are infinitely many points on this circle. 661 00:31:34,800 --> 00:31:38,700 And the likelihood that the point at which things went bad 662 00:31:38,700 --> 00:31:40,380 happened to be at the point where 663 00:31:40,380 --> 00:31:43,860 the circle crossed the x-axis is particularly small. 664 00:31:43,860 --> 00:31:46,200 In other words, speaking from another point of view, 665 00:31:46,200 --> 00:31:50,940 if I have the good fortune to be able to be standing 666 00:31:50,940 --> 00:31:54,130 in the complex number world looking down at this circle, 667 00:31:54,130 --> 00:31:56,190 I see what went wrong. 668 00:31:56,190 --> 00:31:59,700 If my perspective is limited to the x-axis, 669 00:31:59,700 --> 00:32:01,810 I can't see anything that went wrong. 670 00:32:01,810 --> 00:32:04,200 And I can't understand why I can't 671 00:32:04,200 --> 00:32:05,910 expand this circle further. 672 00:32:05,910 --> 00:32:08,985 In other words, the bad spots are on the magnitude 673 00:32:08,985 --> 00:32:14,220 of z equals 1, but not at the point z equals 1 674 00:32:14,220 --> 00:32:16,020 or z equals minus 1. 675 00:32:16,020 --> 00:32:17,340 These were good points. 676 00:32:17,340 --> 00:32:18,750 See, these were good points. 677 00:32:18,750 --> 00:32:20,490 These were the bad ones. 678 00:32:20,490 --> 00:32:22,500 But all you needed was one bad apple 679 00:32:22,500 --> 00:32:24,650 to spoil the whole circle, so to speak. 680 00:32:24,650 --> 00:32:29,190 At any rate, I think that's enough to make my point-- 681 00:32:29,190 --> 00:32:33,780 the point being that, by studying complex series, 682 00:32:33,780 --> 00:32:35,170 we do two things. 683 00:32:35,170 --> 00:32:38,610 One is we get a larger hold on the set of analytic functions, 684 00:32:38,610 --> 00:32:41,220 which help us in things like conformal mappings 685 00:32:41,220 --> 00:32:43,670 and other things, which we haven't gone into. 686 00:32:43,670 --> 00:32:47,010 And secondly, it gives us a new perspective 687 00:32:47,010 --> 00:32:50,700 at series involving real power series, 688 00:32:50,700 --> 00:32:53,280 because we now have an extra dimension from which 689 00:32:53,280 --> 00:32:54,840 to view the power series. 690 00:32:54,840 --> 00:32:58,350 At any rate, we'll talk about other aspects 691 00:32:58,350 --> 00:33:02,400 of complex variables-- namely, integration-- next time. 692 00:33:02,400 --> 00:33:06,030 For the time being, do the exercises in this unit. 693 00:33:06,030 --> 00:33:07,820 And until next time, goodbye. 694 00:33:15,170 --> 00:33:17,570 Funding for the publication of this video 695 00:33:17,570 --> 00:33:22,450 was provided by the Gabriella and Paul Rosenbaum Foundation. 696 00:33:22,450 --> 00:33:26,590 Help OCW continue to provide free and open access to MIT 697 00:33:26,590 --> 00:33:32,025 courses by making a donation at ocw.mit.edu/donate.