Complex Analysis - Maths KU

More documents

Recommendations

Info

ANS-16 Answers to Selected Odd-Numbered Problems 9. − 1 2 ln 3 11. −125� 3 √ 2; −250( √ 2 − 4) � 12; 13. 3; 6; 3 √ 5 15. 21 17. 30 19. 1 21. 1 23. 460 25. 26 9 29. − 8 3 27. − 64 3 31. 0 33. On each curve the line integral has the value 208 3 . 35. With ρ = kx, m = kπ. Exercises 5.2, page 254 1. −28+84i 3. −48 + 736 3 i 5. (2 + π)i 7. πi 9. − 7 1 + i 11. −e − 1 12 12 13. 3 π − 2 4 15. 0 i 19. 0 17. 1 2 21. 4 3 25. 5 4 5 − i 23. − 3 3 3 i 5 12 πe2 31. (a) −11+38i (b) 0 Exercises 5.3, page 262 27. 6 √ 2 9. 2πi 11. 2πi 13. 0 15. (a) 2πi (b) 4πi (c) 0 17. (a) −8πi (b) −6πi 19. −π(1 + i) 21. 0 23. −4πi 25. −6πi Exercises 5.4, page 271 1. −2i 3. 48 + 24i 5. 6+ 26 i 7. 0 3 9. − 7 6 22 1 − i 11. − 3 π 13. 2.3504i 15. 0 17. πi 19. 1 2 i − 1 π i 21. 11.4928 + 0.9667i 23. −0.9056 + 1.7699i 25. √ 2 i 125 2
Answers to Selected Odd-Numbered Problems ANS-15 � √ � 1 25. z = loge 1+ 2 + 2 (2n +1)πi or z = log � √ � 1 e −1+ 2 + (2n − 1)πi, 2 n =0,±1, ±2, ... 27. There are no solutions. 33. cos z sinh z + sin z cosh z 35. i sech 2 (iz − 2) Exercises 4.4, page 221 1. 1 2 (4n +1)π− i log �√ � 1 e 2+1 and 2 (4n − 1)π − i log �√ � e 2 − 1 , n =0, ±1, ±2, ... 3. 1 2 (4n +1)π − i log e �√ 2 ± 1 � , n =0,±1, ±2, ... 5. − 1 1 (4n − 1)π, n =0,±1, ±2, ... 7. (4n +1)πi, n =0,±1, ±2, ... 4 2 9. 1 4 loge 2+ 1 (8n +3)πi, n =0,±1, ±2, ... 8 �√ �� √ 11. (a) −i loge 5 − 1 5 13. (a) 1 2 15. (a) log e � 1 2 (π − arctan 2) + i 1 �√ 2+1 � − 1 2 4 log e Exercises 4.5, page 230 (b) 2 5 1 2 5 (b) − 5 5 i √ 2i πi (b) 1 2 1. φ(x, y) =−x +5 3. φ(x, y) = 5 4 5. φ(x, y) = 12 + 33 π √ 5 3x − y +10 4 25 Arg(sin (z − π)+1)− Arg(sin (z − π) − 1) π 7. φ(x, y) =15− 9 17 Arg(sin (iz) + 1) + Arg(sin (iz) − 1) π π Chapter 4 Review Quiz, page 232 1. true 3. false 5. false 7. true 9. false 11. false 13. false 15. true 17. true 19. true 21. e x cos y, e x sin y 23. log e 2+ 1 6 πi 25. 2nπ − i log e 2, n =0,±1, ±2, ... 27. nonpositive real axis 29. 1 2 loge 2+ 1 4 31. z2 = e 2π (8n +1)πi, n =0,±1, ±2, ... 35. cosh 4 37. sin x cosh y, cos x sinh y 39. ±1 Chapter 5 Exercises 5.1, page 243 1. 64 3 3. − 1 π 5. 1 2 ln 9 7. 8e−1 − 12e −2
Page 1 and 2:
A First Course in Complex Analysis
Page 3:
For Dana, Kasey, and Cody
Page 6 and 7:
vi Contents Chapter 4. Elementary F
Page 9 and 10:
Preface 7.2 Preface Philosophy This
Page 11 and 12:
Preface xi to, but not formally cov
Page 13 and 14:
3π 2π π 0 -π -2π -3π -1 0 1 -
Page 15 and 16:
1.1 Complex Numbers and Their Prope
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
y z 2 z 2 - z 1 or (x 2 - x 1 , y 2
Page 25 and 26:
1.2 Complex Plane 13 From (8) with
Page 27 and 28:
1.2 Complex Plane 15 30. Find an up
Page 29 and 30:
O y θ x = r cos θ (r, θ) or (x,
Page 31 and 32:
1.3 Polar Form of Complex Numbers 1
Page 33 and 34:
1.3 Polar Form of Complex Numbers 2
Page 35 and 36:
1.4 Powers and Roots 23 arg(z) =π/
Page 37 and 38:
√ 4 = 2 and 3 √ 27 = 3 are the
Page 39 and 40:
1.4 Powers and Roots 27 16. Rework
Page 41 and 42:
z 0 ρ ρ |z - z 0 |= Figure 1.15 C
Page 43 and 44:
y Interior Exterior Boundary Figure
Page 45 and 46:
