Complex Analysis - Maths KU

More documents

Recommendations

Info

14 Chapter 1 Complex Numbers and the Complex Plane EXERCISES 1.2 Answers to selected odd-numbered problems begin on page ANS-2. In Problems 1–4, interpret z1 and z2 as vectors. Graph z1, z2, and the indicated sum and difference as vectors. 1. z1 =4+2i, z2 = −2+5i; z1 + z2, z1 − z2 2. z1 =1− i, z2 =1+i; z1 + z2, z1 − z2 3. z1 =5+4i, z2 = −3i; 3z1 +5z2, z1 − 2z2 4. z1 =4− 3i, z2 = −2+3i; 2z1 +4z2, z1 − z2 5. Given that z1 =5− 2i and z2 = −1 − i, find a vector z3 in the same direction as z1 + z2 but four times as long. 6. (a) Plot the points z1 = −2 − 8i, z2 =3i, z3 = −6 − 5i. (b) The points in part (a) determine a triangle with vertices at z1, z2, and z3, respectively. Express each side of the triangle as a difference of vectors. 7. In Problem 6, determine whether the points z1, z2, and z3 are the vertices of a right triangle. 8. The three points z1 =1+5i, z2 = −4 − i, z3 =3+i are vertices of a triangle. Find the length of the median from z1 to the side z3 − z2. In Problems 9–12, find the modulus of the given complex number. 9. (1 − i) 2 10. i(2 − i) − 4 � 1+ 1 4i� 2i 1 − 2i 2 − i 11. 12. + 3 − 4i 1+i 1 − i In Problems 13 and 14, let z = x + iy. Express the given quantity in terms of x and y. 13. |z − 1 − 3i| 2 14. |z +5¯z| In Problems 15 and 16, determine which of the given two complex numbers is closest to the origin. Which is closest to 1 + i? 15. 10+8i, 11 − 6i 16. 1 2 1 2 1 − 4i, 3 + 6i In Problems 17–26, describe the set of points z in the complex plane that satisfy the given equation. 17. Re((1 + i)z − 1) = 0 18. [Im(i¯z)] 2 =2 19. |z − i| = |z − 1| 20. ¯z = z −1 21. Im(z 2 )=2 22. Re(z 2 )= � � √ 3 − i � � 23. |z − 1| =1 24. |z − i| =2|z − 1| 25. |z − 2| = Re(z) 26. |z| =Re(z) In Problems 27 and 28, establish the given inequality. 27. If |z| = 2, then | z +6+8i |≤13. 28. If |z| = 1, then 1 ≤ � � z 2 − 3 � � ≤ 4. 29. Find an upper bound for the modulus of 3z 2 +2z +1if|z| ≤1.
1.2 Complex Plane 15 30. Find an upper bound for the reciprocal of the modulus of z 4 −5z 2 +6 if |z| =2. [Hint: z 4 − 5z 2 +6= � z 2 − 3 �� z 2 − 2 � .] In Problems 31 and 32, find a number z that satisfies the given equation. 31. |z|−z =2+i 32. |z| 2 +1+12i =6z Focus on Concepts 33. (a) Draw the pair of points z = a + ib and ¯z = a − ib in the complex plane if a>0, b > 0; a>0, b < 0; a 0; and a
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 23 and 24: y z 2 z 2 - z 1 or (x 2 - x 1 , y 2
Page 25: 1.2 Complex Plane 13 From (8) with
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: 2.2 Complex Functions as Mappings 6
Page 77 and 78:
4 2 -6 -4 -2 2 4 6 -2 -4 v Figure 2
Page 79 and 80:
2.2 Complex Functions as Mappings 6
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:
Page 446 and 447:
Page 448 and 449:
436 Chapter 7 Conformal Mappings φ
Page 450 and 451:
438 Chapter 7 Conformal Mappings ψ
Page 452 and 453:
Page 454 and 455:
Page 456 and 457:
444 Chapter 7 Conformal Mappings In
Page 458 and 459:
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 473 and 474:
Page 475 and 476:
Page 477 and 478:
Page 479 and 480:
Answers to Selected Odd-Numbered Pr
Page 481 and 482:
Page 483 and 484:
Page 485 and 486:
Page 487 and 488:
Page 489 and 490:
Page 491 and 492:
Page 493 and 494:
Page 495 and 496:
Page 497 and 498:
Page 499 and 500:
Page 501 and 502:
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

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

Delete template?

Save as template?