Real and Complex Analysis (Rudin)

Recommendations

Info

PROLOGUE: THE EXPONENTIAL FUNCTION 3 Take t = 2. The terms of the series (7) then decrease in absolute value (except for the first one) and their signs alternate. Hence cos 2 is less than the sum of the first three terms of (7), with t = 2; thus cos 2 < -t. Since cos 0 = 1 and cos is a continuous real function on the real axis, we conclude that there is a smallest positive number to for which cos to = O. We define n = 2to. It follows from (3) and (S} that sin to = ± 1. Since sin' (t) = cos t > 0 on the segment (0, to) and since sin 0 = 0, we have sin to > 0, hence sin to = 1, and therefore (8) e 1tI / 2 = i. (9) It follows that e 1t1 = i 2 = -1, e21t1 = (_1)2 = 1, and then e 21tln = 1 for every integer n. Also, (e) follows immediately: If z = x + iy, x and y real, then e Z = eXe lY ; hence 1 e Z 1 = eX. If e Z = 1, we therefore must have eX = 1, so that x = 0; to prove that yl2n must be an integer, it is enough to show that ely "# 1 if 0 < y < 2n, by (10). Suppose 0 < y < 2n, and (10) e ly / 4 = u + iv (u and v real). (11) Since 0 < yl4 < n12, we have u > 0 and v > O. Also ely = (u + iV)4 = u4 - 6U 2 V 2 + v4 + 4iuv(u 2 - v 2 ). (12) The right side of (12) is real only if u 2 = v 2 ; since u 2 + v 2 = 1, this happens only when u 2 = v 2 = t, and then (12) shows that ely = -1"# 1. This completes the proof of (d). We already know that t-+ e lt maps the real axis into the unit circle. To prove (f), fix w so that 1 wi = 1; we shall show that w = e lt for some real t. Write w = u + iv, u and v real, and suppose first that u ~ 0 and v ~ O. Since u :::;; 1, the definition of n shows that there exists a t, 0 :::;; t :::;; n12, such that cos t = u; then sin 2 t = 1 - u 2 = v 2 , and since sin t ~ 0 if 0:::;; t :::;; n12, we have sin t = v. Thus w = elt. If u < 0 and v ~ 0, the preceding conditions are satisfied by - iw. Hence - iw = e lt for some real t, and w = e l (t+1t/2). Finally, if v < 0, the preceding two cases show that - w = e lt for some real t, hence w = ei(t+1t). This completes the proof of (f). If w "# 0, put IX = wi 1 w I. Then w = 1 wi IX. By (c), there is a real x such that 1 wi = eX. Since 1 IX 1 = 1, (f) shows that IX = ely for some real y. Hence w = ex+ly. This proves (g) and completes the theorem. IIII
4 REAL AND COMPLEX ANALYSIS We shall encounter the integral of (1 + X 2 )-1 over the real line. To evaluate it, put tp(t) = sin tlcos t in (-nI2, nI2). By (6), tp' = 1 + tp2. Hence tp is a monotonically increasing mapping of ( -nI2, n12) onto ( - 00, (0), and we obtain foo ~ = fft/2 tp'(t) :t = f"/2 dt = n. -00 1 + X -ft/2 1 + tp (t) -ft/2
Page 2: REAL AND COMPLEX ANALYSIS
Page 5 and 6: REAL AND COMPLEX ANALYSIS INTERNATI
Page 8 and 9: CONTENTS Preface xiii Prologue: The
Page 10 and 11: CONTENTS ix Chapter 10 Elementary P
Page 12: CONTENTS xi Appendix: Hausdorff's M
Page 15 and 16: xiv PREFACE Experience with the fir
Page 17: 2 REAL AND COMPLEX ANALYSIS (c) The
Page 21 and 22: 6 REAL AND COMPLEX ANALYSIS for eve
Page 23 and 24: 8 REAL AND COMPLEX ANALYSIS If the
Page 25 and 26: 10 REAL AND COMPLEX ANALYSIS PROOF
Page 29 and 30: 14 REAL AND COMPLEX ANALYSIS and P