1.5 Sets of Points in the Complex P
Page 47 and 48:
1.5 Sets of Points in the Complex P
Page 49 and 50:
Note: The roots z1 and z2 are conju
Page 51 and 52:
1.6 Applications 39 c = 0.The latte
Page 53 and 54:
1.6 Applications 41 Electrical engi
Page 55 and 56:
1.6 Applications 43 In Problems 25
Page 57 and 58:
Chapter 1 Review Quiz 45 Here the s
Page 59 and 60:
Chapter 1 Review Quiz 47 27. The pr
Page 61 and 62:
3i 2i i S 1 2 3 Image of a square u
Page 63 and 64:
2.1 Complex Functions 51 interchang
Page 65 and 66:
2.1 Complex Functions 53 Definition
Page 67 and 68:
2.1 Complex Functions 55 respective
Page 69 and 70:
2.1 Complex Functions 57 Focus on C
Page 71 and 72:
2.2 Complex Functions as Mappings 5
Page 73 and 74:
-4 -4 -3 C 2 3 4 (a) The vertical l
Page 75 and 76:
Page 77 and 78:
4 2 -6 -4 -2 2 4 6 -2 -4 v Figure 2
Page 79 and 80:
Page 81 and 82:
3i 2i i S z 1 2 3 S′ T(z) Figure
Page 83 and 84:
ar Figure 2.12 Magnification C C′
Page 85 and 86:
Note: The order in which you perfor
Page 87 and 88:
2.3 Linear Mappings 75 triangle S4
Page 89 and 90:
2.3 Linear Mappings 77 In Problems
Page 91 and 92:
2.3 Linear Mappings 79 36. (a) Give
Page 93 and 94:
2θ r θ r 2 Figure 2.17 The mappin
Page 95 and 96:
-3 y 3 2 1 -2 -1 1 2 3 -1 -2 -3 (a)
Page 97 and 98:
y 2 1.5 1 0.5 -2 -1.5 -1 -0.5 -0.5
Page 99 and 100:
Note ☞ 2.4 Special Power Function
Page 101 and 102:
Solve the equation z = f(w) for w t
Page 103 and 104:
Remember: Arg(z) is in the interval
Page 105 and 106:
S y (a) A circular sector 3π/8 v 3
Page 107 and 108:
B y y A w = z 2 x (a) A maps onto A
Page 109 and 110:
3π 2π π 0 -π -2π -3π -1 0 1 -
Page 111 and 112:
2.4 Special Power Functions 99 43.
Page 113 and 114:
z c(z) Figure 2.41 Complex conjugat
Page 115 and 116:
w = 1/z Figure 2.43 The reciprocal
Page 117 and 118:
2.5 Reciprocal Function 105 of the
Page 119 and 120:
2.5 Reciprocal Function 107 underst
Page 121 and 122:
2.5 Reciprocal Function 109 24. Acc
Page 123 and 124:
L y y = L + ε y = L - ε y = f(x)
Page 125 and 126:
2.6 Limits and Continuity 113 Crite
Page 127 and 128:
2.6 Limits and Continuity 115 Befor
Page 129 and 130:
2.6 Limits and Continuity 117 Theor
Page 131 and 132:
2.6 Limits and Continuity 119 See (
Page 133 and 134:
-1 y z = e iθ Figure 2.54 Figure f
Page 135 and 136:
2.6 Limits and Continuity 123 It th
Page 137 and 138:
2.6 Limits and Continuity 125 While
Page 139 and 140:
y Values of arg decreasing Values o
Page 141 and 142:
2.6 Limits and Continuity 129 In Pr
Page 143 and 144:
2.6 Limits and Continuity 131 In Pr
Page 145 and 146:
-4 -4 -3 -2 (-2, -1) 4 3 2 1 y -1 1
Page 147 and 148:
Note: Normalized vector fields shou
Page 149 and 150:
4 2 y -4 -2 2 4 -2 -4 Figure 2.63 S
Page 151 and 152:
Chapter 2 Review Quiz 139 10. The l
Page 153 and 154:
3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3 L
Page 155 and 156:
3.1 Differentiability and Analytici
Page 157 and 158:
☞ 3.1 Differentiability and Analy
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
3.2 Cauchy-Riemann Equations 153 We
Page 167 and 168:
3.2 Cauchy-Riemann Equations 155 EX
Page 169 and 170:
3.2 Cauchy-Riemann Equations 157 EX
