00:00 Intro

00:25 Exponential Function

03:27 Logarithm Function

05:00 Polynomials

05:33 Power Series

08:49 Credits

1 00:00:00,430 –> 00:00:02,329 Hello and welcome back to

2 00:00:02,339 –> 00:00:03,880 real analysis.

3 00:00:04,570 –> 00:00:05,969 And first many, many thanks

4 00:00:05,980 –> 00:00:07,179 to all the nice supporters

5 00:00:07,190 –> 00:00:08,670 on steady or paypal.

6 00:00:09,250 –> 00:00:10,630 Now, in this video today,

7 00:00:10,640 –> 00:00:11,840 you will get to know some

8 00:00:11,850 –> 00:00:13,180 important continuous

9 00:00:13,189 –> 00:00:13,869 functions.

10 00:00:14,630 –> 00:00:16,180 Some of them are so important

11 00:00:16,190 –> 00:00:17,319 that they are often called

12 00:00:17,329 –> 00:00:18,780 elementary functions.

13 00:00:19,489 –> 00:00:21,159 Indeed, all of them we will

14 00:00:21,170 –> 00:00:22,620 consider again later when

15 00:00:22,629 –> 00:00:24,040 we differentiate and

16 00:00:24,049 –> 00:00:24,899 integrate.

17 00:00:25,440 –> 00:00:26,579 Hence, let’s immediately

18 00:00:26,590 –> 00:00:28,000 start with the most important

19 00:00:28,010 –> 00:00:28,420 one.

20 00:00:28,430 –> 00:00:29,879 The exponential function

21 00:00:30,540 –> 00:00:31,709 we shorten the name with

22 00:00:31,719 –> 00:00:33,669 X and the domain is the whole

23 00:00:33,680 –> 00:00:34,700 real number line.

24 00:00:35,119 –> 00:00:36,369 And we have already learned

25 00:00:36,380 –> 00:00:37,759 in the former video that

26 00:00:37,770 –> 00:00:39,619 it is defined by a series,

27 00:00:40,400 –> 00:00:42,029 namely we have X to the power

28 00:00:42,040 –> 00:00:43,540 K divided by K

29 00:00:43,549 –> 00:00:44,580 factorial.

30 00:00:45,290 –> 00:00:46,669 And now you might already

31 00:00:46,680 –> 00:00:48,509 know at the point X is equal

32 00:00:48,520 –> 00:00:49,810 to one, we find the

33 00:00:49,819 –> 00:00:51,540 so-called Euler number.

34 00:00:52,330 –> 00:00:53,830 Also it’s possible to show

35 00:00:53,840 –> 00:00:55,389 that this number can be defined

36 00:00:55,400 –> 00:00:56,630 by another sequence

37 00:00:57,380 –> 00:00:59,000 namely by the limit one

38 00:00:59,009 –> 00:01:00,599 plus one over N to the

39 00:01:00,610 –> 00:01:02,430 power N which is

40 00:01:02,439 –> 00:01:02,939 roughly

41 00:01:02,950 –> 00:01:04,910 2.718.

42 00:01:05,559 –> 00:01:07,040 Now this exponential function

43 00:01:07,050 –> 00:01:08,779 shouldn’t be new to you because

44 00:01:08,790 –> 00:01:10,099 we have already discussed

45 00:01:10,110 –> 00:01:11,500 it in part 22

46 00:01:12,099 –> 00:01:13,260 there, we have shown the

47 00:01:13,269 –> 00:01:14,860 fundamental multiplicative

48 00:01:14,870 –> 00:01:16,580 identity for the exponential

49 00:01:16,589 –> 00:01:18,000 function which

50 00:01:18,010 –> 00:01:19,339 translates the addition

51 00:01:19,349 –> 00:01:20,489 inside into a

52 00:01:20,500 –> 00:01:22,120 multiplication outside.

53 00:01:22,580 –> 00:01:23,879 And because we have this

54 00:01:23,889 –> 00:01:25,559 property, we can immediately

55 00:01:25,569 –> 00:01:26,680 explain the name

56 00:01:26,690 –> 00:01:28,559 exponential for this function.

