Advanced Calculus fi..

More documents

Recommendations

Info

560 Advanced Calculus, Fifth EditionFor the theory of series of complex numbers, see Section 6.19. The followingfundamental theorem for complex power series is proved just as for real series(Theorem 35 in Section 6.15).THEOREM 18 Every power series ESPEocn(z - zo)" has a radius of convergencer* such that the series converges absolutely when (z- zo( < r* and diverges whenlz -zoI > r*. The series converges uniformly for lz - zoI 5 rl, provided that rl < r*.The number r* can be zero, in which case the series converges only for z = zo,a positive number, or oo, in which case the series converges for all z.The number r* can be evaluated as follows:1% r* = lim 1, if the limit exists."-+O0 Cn+lI .I -+1r* = lim -2.- if the limit exists,n+w m'(8.54)and in any case by the formulaAs for real variables, no general statement can be made about convergence onthe boundary of the domain of convergence. This boundary (when r* # 0, r* # oo)is a circle )z - zol = r*, termed the circle of convergence (Fig. 8.10). The series mayconverge at some points, all points, or no points of this circle.EXAMPLE 1Cz,(zn/n2). The first formula (8.54) givesThe series converges absolutely on the circle of convergence, for when lzl = 1, theseries of absolute value is the convergent series C(l/n2).ConvergenceD~vergenceFigure 8.10Circle of convergence of a power series.P C J
Chapter 8 Functions of a Complex Variable 561EXAMPLE 2 CEO zn. This complex geometric series converges for JzJ < 1, as(8.54) shows. We have furtheras for real variables. On the circle of convergence the series diverges everywhere,since the nth term fails to converge to zero.The following theorems are proved as for real variables.THEOREM 19 A power series with nonzero convergence radius represents acontinuous function within the circle of convergence.THEOREM 20 A power series can be integrated term by term within the circleof convergence; that is, if r* # 0 and(lz - zol < r*),then, for every path C inside the circle of convergence,or in terms of indefinite integrals,doJ f(z) dz = c cnn=O(z- zo)"+l+n+lconst (Iz - 201 < r*).ITHEOREM 21 A power series can be differentiated term by term; that is, if r* # 0andthen
Page 1 and 2:
L'' . ' '(SZilt6i) (SX156i).'Advanc
Page 3:
This edition differs from the previ
Page 6 and 7:
ContentsVectors and Matrices1.1 Int
Page 8 and 9:
3.6 Combined Operations 183*3.7 Cur
Page 10 and 11:
*6.24 Principal Value of Improper I
Page 12 and 13:
Case of Two Particles 662Case of N
Page 14 and 15:
2 Advanced Calculus, Fifth EditionF
Page 16 and 17:
4 Advanced Calculus, Fifth EditionF
Page 18 and 19:
6 Advanced Calculus. Fifth Editiona
Page 20 and 21:
Advanced Calculus. Fifth EditionFig
Page 22 and 23:
10 Advanced Calculus. Fifth Edition
Page 24 and 25:
12 Advanced Calculus, Fifth Edition
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Advanced Calculus. Fifth Editionhas
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Advanced Calculus, Fifth Edition8.
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Advanced Calculus, Fifth EditionBy
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 85 and 86:
Differential Calculusof Functionsof
Page 87 and 88:
Chapter 2 Differential Calculus of
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
PROBLEMSChapter 2 Differential Calc
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Chapter 2 Differential ~alculus of
Page 149 and 150:
defined analogously:Chapter 2 Diffe
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185:
Page 188 and 189:
176 Advanced Calculus, Fifth Editio
Page 190 and 191:
magnitudeand the force is inversely
Page 192 and 193:
Page 194 and 195:
i, -A f1 82 Advanced Calculus, Fift
Page 196 and 197:
2&Advanced Calculus, Fifth EditionT
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Advanced Calculus, Fifth Edition: s
Page 206 and 207:
TI1 194 Advanced Calculus, Fifth Ed
Page 208 and 209:
196 Advanced Calculus. Fifth Editio
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
f4202 Advanced Calculus, Fifth Edit
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
i208 Advanced Calculus, Fifth Editi
Page 222 and 223:
Page 224 and 225:
21 2 Advanced Calculus, Fifth Editi
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
*I. 421 8 Advanced Calculus, Fifth
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 246 and 247:
, '234 Advanced Calculus, Fifth Edi
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
256 Advanced Calculus. Fifth Editio
Page 270 and 271:
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Advanced Calculus, Fifth EditionFig
Page 279 and 280:
Vector Integral CalculusPART I.TWO-
Page 281 and 282:
Chapter 5 Vector Integral Calculus
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
3. Evaluate the following line inte
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
Page 299 and 300:
5. Evaluate by Green's theorem:a) $
Page 301 and 302:
Page 303 and 304:
Page 305 and 306:
Page 307 and 308:
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Chapter 5 Vector Integral' Calculus
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Chapter 5 Vector Integral CalculusT
Page 329 and 330:
Chapter 5 Vector Integral CalculusN
Page 331 and 332:
Chapter 5 Vector Integral Calculusn
Page 333 and 334:
Page 335 and 336:
Chapter 5 Vector Integral CalculusF
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Green's theorem now givesChapter 5
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
*5.15 PHYSICAL APPLICATIONSChapter
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
Page 375 and 376:
Page 377 and 378:
Page 379 and 380:
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Infinite Series6.1 INTRODUCTIONAn i
Page 389 and 390:
Chapter 6 Infinite Series 377These
Page 391 and 392:
Chapter 6 Infinite Series 379If a s
Page 393 and 394:
Chapter 6 Infinite Series 381Every
Page 395 and 396:
Chapter 6 Infinite Series 383PROBLE
Page 397 and 398:
Chapter 6 Infinite Series 385EXAMPL
Page 399 and 400:
Chapter 6 Infinite Series 387If ELP
Page 401 and 402:
Chapter 6 Infinite Seriesi ,Figure
Page 403 and 404:
n Chapter 6 Infinite Series 391If L
Page 405 and 406:
Chapter 6 Infinite Series 393of con
Page 407 and 408:
EXAMPLE 8zz, 5. Again the ratio tes
Page 409 and 410:
12. Determine convergence or diverg
Page 411 and 412:
The theorem has a counterpart in te
Page 413 and 414:
Chapter 6 Infinite Series 401Proof.
Page 415 and 416:
Chapter 6 Infinite Series 403as alg
Page 417 and 418:
Chapter 6 Infinite Series 405b) Sho
Page 419 and 420:
at. Chapter 6 Infinite Series 407Fi
Page 421 and 422:
Chapter 6 Infinite Series 409Furthe
Page 423 and 424:
F; Chapter 6 .Infinite Series 41 1C
Page 425 and 426:
Chapter 6 Infinite Series 41 3Figur
Page 427 and 428:
for when x = $, the numerator Ixn 1
Page 429 and 430:
EXAMPLE 1Ezl 5. Here the ratio test
Page 431 and 432:
Chapter 6 Infinite Series 41 9Figur
Page 433 and 434:
Chapter 6 Infinite Series 421This h
Page 435 and 436:
. ,. Chapter 6 Infinite Series 423D
Page 437 and 438:
, Chapter 6 Infinite Series 425by T
Page 439 and 440:
Chapter 6 Infinite Series 427THEORE
Page 441 and 442:
nL: Chapter 6 Infinite Series 429ha
Page 443 and 444:
Chapter 6 Infinite Series 431the Ta
Page 445 and 446:
I .Chapter 6 Infinite Series 433EXA
Page 447 and 448:
I:I Chapter 6 Infinite Series 435An
Page 449 and 450:
[Hint: Use induction. For n = 0 the
Page 451 and 452:
Chapter 6 Infinite Series 439%Let z
Page 453 and 454:
Chapter 6 Infinite Series 441is not
Page 455 and 456:
Chapter 6 Infinite Series 443Figure
Page 457 and 458:
Chapter 6 Infinite Series 445The th
Page 459 and 460:
Chapter 6 Infinite Series 447d 2 =
Page 461 and 462:
Page 463 and 464:
Page 465 and 466:
6Chapter 6 Infinite Series 453diver
Page 467 and 468:
, .rJii' Chapter 6 Infinite SeriesE
Page 469 and 470:
.nr~. Chapter 6 Infinite Series 457
Page 471 and 472:
Chapter 6 Infinite Series 459Figure
Page 473 and 474:
Chapter 6 Infinite Series 461b) Let
Page 475 and 476:
Chapter 6 Infinite Series 463By ana
Page 477 and 478:
Therefore R3 lies in some annulus h
Page 479 and 480:
Fourier Series andOrthogonal Functi
Page 481 and 482:
Chapter 7 Fourier Series and Orthog
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:
Page 505 and 506:
Page 507 and 508:
Page 509 and 510:
Page 511 and 512:
Chapter 7Fourier Series and Orthogo
Page 513 and 514:
Page 515 and 516:
Page 517 and 518:
Page 519 and 520:
Page 521 and 522: Chapter 7 Fourier Series and Orthog
Page 523 and 524: y the Schwarz inequality (7.46). By
Page 541 and 542: IiChapter 7 Fourier Series and Orth
Page 543 and 544: Functions of aComplex VariableWe as
Page 545 and 546: Chapter 8 Functions of a Complex Va
Page 547 and 548: TChapter 8 Functions of a Complex V
Page 561 and 562: PROBLEMSChapter 8 Functions of a Co
Page 571: 2f 'c) J, zeZ dz on the line segmen
Page 591 and 592: Rule IV could also be used, with A
Page 593 and 594: RULE V If f (z) has a zero of first
Page 603 and 604: +aJ 12. Evaluate the following inte
Page 623 and 624:
Chapter 8 Functions of a Complex Va
Page 625 and 626:
Page 627 and 628:
The function U is Airy's stress fun
Page 629 and 630:
'Chapter 8 Functions of a Complex V
Page 631 and 632:
Page 633 and 634:
PassibleChapter 8 Functions of a Co
Page 635 and 636:
C) the first quadrant of the w plan
Page 637 and 638:
Ordinary DifferentialEquations9.1 D
Page 639 and 640:
Chapter 9 Ordinary Differential Equ
Page 641 and 642:
Page 643 and 644:
Page 645 and 646:
Page 647 and 648:
Page 649 and 650:
Page 651 and 652:
Page 653 and 654:
Page 655 and 656:
Page 657 and 658:
Here we make a change of variable,
Page 659 and 660:
Page 661 and 662:
Page 663 and 664:
Page 665 and 666:
Page 667 and 668:
Page 669 and 670:
Page 671 and 672:
Partial DifferentialEquationsA part
Page 673 and 674:
Chapter 10 Partial Differential Equ
Page 675 and 676:
Page 677 and 678:
Page 679 and 680:
Page 681 and 682:
Page 683 and 684:
Page 685 and 686:
Page 687 and 688:
Chapter 10Partial Differential Equa
Page 689 and 690:
Page 691 and 692:
Equations (10.66) and (10.67) then
Page 693 and 694:
Page 695 and 696:
Page 697 and 698:
Page 699 and 700:
For t = 0 the series (10.99), if co
Page 701 and 702:
These are easily solved for A and B
Page 703 and 704:
Page 705 and 706:
Page 707 and 708:
Page 709 and 710:
Page 711 and 712:
Accordingly,Chapter 10 Partial Diff
Page 713 and 714:
Page 715 and 716:
Page 717 and 718:
Page 719 and 720:
Page 721 and 722:
Page 723 and 724:
Page 726 and 727:
Page 728 and 729:
Page 730 and 731:
Page 732 and 733:
Page 734 and 735:
Advanced Calculus, Fifth EditionSec
Page 736 and 737:
Page 738 and 739:
726 Advhhced Calculus, Fifth Editio
Page 740 and 741:
Page 743:
Section 10.12, page 6941 - (-1)"1.
Page 746 and 747:
Page 748 and 749:
Page 750 and 751:
Page 752 and 753:
show all

Advanced Calculus fi..

Create successful ePaper yourself

Delete template?

Save as template?