Page 171 and 172:
3.3 Harmonic Functions 159 then sol
Page 173 and 174:
3.3 Harmonic Functions 161 EXAMPLE
Page 175 and 176:
3.3 Harmonic Functions 163 In Probl
Page 177 and 178:
3.4 Applications 165 tangent is the
Page 179 and 180:
Lines of force ψ = c2 Lower potent
Page 181 and 182:
In Section 4.5, for purposes that w
Page 183 and 184:
y a b φ = k 1 x φ = k 0 Figure 3.
Page 185:
Chapter 3 Review Quiz 173 11. The C
Page 188 and 189:
176 Chapter 4 Elementary Functions
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 247 and 248:
Normalized velocity vector field fo
Page 249 and 250:
5.1 Real Integrals 237 3. Let �P
Page 251 and 252:
y π t = gives 2 (0,4) C t = 0 give
Page 253 and 254:
(-1, -1) y C (2, 8) x Figure 5.5 Gr
Page 255 and 256:
5.1 Real Integrals 243 denotes the
Page 257 and 258:
5.2 Complex Integrals 245 In Proble
Page 259 and 260:
z 0 C z k * z 1 * z 2 * z 2 z 1 z k
Page 261 and 262:
These properties are important in t
Page 263 and 264:
5.2 Complex Integrals 251 Now in vi
Page 265 and 266:
5.2 Complex Integrals 253 Proof It
Page 267 and 268:
y 1 + i 1 Figure 5.21 Figure for Pr
Page 269 and 270:
D C Figure 5.26 Simply connected do
Page 271 and 272:
D D A C C (a) (b) B C 1 C 1 Figure
Page 273 and 274:
C 1 C 1 C (a) C (b) Figure 5.31 Tri
Page 275 and 276:
C C y Figure 5.34 Figure for Proble
Page 277 and 278:
z 0 C 1 D z 1 C Figure 5.38 If f is
Page 279 and 280:
5.4 Independence of Path 267 and z(
Page 281 and 282:
Be careful when using Ln z as an an
Page 283 and 284:
5.4 Independence of Path 271 (ii) I
Page 285 and 286:
5.5 Cauchy’s Integral Formulas an
Page 287 and 288:
3i -3i y Figure 5.44 Contour for Ex
Page 289 and 290:
y i 0 C 2 C 1 Figure 5.45 Contour f
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
5.6 Applications 285 A B A B A B (a
Page 299 and 300:
5.6 Applications 287 Indeed, by rew
Page 301 and 302:
Note ☞ 5.6 Applications 289 If yo
Page 303 and 304:
-2 -1 2 1 -1 -2 y Figure 5.51 Zero
Page 305 and 306:
-1 -0.5 1 0.5 -0.5 -1 y 0.5 Figure
Page 307 and 308:
5.6 Applications 295 In Problems 5-
Page 309 and 310:
C 1 y 2i C 2 -2 2 Figure 5.58 Figur
Page 311:
Chapter 5 Review Quiz 299 � z 38.
Page 314 and 315:
302 Chapter 6 Series and Residues y
Page 316 and 317:
304 Chapter 6 Series and Residues Y
Page 318 and 319:
306 Chapter 6 Series and Residues D
Page 320 and 321:
308 Chapter 6 Series and Residues E
Page 322 and 323:
310 Chapter 6 Series and Residues R
Page 324 and 325:
312 Chapter 6 Series and Residues 3
Page 326 and 327:
314 Chapter 6 Series and Residues I
Page 328 and 329:
316 Chapter 6 Series and Residues D
Page 330 and 331:
318 Chapter 6 Series and Residues N
Page 332 and 333:
Page 334 and 335:
Page 336 and 337:
Page 338 and 339:
326 Chapter 6 Series and Residues a
Page 340 and 341:
328 Chapter 6 Series and Residues N
Page 342 and 343:
330 Chapter 6 Series and Residues T
Page 344 and 345:
Page 346 and 347:
334 Chapter 6 Series and Residues R
Page 348 and 349:
336 Chapter 6 Series and Residues i
Page 350 and 351:
338 Chapter 6 Series and Residues A
Page 352 and 353:
340 Chapter 6 Series and Residues S
Page 354 and 355:
Page 356 and 357:
344 Chapter 6 Series and Residues T
Page 358 and 359:
346 Chapter 6 Series and Residues (
Page 360 and 361:
Page 362 and 363:
Page 364 and 365:
Page 366 and 367:
354 Chapter 6 Series and Residues H
Page 368 and 369:
356 Chapter 6 Series and Residues -
Page 370 and 371:
358 Chapter 6 Series and Residues o
Page 372 and 373:
360 Chapter 6 Series and Residues -
Page 374 and 375:
362 Chapter 6 Series and Residues z
Page 376 and 377:
364 Chapter 6 Series and Residues f
Page 378 and 379:
366 Chapter 6 Series and Residues v
Page 380 and 381:
368 Chapter 6 Series and Residues
Page 382 and 383:
Page 384 and 385:
Page 386 and 387:
Page 388 and 389:
Page 390 and 391:
378 Chapter 6 Series and Residues C
Page 392 and 393:
380 Chapter 6 Series and Residues s
Page 394 and 395:
Page 396 and 397:
Page 398 and 399:
386 Chapter 6 Series and Residues P
Page 400 and 401:
Page 402 and 403:
390 Chapter 7 Conformal Mappings y
Page 404 and 405:
Page 406 and 407:
394 Chapter 7 Conformal Mappings π
Page 408 and 409:
396 Chapter 7 Conformal Mappings Re
Page 410 and 411:
398 Chapter 7 Conformal Mappings 15
Page 412 and 413:
400 Chapter 7 Conformal Mappings De
Page 414 and 415:
402 Chapter 7 Conformal Mappings Th
Page 416 and 417:
404 Chapter 7 Conformal Mappings v
Page 418 and 419:
406 Chapter 7 Conformal Mappings Re
Page 420 and 421:
408 Chapter 7 Conformal Mappings No
Page 422 and 423:
Page 424 and 425:
412 Chapter 7 Conformal Mappings x
Page 426 and 427:
414 Chapter 7 Conformal Mappings A
Page 428 and 429:
416 Chapter 7 Conformal Mappings fo
Page 430 and 431:
418 Chapter 7 Conformal Mappings EX
Page 432 and 433:
420 Chapter 7 Conformal Mappings
Page 434 and 435:
Page 436 and 437:
Page 438 and 439:
Page 440 and 441:
428 Chapter 7 Conformal Mappings (a
Page 442 and 443:
Page 444 and 445:
432 Chapter 7 Conformal Mappings 3
Page 446 and 447: 434 Chapter 7 Conformal Mappings y
Page 448 and 449: 436 Chapter 7 Conformal Mappings φ
Page 450 and 451: 438 Chapter 7 Conformal Mappings ψ
Page 452 and 453: 440 Chapter 7 Conformal Mappings y
Page 454 and 455: 442 Chapter 7 Conformal Mappings 4
Page 456 and 457: 444 Chapter 7 Conformal Mappings In
Page 458 and 459: 446 Chapter 7 Conformal Mappings 24
Page 460 and 461: 448 Chapter 7 Conformal Mappings In
Page 463 and 464: Appendixes 1
Page 465 and 466: Appendix II Proof of the Cauchy-Gou
Page 467 and 468: Integrals � � � FE , GF , EG
Page 469 and 470: Appendix I Proof of Theorem 2.1 APP
Page 471 and 472: Appendix III Table of Conformal Map
Page 479 and 480: Answers to Selected Odd-Numbered Pr
Page 495: Answers to Selected Odd-Numbered Pr
Page 503 and 504: Indexes 1
Page 505 and 506: Word Index IND-7 Convergence of an
Page 507 and 508: Word Index IND-5 Cauchy-Riemann equ
Page 509 and 510: Symbol Index IND-3 z α , 194 sin z
Page 511 and 512: Word Index IND-9 principal square r
Page 513 and 514: IND-14 Word Index partial sums of,
Page 515 and 516: IND-12 Word Index Partial fractions
Page 517: Word Index IND-15 mapping by, 206-2
show all

Complex Analysis - Maths KU

Create successful ePaper yourself

Delete template?

Save as template?