57 00:01:29,160 –> 00:01:30,989 For example, if we have X

58 00:01:31,000 –> 00:01:32,870 of two, we can

59 00:01:32,879 –> 00:01:34,669 write this as X of one

60 00:01:34,680 –> 00:01:35,650 plus one

61 00:01:36,430 –> 00:01:37,870 which is by our fundamental

62 00:01:37,879 –> 00:01:39,489 multiplicative identity here

63 00:01:39,500 –> 00:01:41,300 exactly X of one

64 00:01:41,309 –> 00:01:42,580 times X of one.

65 00:01:43,110 –> 00:01:44,419 So it’s Euler number

66 00:01:44,430 –> 00:01:45,209 squared.

67 00:01:45,849 –> 00:01:47,790 So you see the left here

68 00:01:47,800 –> 00:01:49,400 and the right hand side are

69 00:01:49,410 –> 00:01:49,989 the same.

70 00:01:50,769 –> 00:01:52,209 And indeed, we can show that

71 00:01:52,220 –> 00:01:53,699 in general, the exponential

72 00:01:53,709 –> 00:01:55,430 function X of X is

73 00:01:55,440 –> 00:01:57,239 exactly Euler number to the

74 00:01:57,250 –> 00:01:57,989 power X.

75 00:01:58,739 –> 00:01:59,959 Therefore, the name makes

76 00:01:59,970 –> 00:02:01,529 sense because we can rewrite

77 00:02:01,540 –> 00:02:03,019 a function such that we have

78 00:02:03,029 –> 00:02:04,139 the exponent here.

79 00:02:04,849 –> 00:02:05,980 And of course, we can show

80 00:02:05,989 –> 00:02:07,750 even more properties of this

81 00:02:07,760 –> 00:02:09,029 important function here.

82 00:02:09,839 –> 00:02:10,860 The first part might not

83 00:02:10,869 –> 00:02:11,779 surprise you.

84 00:02:11,789 –> 00:02:13,029 The exponential function

85 00:02:13,039 –> 00:02:14,619 is a continuous function.

86 00:02:15,119 –> 00:02:16,820 So it’s continuous at all

87 00:02:16,830 –> 00:02:17,960 points in R.

88 00:02:18,710 –> 00:02:20,149 Then the next property is

89 00:02:20,160 –> 00:02:21,580 that this function is at

90 00:02:21,589 –> 00:02:23,289 all points increasing

91 00:02:23,570 –> 00:02:24,750 more concretely, we would

92 00:02:24,759 –> 00:02:26,300 say the exponential function

93 00:02:26,309 –> 00:02:27,960 is strictly monotonically

94 00:02:27,970 –> 00:02:29,929 increasing, which simply

95 00:02:29,940 –> 00:02:31,699 means if we look at two points

96 00:02:31,710 –> 00:02:33,500 X and Y where Y is

97 00:02:33,509 –> 00:02:35,369 greater than X, then also

98 00:02:35,380 –> 00:02:36,990 the values fulfill the same

99 00:02:37,000 –> 00:02:37,889 inequality.

100 00:02:38,139 –> 00:02:40,029 So X of Y is greater

101 00:02:40,039 –> 00:02:41,229 than X of X.

102 00:02:41,800 –> 00:02:43,070 In addition, we can also

103 00:02:43,080 –> 00:02:44,610 say what happens in the limits

104 00:02:44,619 –> 00:02:46,559 X to infinity and X

105 00:02:46,570 –> 00:02:47,929 to minus infinity

106 00:02:48,100 –> 00:02:49,889 for plus infinity, we

107 00:02:49,899 –> 00:02:51,369 exceed all bounds.

108 00:02:51,380 –> 00:02:52,960 Therefore, we go to plus

109 00:02:52,970 –> 00:02:53,649 infinity.

110 00:02:54,410 –> 00:02:56,119 However, for minus infinity,

111 00:02:56,130 –> 00:02:57,360 it’s different because we

112 00:02:57,369 –> 00:02:59,160 stay positive the whole time

113 00:02:59,169 –> 00:03:00,960 we reach zero in the limit.

114 00:03:01,429 –> 00:03:01,800 OK.

115 00:03:01,809 –> 00:03:03,110 So these are the important

116 00:03:03,119 –> 00:03:04,399 properties you really should

117 00:03:04,410 –> 00:03:06,270 remember for the exponential

118 00:03:06,279 –> 00:03:06,800 function.

