Complex Analysis - Maths KU

More documents

Recommendations

Info

206 Chapter 4 Elementary Functions function, the derivative of the complex sine function is the complex cosine function: � � iz −iz d d e − e sin z = = dz dz 2i ieiz + ie−iz = 2i eiz + e−iz , 2 or d sin z = cos z. dz Since thisderivative isdefined for all complex z, sinzisan entire function. In a similar manner, we find d cos z = − sin z. dz The derivativesof sin z and cos z can then be used to show that derivatives of all of the complex trigonometric functionsare the same asderivativesof the real trigonometric functions. The derivatives of the six complex trigonometric functionsare summarized in the following. Derivatives of <strong>Complex</strong> Trigonometric Functions d sin z = cos z dz d dz tan z =sec2z d sec z =secztan z dz d cos z = − sin z dz d dz cot z = − csc2 z d csc z = − csc z cot z dz The sine and cosine functions are entire, but the tangent, cotangent, secant, and cosecant functions are only analytic at those points where the denominator isnonzero. From (20) and (21), it then followsthat the tangent and secant functions have singularities at z =(2n +1)π/2 for n =0,±1, ±2 ... , whereasthe cotangent and cosecant functionshave singularitiesat z = nπ for n =0,±1, ±2 ... . Trigonometric Mapping We will now discuss the complex mapping w =sinz of the z-plane onto the w-plane. Because sin z isperiodic with a real period of 2π, thisfunction takeson all valuesin any infinite vertical strip x0
4.3 Trigonometric and Hyperbolic Functions 207 we need only consider the mapping w =sinz on the region −π/2 ≤ x ≤ π/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 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

Create successful ePaper yourself

Delete template?

Save as template?