Page 31 and 32: 16 REAL AND COMPLEX ANAl-YSIS Each
Page 33 and 34: 18 REAL AND COMPLEX ANALYSIS This i
Page 35 and 36: 20 REAL AND COMPLEX ANALYSIS (c) If
Page 37 and 38: 22 REAL AND COMPLEX ANALYSIS Let n-
Page 39 and 40: 24 REAL AND COMPLEX ANALYSIS It now
Page 41 and 42: 26 REAL AND COMPLEX ANALYSIS That (
Page 43 and 44: 28 REAL AND COMPLEX ANALYSIS Thus,
Page 45 and 46: 30 REAL AND COMPLEX ANALYSIS Note t
Page 47 and 48: 32 REAL AND COMPLEX ANALYSIS 5 (a)
Page 49 and 50: 34 REAL AND COMPLEX ANALYSIS The pr
Page 51 and 52: 36 REAL AND COMPLEX ANALYSIS "neigh
Page 53 and 54: 38 REAL AND COMPLEX ANALYSIS (a) Ch
Page 55 and 56: 40 REAL AND COMPLEX ANALYSIS f(x) =
Page 57 and 58: 42 REAL AND COMPLEX ANALYSIS If VI
Page 63 and 64: 48 REAL AND COMPLEX ANALYSIS 2.17 T
Page 65 and 66: SO REAL AND COMPLEX ANALYSIS If E C
Page 67 and 68: 52 REAL AND COMPLEX ANALYSIS To pro
Page 69 and 70:
54 REAL AND COMPLEX ANALYSIS there
Page 71 and 72:
56 REAL AND COMPLEX ANALYSIS except
Page 73 and 74:
58 REAL AND COMPLEX ANALYSIS 3 Let
Page 75 and 76:
60 REAL AND COMPLEX ANALYSIS 22 Sup
Page 77 and 78:
62 REAL AND COMPLEX ANALYSIS PROOF
Page 79 and 80:
Page 81 and 82:
66 REAL AND COMPLEX ANALYSIS and si
Page 83 and 84:
68 REAL AND COMPLEX ANALYSIS Put k
Page 85 and 86:
70 REAL AND COMPLEX ANALYSIS If 1 S
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
CHAPTER FOUR ELEMENTARY HILBERT SPA
Page 93 and 94:
78 REAL AND COMPLEX ANALYSIS (b) If
Page 95 and 96:
80 REAL AND COMPLEX ANALYSIS In oth
Page 97 and 98:
Page 99 and 100:
84 REAL AND COMPLEX ANALYSIS A mome
Page 101 and 102:
Page 103 and 104:
88 REAL AND COMPLEX ANALYSIS Trigon
Page 105 and 106:
90 REAL AND COMPLEX ANALYSIS If we
Page 107 and 108:
92 REAL AND COMPLEX ANALYSIS whenev
Page 109 and 110:
94 REAL AND COMPLEX ANALYSIS 15 Com
Page 111 and 112:
96 REAL AND COMPLEX ANALYSIS For in
Page 113 and 114:
98 REAL AND COMPLEX ANALYSIS This f
Page 115 and 116:
100 REAL AND COMPLEX ANALYSIS Fix y
Page 117 and 118:
102 REAL AND COMPLEX ANALYSIS Since
Page 119 and 120:
104 REAL AND COMPLEX ANALYSIS Let C
Page 121 and 122:
106 REAL AND COMPLEX ANALYSIS Under
Page 123 and 124:
lOS REAL AND COMPLEX ANALYSIS PROOF
Page 125 and 126:
110 REAL AND COMPLEX ANALYSIS Note
Page 127 and 128:
112 REAL AND COMPLEX ANALYSIS 5.25
Page 129 and 130:
114 REAL AND COMPLEX ANALYSIS 14 Le
Page 131 and 132:
CHAPTER SIX COMPLEX MEASURES Total
Page 133 and 134:
Page 135 and 136:
120 REAL AND COMPLEX ANALYSIS Absol
Page 137 and 138:
122 REAL AND COMPLEX ANALYSIS The e
Page 139 and 140:
124 REAL AND COMPLEX ANALYSIS measu
Page 141 and 142:
126 REAL AND COMPLEX ANALYSIS A u B
Page 143 and 144:
128 REAL AND COMPLEX ANALYSIS since
Page 145 and 146:
130 REAL AND COMPLEX ANALYSIS whene
Page 147 and 148:
132 REAL AND COMPLEX ANALYSIS There
Page 149 and 150:
134 REAL AND COMPLEX ANALYSIS Prove
Page 151 and 152:
136 REAL AND COMPLEX ANALYSIS for e
Page 153 and 154:
138 REAL AND COMPLEX ANALYSIS 7.S W
Page 155 and 156:
140 REAL AND COMPLEX ANALYSIS We sh
Page 157 and 158:
142 REAL AND COMPLEX ANALYSIS Havin
Page 159 and 160:
144 REAL AND COMPLEX ANALYSIS The F
Page 161 and 162:
146 REAL AND COMPLEX ANALYSIS for a
Page 163 and 164:
148 REAL AND COMPLEX ANALYSIS [F is
Page 165 and 166:
156 REAL AND COMPLEX ANALYSIS If f
Page 167 and 168:
152 REAL AND COMPLEX ANALYSIS one i
Page 169 and 170:
154 REAL AND COMPLEX ANALYSIS (ii)
Page 171 and 172:
Page 173 and 174:
158 REAL AND COMPLEX ANALYSIS IS Co
Page 175 and 176:
CHAPTER EIGHT INTEGRATION ON PRODUC
Page 177 and 178:
162 REAL AND COMPLEX ANALYSIS Fix P
Page 179 and 180:
164 REAL AND COMPLEX ANALYSIS For f
Page 181 and 182:
166 REAL AND COMPLEX ANALYSIS I + ~
Page 183 and 184:
168 REAL AND COMPLEX ANALYSIS this
Page 185 and 186:
170 REAL AND COMPLEX ANALYSIS Let N
Page 187 and 188:
172 REAL AND COMPLEX ANALYSIS Distr
Page 189 and 190:
174 REAL AND COMPLEX ANALYSIS We no
Page 191 and 192:
176 REAL AND COMPLEX ANALYSIS Prove
Page 193 and 194:
CHAPTER NINE FOURIER TRANSFORMS For
Page 195 and 196:
180 REAL AND COMPLEX ANALYSIS where
Page 197 and 198:
182 REAL AND COMPLEX ANALYSIS [It s
Page 199 and 200:
Page 201 and 202:
186 REAL AND COMPLEX ANALYSIS trans
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
192 REAL AND COMPLEX ANALYSIS By Th
Page 209 and 210:
194 REAL AND COMPLEX ANALYSIS S If
Page 211 and 212:
CHAPTER TEN ELEMENTARY PROPERTIES O
Page 213 and 214:
198 REAL AND COMPLEX ANALYSIS where
Page 215 and 216:
Page 217 and 218:
202 REAL AND COMPLEX ANALYSIS From
Page 219 and 220:
204 REAL AND COMPLEX ANALYSIS By Th
Page 221 and 222:
206 REAL AND COMPLEX ANALYSIS such
Page 223 and 224:
Page 225 and 226:
210 REAL AND COMPLEX ANALYSIS In ot
Page 227 and 228:
Page 229 and 230:
214 REAL AND COMPLEX ANALYSIS It is
Page 231 and 232:
216 REAL AND COMPLEX ANALYSIS By (1
Page 233 and 234:
218 REAL AND COMPLEX ANALYSIS The o
Page 235 and 236:
220 REAL AND COMPLEX ANALYSIS Next,
Page 237 and 238:
111 REAL AND COMPLEX ANALYSIS We co
Page 239 and 240:
224 REAL AND COMPLEX ANALYSIS The C
Page 241 and 242:
226 REAL AND COMPLEX ANALYSIS Let A
Page 243 and 244:
228 REAL AND COMPLEX ANALYSIS and h
Page 245 and 246:
230 REAL AND COMPLEX ANALYSIS and (
Page 247 and 248:
Page 249 and 250:
234 REAL AND COMPLEX ANALYSIS Iffis
Page 251 and 252:
236 REAL AND COMPLEX ANALYSIS We ma
Page 253 and 254:
238 REAL AND COMPLEX ANALYSIS Suppo
Page 255 and 256:
240 REAL AND COMPLEX ANALYSIS we ob
Page 257 and 258:
242 REAL AND COMPLEX ANALYSIS We wi
Page 259 and 260:
244 REAL AND COMPLEX ANALYSIS lim s
Page 261 and 262:
Page 263 and 264:
Page 265 and 266:
Page 267 and 268:
252 REAL AND COMPLEX ANALYSIS Sugge
Page 269 and 270:
254 REAL AND COMPLEX ANALYSIS The S
Page 271 and 272:
Page 273 and 274:
1S8 REAL AND COMPLEX ANALYSIS for a
Page 275 and 276:
260 REAL AND COMPLEX ANAL YSlS for
Page 277 and 278:
161 REAL AND COMPLEX ANALYSIS The B
Page 279 and 280:
264 REAL AND COMPLEX ANALYSIS Exerc
Page 281 and 282:
CHAPTER THIRTEEN APPROXIMATIONS BY
Page 283 and 284:
Page 285 and 286:
270 REAL AND COMPLEX ANALYSIS Runge
Page 287 and 288:
272 REAL AND COMPLEX ANALYSIS m = m
Page 289 and 290:
274 REAL AND COMPLEX ANALYSIS is ho
Page 291 and 292:
276 REAL AND COMPLEX ANALYSIS The f
Page 293 and 294:
CHAPTER FOURTEEN CONFORMAL MAPPING
Page 295 and 296:
280 REAL AND COMPLEX ANALYSIS (c) H
Page 297 and 298:
282 REAL AND COMPLEX ANALYSIS (Some