119 00:03:07,369 –> 00:03:08,919 Now, one important conclusion

120 00:03:08,929 –> 00:03:10,320 here is what we will need

121 00:03:10,330 –> 00:03:11,669 for the next example.

122 00:03:12,250 –> 00:03:13,720 Namely now you are able to

123 00:03:13,729 –> 00:03:15,380 show if we change the codomain

124 00:03:15,389 –> 00:03:16,619 to zero

125 00:03:16,630 –> 00:03:17,509 infinity.

126 00:03:17,929 –> 00:03:19,830 So the interval of the positive

127 00:03:19,839 –> 00:03:21,679 numbers, then we

128 00:03:21,690 –> 00:03:23,119 get a bijective map.

129 00:03:23,740 –> 00:03:25,110 Indeed, that’s not hard to

130 00:03:25,119 –> 00:03:26,630 show when you know all the

131 00:03:26,639 –> 00:03:27,410 other things here.

132 00:03:28,089 –> 00:03:29,509 And of course, now you already

133 00:03:29,520 –> 00:03:31,490 know a bijective map has

134 00:03:31,500 –> 00:03:32,589 an inverse map.

135 00:03:33,050 –> 00:03:34,339 And this inverse is what

136 00:03:34,350 –> 00:03:36,149 we usually call the logarithm

137 00:03:36,160 –> 00:03:36,830 function.

138 00:03:37,240 –> 00:03:38,770 So you see this is a simple

139 00:03:38,779 –> 00:03:40,259 definition and then we can

140 00:03:40,270 –> 00:03:41,600 ask the question, what are

141 00:03:41,610 –> 00:03:43,190 the properties of the logarithm

142 00:03:43,199 –> 00:03:43,669 function?

143 00:03:44,300 –> 00:03:46,050 And of course, not so surprising,

144 00:03:46,059 –> 00:03:47,589 we can deduce a lot of

145 00:03:47,600 –> 00:03:49,360 properties from the exponential

146 00:03:49,369 –> 00:03:49,880 function.

147 00:03:50,179 –> 00:03:51,309 So maybe let’s look at the

148 00:03:51,320 –> 00:03:52,960 graphs to see what we already

149 00:03:52,970 –> 00:03:53,470 have.

150 00:03:54,350 –> 00:03:55,690 So in blue, you see the graph

151 00:03:55,699 –> 00:03:56,990 of the exponential function

152 00:03:57,050 –> 00:03:58,809 and then we reflect it where

153 00:03:58,820 –> 00:04:00,169 the mirror line is this

154 00:04:00,179 –> 00:04:01,690 diagonal here.

155 00:04:01,699 –> 00:04:03,449 Please recall this reflecting

156 00:04:03,460 –> 00:04:04,500 follows immediately by the

157 00:04:04,509 –> 00:04:06,089 definition of the graph and

158 00:04:06,100 –> 00:04:07,089 the inverse function

159 00:04:07,910 –> 00:04:08,250 HC.

160 00:04:08,259 –> 00:04:09,580 And we, you find the graph

161 00:04:09,589 –> 00:04:11,130 of the logarithm function.

162 00:04:11,550 –> 00:04:12,960 Now what we see is that the

163 00:04:12,970 –> 00:04:14,589 logarithm function is also

164 00:04:14,600 –> 00:04:15,990 strictly monotonically

165 00:04:16,000 –> 00:04:17,450 increasing and it should

166 00:04:17,459 –> 00:04:19,029 still be continuous.

167 00:04:19,559 –> 00:04:20,890 Indeed showing these two

168 00:04:20,899 –> 00:04:22,890 things is a very good exercise.

169 00:04:23,059 –> 00:04:24,700 First, you show the increasing

170 00:04:24,709 –> 00:04:25,910 part and then you use the

171 00:04:25,920 –> 00:04:27,519 epsilon delta criterion to

172 00:04:27,529 –> 00:04:28,799 show the continuity.

173 00:04:29,359 –> 00:04:29,859 OK.

174 00:04:29,869 –> 00:04:31,239 However, the most important

175 00:04:31,250 –> 00:04:32,459 property for the logarithm

176 00:04:32,470 –> 00:04:34,049 function follows immediately

177 00:04:34,059 –> 00:04:35,899 from the fundamental multiplicative

178 00:04:35,910 –> 00:04:37,690 identity for the exponential

179 00:04:37,700 –> 00:04:38,200 function.

180 00:04:38,720 –> 00:04:39,000 OK.

181 00:04:39,010 –> 00:04:40,320 Now, because the logarithm

182 00:04:40,329 –> 00:04:42,160 function is the inverse function,

183 00:04:42,170 –> 00:04:43,839 this identity goes the other

184 00:04:43,850 –> 00:04:44,720 way around.

185 00:04:45,309 –> 00:04:47,179 So the product inside gets

186 00:04:47,190 –> 00:04:48,739 translated to the sum

187 00:04:48,750 –> 00:04:49,500 outside.

188 00:04:50,160 –> 00:04:50,519 OK?

189 00:04:50,529 –> 00:04:52,200 I can tell you this identity

190 00:04:52,209 –> 00:04:53,380 is very important.

191 00:04:53,390 –> 00:04:54,760 So please remember it.

192 00:04:55,309 –> 00:04:56,690 Now, with this, we go to

193 00:04:56,700 –> 00:04:58,589 our next examples which are

194 00:04:58,600 –> 00:04:59,720 polynomials

195 00:05:00,350 –> 00:05:00,839 for them.

196 00:05:00,850 –> 00:05:01,829 I don’t have to tell you

197 00:05:01,839 –> 00:05:03,459 a lot because you already

198 00:05:03,470 –> 00:05:04,079 know them.

199 00:05:04,760 –> 00:05:05,959 The important part here is

200 00:05:05,970 –> 00:05:07,579 the domain of a polynomial

201 00:05:07,589 –> 00:05:08,970 is the whole real number

202 00:05:08,980 –> 00:05:10,910 line and defined as it by

203 00:05:10,920 –> 00:05:12,910 giving finitely many coefficients

204 00:05:12,920 –> 00:05:13,279 A.

205 00:05:13,950 –> 00:05:15,049 And in the case that the

206 00:05:15,059 –> 00:05:16,970 last one am here is

207 00:05:16,980 –> 00:05:17,950 nonzero.

208 00:05:17,959 –> 00:05:19,570 We say that the polynomial

209 00:05:19,579 –> 00:05:21,519 has to create M as a

210 00:05:21,529 –> 00:05:22,839 reminder, we have already

211 00:05:22,850 –> 00:05:24,239 showed that the polynomials

212 00:05:24,250 –> 00:05:25,970 are continuous functions.

213 00:05:26,549 –> 00:05:27,929 So nothing new here.

214 00:05:27,940 –> 00:05:29,890 But we need to know the polynomials

215 00:05:29,899 –> 00:05:31,450 to define the next class

216 00:05:31,459 –> 00:05:32,529 of examples

217 00:05:33,350 –> 00:05:33,660 here.

218 00:05:33,670 –> 00:05:35,029 Please recall for a

219 00:05:35,040 –> 00:05:36,600 polynomial and important

220 00:05:36,609 –> 00:05:38,309 ingredient was that we have

221 00:05:38,320 –> 00:05:39,809 a highest power M here.

222 00:05:40,279 –> 00:05:41,600 If we don’t want to have

223 00:05:41,609 –> 00:05:42,910 that, we get something we

224 00:05:42,920 –> 00:05:44,549 call a power series.

225 00:05:44,970 –> 00:05:46,660 In fact, this is a very general

226 00:05:46,670 –> 00:05:48,209 term where we already know

227 00:05:48,220 –> 00:05:49,220 one example,

228 00:05:49,790 –> 00:05:51,630 namely the exponential function

229 00:05:51,640 –> 00:05:53,540 from above was defined using

230 00:05:53,549 –> 00:05:54,640 a power series.

231 00:05:55,149 –> 00:05:56,880 Now, in general a power series

232 00:05:56,890 –> 00:05:58,540 does not need to be defined

233 00:05:58,549 –> 00:05:59,730 on the whole real number

234 00:05:59,739 –> 00:06:00,179 line.

235 00:06:00,589 –> 00:06:02,149 So we could have a smaller

236 00:06:02,160 –> 00:06:03,109 domain D.

237 00:06:03,850 –> 00:06:05,320 However, otherwise it looks

238 00:06:05,329 –> 00:06:06,750 like a polynomial where we

239 00:06:06,760 –> 00:06:08,579 have coefficients A K here.