Page 299 and 300:
284 REAL AND COMPLEX ANALYSIS For I
Page 301 and 302:
Page 303 and 304:
Page 305 and 306:
Page 307 and 308:
292 REAL AND COMPLEX ANALYSIS also
Page 309 and 310:
Page 311 and 312:
296 REAL AND COMPLEX ANALYSIS 27 Pr
Page 313 and 314:
CHAPTER FIFTEEN ZEROS OF HOLOMORPHI
Page 315 and 316:
300 REAL AND COMPLEX ANALYSIS Choos
Page 317 and 318:
302 REAL AND COMPLEX ANALYSIS 1 - E
Page 319 and 320:
304 REAL AND COMPLEX ANALYSIS as n-
Page 321 and 322:
Page 323 and 324:
308 REAL AND COMPLEX ANALYSIS This
Page 325 and 326:
310 REAL AND COMPLEX ANALYSIS then
Page 327 and 328:
312 REAL AND COMPLEX ANALYSIS Our a
Page 329 and 330:
314 REAL AND COMPLEX ANALYSIS So as
Page 331 and 332:
316 REAL AND COMPLEX ANALYSIS holds
Page 333 and 334:
318 REAL AND COMPLEX ANALYSIS 18 Su
Page 335 and 336:
Page 337 and 338:
322 REAL AND COMPLEX ANAL YSlS The
Page 339 and 340:
324 REAL AND COMPLEX ANALYSIS If (I
Page 341 and 342:
Page 343 and 344:
328 REAL AND COMPLEX ANALYSIS Const
Page 345 and 346:
330 REAL AND COMPLEX ANALYSIS every
Page 347 and 348:
332 REAL AND COMPLEX ANALYSIS value
Page 349 and 350:
334 REAL AND COMPLEX ANALYSIS II Fo
Page 351 and 352:
Page 353 and 354:
338 REAL AND COMPLEX ANALYSIS If 0
Page 355 and 356:
340 REAL AND COMPLEX ANALYSIS We ca
Page 357 and 358:
342 REAL AND COMPLEX ANALYSIS as in
Page 359 and 360:
Page 361 and 362:
346 REAL AND COMPLEX ANAL YSlS and
Page 363 and 364:
348 REAL AND COMPLEX ANALYSIS This
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
354 REAL AND COMPLEX ANALYSIS 24 Th
Page 371 and 372:
CHAPTER EIGHTEEN ELEMENTARY THEORY
Page 373 and 374:
358 REAL AND COMPLEX ANALYSIS it fo
Page 375 and 376:
360 REAL AND COMPLEX ANALYSIS An al
Page 377 and 378:
362 REAL AND COMPLEX ANALYSIS and {
Page 379 and 380:
364 REAL AND COMPLEX ANALYSIS If X
Page 381 and 382:
366 REAL AND COMPLEX ANALYSIS Befor
Page 383 and 384:
368 REAL AND COMPLEX ANALYSIS and/(
Page 385 and 386:
370 REAL AND COMPLEX ANALYSIS 14 Su
Page 387 and 388:
372 REAL AND COMPLEX ANALYSIS Let U
Page 389 and 390:
Page 391 and 392:
376 REAL AND COMPLEX ANALYSIS cl>,,
Page 393 and 394:
378 REAL AND COMPLEX ANALYSIS These
Page 395 and 396:
380 REAL AND COMPLEX ANALYSIS We ar
Page 397 and 398:
382 REAL AND COMPLEX ANAL YSlS PROO
Page 399 and 400:
384 REAL AND COMPLEX ANALYSIS if r(
Page 401 and 402:
CHAPTER TWENTY UNIFORM APROXIMATION
Page 403 and 404:
Page 405 and 406:
390 REAL AND COMPLEX ANALYSIS Put F
Page 407 and 408:
392 REAL AND COMPLEX ANALYSIS A E C
Page 409 and 410:
394 REAL AND COMPLEX ANALYSIS and H
Page 411 and 412:
396 REAL AND COMPLEX ANALYSIS Let r
Page 413 and 414:
398 REAL AND COMPLEX ANALYSIS Sec.
Page 415 and 416:
400 REAL AND COMPLEX ANALYSIS Chapt
Page 417 and 418:
402 REAL AND COMPLEX ANALYSIS one,
Page 420 and 421:
BIBLIOGRAPHY 1. L. V. Ahlfors: "Com
Page 422 and 423:
LIST OF SPECIAL SYMBOLS AND ABBREVI
Page 424 and 425:
INDEX Absolute continuity, 120 of f
Page 426 and 427:
INDEX 411 Double integral, 165 Dual
Page 428 and 429:
INDEX 413 Lower limit, 14 Lower sem
Page 430 and 431:
INDEX 415 Separable space, 92, 247
show all

Real and Complex Analysis (Rudin)

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?