240 00:06:08,609 –> 00:06:10,510 But now we could have infinitely

241 00:06:10,519 –> 00:06:11,040 many.

242 00:06:11,559 –> 00:06:12,880 In other words, for each

243 00:06:12,890 –> 00:06:14,799 sequence A K, we get out

244 00:06:14,809 –> 00:06:16,239 a function we call a power

245 00:06:16,250 –> 00:06:16,880 series.

246 00:06:17,570 –> 00:06:18,829 In this case, the domain

247 00:06:18,839 –> 00:06:20,399 of definition is given by

248 00:06:20,410 –> 00:06:21,940 all the points X in

249 00:06:21,950 –> 00:06:23,899 R that satisfy

250 00:06:23,910 –> 00:06:25,589 that this series is indeed

251 00:06:25,600 –> 00:06:26,809 a convergent one.

252 00:06:27,649 –> 00:06:28,790 And of course, you already

253 00:06:28,799 –> 00:06:30,269 know a lot of criteria you

254 00:06:30,279 –> 00:06:31,709 can use here to check for

255 00:06:31,720 –> 00:06:32,510 convergence.

256 00:06:33,040 –> 00:06:33,279 OK.

257 00:06:33,290 –> 00:06:34,489 Let’s immediately discuss

258 00:06:34,500 –> 00:06:36,329 a very famous example here

259 00:06:36,339 –> 00:06:37,899 which means we fix a special

260 00:06:37,910 –> 00:06:38,420 sequence.

261 00:06:38,429 –> 00:06:40,260 A K let’s start a

262 00:06:40,269 –> 00:06:41,549 sequence with the zeroth

263 00:06:41,559 –> 00:06:42,839 member which should be

264 00:06:42,850 –> 00:06:43,570 zero.

265 00:06:44,109 –> 00:06:45,679 And then I want one, but

266 00:06:45,690 –> 00:06:47,670 let’s write it as one divided

267 00:06:47,679 –> 00:06:49,010 by one factorial,

268 00:06:49,700 –> 00:06:51,339 then afterwards comes zero

269 00:06:51,350 –> 00:06:52,910 again and then

270 00:06:52,920 –> 00:06:54,720 minus one divided by

271 00:06:54,730 –> 00:06:56,230 three factorial.

272 00:06:56,809 –> 00:06:58,109 Then not surprising we have

273 00:06:58,119 –> 00:06:59,820 zero again and then one

274 00:06:59,829 –> 00:07:01,820 divided by five factorial,

275 00:07:02,230 –> 00:07:03,809 then zero and then

276 00:07:03,820 –> 00:07:05,570 minus one divided by

277 00:07:05,579 –> 00:07:06,820 seven factorial.

278 00:07:07,540 –> 00:07:07,829 OK.

279 00:07:07,839 –> 00:07:09,109 With this, I think you see

280 00:07:09,119 –> 00:07:10,440 what the whole wall is and

281 00:07:10,450 –> 00:07:11,829 that you can continue it.

282 00:07:12,679 –> 00:07:14,279 And now this sequence gives

283 00:07:14,290 –> 00:07:16,070 us a power series we usually

284 00:07:16,079 –> 00:07:17,899 call sine of X.

285 00:07:18,299 –> 00:07:19,779 Also, it’s not hard to check

286 00:07:19,790 –> 00:07:21,019 that the whole real number

287 00:07:21,029 –> 00:07:22,600 line is the domain D

288 00:07:23,019 –> 00:07:24,899 indeed, a lot of common functions

289 00:07:24,910 –> 00:07:26,339 can be defined with such

290 00:07:26,350 –> 00:07:27,420 a power series.

291 00:07:27,940 –> 00:07:29,299 And for this reason, I give

292 00:07:29,309 –> 00:07:31,119 you here a general result.

293 00:07:31,440 –> 00:07:33,160 Now, for any power series,

294 00:07:33,170 –> 00:07:34,529 given the coefficients A

295 00:07:34,540 –> 00:07:36,079 K, you find the

296 00:07:36,089 –> 00:07:38,000 maximum radius R

297 00:07:38,660 –> 00:07:40,500 where also infinity as a

298 00:07:40,510 –> 00:07:41,959 symbol is possible,

299 00:07:42,609 –> 00:07:43,720 which indeed would be the

300 00:07:43,730 –> 00:07:45,170 best case scenario.

301 00:07:45,700 –> 00:07:47,019 Now, this maximum radius

302 00:07:47,029 –> 00:07:48,910 fulfills that the open interval

303 00:07:48,920 –> 00:07:50,890 from minus R to R lies

304 00:07:50,899 –> 00:07:52,350 completely in the domain

305 00:07:52,359 –> 00:07:52,760 D.

306 00:07:53,329 –> 00:07:54,829 Now I can also tell you we

307 00:07:54,839 –> 00:07:55,989 immediately get that the

308 00:07:56,000 –> 00:07:57,700 power series is a continuous

309 00:07:57,709 –> 00:07:59,510 function on this interval.

310 00:07:59,670 –> 00:08:01,100 So in the best case scenario

311 00:08:01,109 –> 00:08:02,640 like for the sine function,

312 00:08:02,649 –> 00:08:04,239 we have a continuous function

313 00:08:04,250 –> 00:08:05,549 defined on the whole real

314 00:08:05,559 –> 00:08:06,320 number line.

315 00:08:06,910 –> 00:08:08,649 Now even better for the theorem,

316 00:08:08,660 –> 00:08:10,570 we have a formula to calculate

317 00:08:10,579 –> 00:08:11,690 this number R.

318 00:08:12,170 –> 00:08:13,609 Indeed this comes out of

319 00:08:13,619 –> 00:08:15,579 the root criterion for the.

320 00:08:15,730 –> 00:08:16,989 Therefore, we can calculate

321 00:08:17,000 –> 00:08:18,609 the lim sup of the cave

322 00:08:18,619 –> 00:08:20,309 root of the

323 00:08:20,320 –> 00:08:22,109 absolute value of A K.

324 00:08:22,609 –> 00:08:24,269 And this gives us the number

325 00:08:24,279 –> 00:08:25,549 one over R.

326 00:08:26,119 –> 00:08:27,549 And here in this case for

327 00:08:27,559 –> 00:08:29,230 zero and infinity, we

328 00:08:29,239 –> 00:08:31,089 use this symbolic calculation

329 00:08:31,589 –> 00:08:33,030 using this, we have this

330 00:08:33,039 –> 00:08:34,390 formula in general.

331 00:08:34,960 –> 00:08:35,308 OK.

332 00:08:35,320 –> 00:08:36,789 Now this nice formula is

333 00:08:36,799 –> 00:08:38,409 often called the Cauchy Hadamard

334 00:08:38,679 –> 00:08:39,058 theorem.

335 00:08:39,719 –> 00:08:41,130 With this, I would say we’ve

336 00:08:41,140 –> 00:08:42,558 learned a lot today and we

337 00:08:42,570 –> 00:08:44,140 can start with derivatives

338 00:08:44,150 –> 00:08:45,200 in the next video.

339 00:08:45,780 –> 00:08:46,900 Therefore, I hope I see you

340 00:08:46,909 –> 00:08:48,270 there and have a nice day.

341 00:08:48,400 –> 00:08:49,070 Bye.

Q1: What is not a property of the exponential function $\exp: \mathbb{R} \rightarrow \mathbb{R}$?

A1: $\exp$ is continuous.

A2: $\exp(x+y) = \exp(x) \exp(y)$

A3: $\exp$ is strictly monotonically increasing.

A4: $\exp(2) = e^2$

A5: $\exp: \mathbb{R} \rightarrow [0,\infty)$ is bijective.

Q2: What is not a property of the logarithm function $\log: (0, \infty) \rightarrow \mathbb{R}$?

A1: $\log$ is continuous.

A2: $\log(x+y) = \log(x) \log(y)$

A3: $\log$ is strictly monotonically increasing.

A4: $\log(1) = 0$

A5: $\log$ is bijective.

Q3: Are polynomials continuous?

A1: Yes!

A2: No!

Q4: Which statement is true for each power series $\sum_{k=0}^\infty a_k x^k$ with domain $\mathcal{D}$?

A1: The point $x=0$ lies in $\mathcal{D}$.

A2: The domain is $\mathcal{D} = \mathbb{R}$.